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• Bernoulli-Euler Gamma function
  ....de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html#id1 Is the Gamma function misdefined?]
  128 bytes (16 words) - 22:45, 13 January 2015
• Heaviside step function
  The Heaviside step function is
  148 bytes (22 words) - 18:32, 24 May 2016
• Sinc function
  18 bytes (2 words) - 04:41, 19 January 2015
• Beta function
  18 bytes (2 words) - 06:53, 19 January 2015
• Dedekind eta function
  26 bytes (3 words) - 07:08, 19 January 2015
• Digamma function
  21 bytes (2 words) - 07:08, 19 January 2015
• Exponential function
  25 bytes (2 words) - 07:15, 19 January 2015
• Gamma function
  19 bytes (2 words) - 07:21, 19 January 2015
• Generalized hypergeometric function
  31 bytes (3 words) - 07:30, 19 January 2015
• Thomae function
  Thomae's function (sometimes called the popcorn function, raindrop function, Stars over Babylon) is given by the formula File:Thomae.png|Plot of the Thomae function.
  1 KB (150 words) - 00:36, 9 December 2016
• Polygamma function
  23 bytes (2 words) - 07:52, 19 January 2015
• Reciprocal gamma function
  30 bytes (3 words) - 07:56, 19 January 2015
• Lambert W function
  23 bytes (3 words) - 08:01, 19 January 2015
• Dirichlet beta function
  28 bytes (3 words) - 08:11, 19 January 2015
• Dirichlet eta function
  27 bytes (3 words) - 08:14, 19 January 2015
• Hadamard's gamma function
  28 bytes (3 words) - 08:17, 19 January 2015
• Minkowski question mark function
  37 bytes (4 words) - 08:25, 19 January 2015
• Partition function
  23 bytes (2 words) - 08:30, 19 January 2015
• Riemann xi function
  24 bytes (3 words) - 08:35, 19 January 2015
• Planck's radiation function
  30 bytes (3 words) - 08:42, 19 January 2015
• Q-Gamma function
  21 bytes (2 words) - 10:12, 19 January 2015
• Q-Binomial function
  24 bytes (2 words) - 10:47, 19 January 2015
• Hurwitz zeta function
  26 bytes (3 words) - 11:25, 19 January 2015
• Prime zeta function
  26 bytes (4 words) - 11:42, 19 January 2015
• Mangoldt function
  22 bytes (2 words) - 11:48, 19 January 2015
• Möbius function
  21 bytes (3 words) - 06:35, 22 June 2016
• Sum of positive divisors function
  29 bytes (4 words) - 11:56, 19 January 2015
• Prime counting function
  28 bytes (3 words) - 12:00, 19 January 2015
• Liouville function
  30 bytes (3 words) - 12:09, 19 January 2015
• Bessel function
  20 bytes (2 words) - 01:31, 17 March 2015
• Secant zeta function
  [[Absolute convergence of secant zeta function]]
  295 bytes (40 words) - 06:10, 16 June 2016
• Cotangent zeta function
  200 bytes (30 words) - 06:48, 16 June 2016
• Struve function
  [[Relationship between Struve function and hypergeometric pFq]]<br /> [[Relationship between Weber function 0 and Struve function 0]]<br />
  1 KB (218 words) - 01:09, 21 December 2017
• Error function
  The (normalized) error function $\mathrm{erf}$ is defined by where $\pi$ denotes [[pi]] and $e^{-\tau^2}$ denotes the [[exponential]] function.
  2 KB (271 words) - 00:43, 25 June 2017
• Closed form for partition function with sinh
  a [[sawtooth function]], define where $p$ denotes the [[partition]] function, $\pi$ denotes [[pi]], and $\sinh$ denotes the [[sinh|hyperbolic sine]].
  1 KB (172 words) - 20:40, 26 June 2016
• Pure recurrence relation for partition function
  ...the [[partition]] function and $\sigma_1$ denotes the [[sum of divisors]] function. ...rm for partition function with sinh|next=Recurrence relation for partition function with sum of divisors}}: $24.2.1 \mathrm{II}.A.$
  581 bytes (80 words) - 20:41, 26 June 2016
• Ihara zeta function
  Let $X$ be a finite [[graph]]. The Ihara zeta function is given by the formula ...an analogue of the [[Euler product]] representation of the [[Riemann zeta function]].
  765 bytes (122 words) - 18:52, 24 May 2016
• Cosine function
  20 bytes (2 words) - 15:52, 4 October 2014
• Book:Edward Charles Titchmarsh/The Zeta-Function of Riemann
  {{Book|The Zeta-Function of Riemann|1930|Cambridge University Press||Edward Charles Titchmarch}} ...r primes|$(2')$]] (and [[Series for log(Riemann zeta) in terms of Mangoldt function|$(2')$]])
  1 KB (170 words) - 15:23, 18 March 2017
• Q-Beta function
  272 bytes (50 words) - 18:54, 24 May 2016
• Sine function
  18 bytes (2 words) - 15:52, 4 October 2014
• Tangent function
  21 bytes (2 words) - 15:53, 4 October 2014
• Mobius function
  #REDIRECT [[Möbius function]]
  30 bytes (4 words) - 16:13, 4 October 2014
• Mertens function
  Merten's function is defined by the formula where $\mu$ is the [[Möbius function]].
  129 bytes (22 words) - 16:13, 4 October 2014
• Euler E generating function
  where $e^{xt}$ denotes the [[exponential function]] and $E_n$ denotes an [[Euler E]] polynomial.
  519 bytes (75 words) - 01:05, 4 March 2018
• Q-Polygamma function
  ...a functions of order $m$, $\psi_q^{(m)}$, are analogues of the [[polygamma function]] defined by ...ma_q(z).$ Here the function $\Gamma_q$ is the [[q-Gamma function|$q$-Gamma function]].
  368 bytes (60 words) - 18:56, 24 May 2016
• Q-theta function
  235 bytes (39 words) - 00:56, 19 October 2014
• Riemann zeta function
  26 bytes (3 words) - 19:40, 9 June 2016
• Dedekind zeta function
  ...finite [[field extension]] of the [[rational numbers]]. The Dedekind zeta function of $F$ is
  896 bytes (130 words) - 18:52, 24 May 2016
• Dirichlet L-function
  Let $\chi$ be a [[Dirichlet character]]. The Dirichlet $L$-function associated with $\chi$ is ...ac.uk/people/staff/mrwatkin//zeta/devlin.pdf How Euler discovered the zeta function]
  379 bytes (58 words) - 19:27, 17 November 2016
• Riemann-Siegel theta function
  The Riemann-Siegel theta function is defined by where $\Gamma$ denotes the [[Gamma function]] and $\log$ denotes the [[logarithm]].
  229 bytes (33 words) - 00:27, 21 March 2015
• Reciprocal zeta function
  ...eta function is $\dfrac{1}{\zeta(s)}$, where $\zeta$ is the [[Riemann zeta function]]. ...eta$ denotes the [[Riemann zeta function]], and $\mu$ denotes the [[Mobius function]].
  587 bytes (73 words) - 18:53, 24 May 2016
• Lefschetz zeta function
  Let $f$ be a function. The Lefschetz zeta function is
  244 bytes (44 words) - 18:52, 24 May 2016
• P-adic zeta function
  The $p$-adic zeta function is
  95 bytes (14 words) - 18:53, 24 May 2016
• Relationship between the exponential integral and upper incomplete gamma function
  ...integral E]] and $\Gamma$ denotes the [[incomplete gamma|incomplete gamma function]].
  268 bytes (36 words) - 00:16, 8 August 2016
• Partial derivative of beta function
  ...$\Gamma$ denotes the [[gamma function]], and $\psi$ denotes the [[digamma function]].
  397 bytes (62 words) - 15:32, 23 June 2016
• Arithmetic zeta function
  Let $X$ be a [[scheme]]. The arithmetic zeta function over $X$ is defined by
  166 bytes (26 words) - 18:51, 24 May 2016
• Artin-Mazur zeta function
  ...e the [[cardinality]] of the set $\mathrm{Fix}(f^n)$. The Artin-Mazur zeta function is
  441 bytes (74 words) - 18:52, 24 May 2016
• P-adic L function
  73 bytes (10 words) - 04:11, 12 April 2015
• Arakawa-Kaneko zeta function
  where $\Gamma$ denotes the [[gamma function]] and $\mathrm{Li}_k$ denotes the [[polylogarithm]]. ...$ and $\xi_k$ has [[analytic continuation]] to $\mathbb{C}$ as an [[entire function]].
  1 KB (140 words) - 17:22, 24 June 2016
• Barnes zeta function
  The Barnes zeta function is
  184 bytes (32 words) - 18:52, 24 May 2016
• Takagi function
  The Takagi function (also called the blancmange function) is defined by where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function.
  740 bytes (97 words) - 03:33, 6 July 2016
• Relationship between the exponential integral and incomplete gamma function
  ...[[Relationship between the exponential integral and upper incomplete gamma function]]
  95 bytes (11 words) - 03:42, 23 April 2015
• Meijer G-function
  ....youtube.com/watch?v=rZngIxZsxeA Mellin Barnes Integral for an Exponential Function]<br />
  111 bytes (16 words) - 02:03, 30 April 2015
• Relationship between Hurwitz zeta and gamma function
  ...a$ denotes the [[gamma function]] and $\zeta$ denotes the [[Hurwitz zeta]] function.
  310 bytes (42 words) - 07:13, 16 June 2016
• Laurent series of the Riemann zeta function
  where $\zeta$ denotes the [[Riemann zeta]] function and $\lambda_k$ denotes the [[Stieltjes constants]].
  328 bytes (42 words) - 05:02, 16 September 2016
• Complex conjugate of argument of error function
  where $\mathrm{erf}$ denotes the [[error function]] and $\overline{z}$ denotes the [[complex conjugate]].
  278 bytes (33 words) - 00:22, 8 August 2016
• Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta
  ..., $\log$ denotes the [[logarithm]], and $\zeta$ denotes the [[Riemann zeta function]].
  359 bytes (48 words) - 19:31, 15 June 2016
• Takagi function is nowhere differentiable
  The [[Takagi function]] is [[nowhere differentiable]].
  137 bytes (13 words) - 03:16, 6 July 2016
• Doubly periodic function
  ...\rightarrow \mathbb{C}$ is called doubly periodic if it has two [[periodic function|periods]] $\omega_1$ and $\omega_2$ such that $\dfrac{\omega_1}{\omega_2}$
  207 bytes (32 words) - 21:04, 6 June 2015
• Cantor function
  31 bytes (3 words) - 20:18, 21 May 2015
• Periodic function
  ...omega \in X$. This concept is related to the notion of a [[doubly periodic function]].
  215 bytes (39 words) - 21:03, 6 June 2015
• Weierstrass nowhere differentiable function
  The Weierstrass function is [[Weierstrass function is continuous]]<br />
  473 bytes (63 words) - 17:54, 25 June 2017
• Weierstrass function
  #REDIRECT [[Weierstrass nowhere differentiable function]]
  57 bytes (5 words) - 02:30, 22 May 2015
• Chebyshev theta function
  The Chebyshev $\vartheta$ function is
  531 bytes (72 words) - 02:18, 28 November 2016
• Chebyshev psi function
  The Chebyshev $\psi$ function is where $\Lambda$ denotes the [[Mangoldt function]].
  418 bytes (54 words) - 03:47, 22 June 2016
• Elliptic function
  A function $f$ is called elliptic if it is a [[doubly periodic function]] and it is [[meromorphic]]. <strong>Theorem:</strong> A nonconstant [[elliptic function]] has a [[fundamental pair of periods]].
  2 KB (259 words) - 00:00, 23 December 2016
• Anger function
  Let $\nu \in \mathbb{C}$. The Anger function $\mathbf{J}_{\nu}$ is defined by [[Relationship between Anger function and Bessel J sub nu]]<br />
  1 KB (193 words) - 04:05, 6 June 2016
• Relationship between Anger function and Bessel J
  ...{J}_n$ denotes an [[Anger function]] and $J_n$ denotes a [[Bessel J|Bessel function of the first kind]].
  278 bytes (42 words) - 20:27, 27 June 2016
• Weber function
  The Weber function is defined by [[Relationship between Weber function and Anger function]]<br />
  781 bytes (98 words) - 04:13, 6 June 2016
• Relationship between Weber function and Anger function
  ...u}$ denotes a [[Weber function]] and $\mathbf{J}_{\nu}$ denotes an [[Anger function]]. ...unction and Weber function|next=Relation between Weber function and Struve function}}: 12.3.5
  494 bytes (70 words) - 04:16, 6 June 2016
• Relationship between Anger function and Weber function
  ...u}$ denotes an [[Anger function]] and $\mathbf{E}_{\nu}$ denotes a [[Weber function]]. ...gun|prev=Weber function|next=Relationship between Weber function and Anger function}}: 12.3.4
  456 bytes (65 words) - 04:15, 6 June 2016
• Relationship between Weber function 0 and Struve function 0
  ...f{E}_0$ denotes a [[Weber function]] and $\mathbf{H}_0$ denotes a [[Struve function]].
  249 bytes (32 words) - 13:18, 25 June 2016
• Relationship between Weber function 1 and Struve function 1
  ...f{E}_1$ denotes a [[Weber function]] and $\mathbf{H}_1$ denotes a [[Struve function]].
  263 bytes (34 words) - 13:19, 25 June 2016
• Relationship between Weber function and Struve function
  #REDIRECT [[Relationship between Weber function 0 and Struve function 0]]
  73 bytes (8 words) - 18:20, 28 June 2015
• Relationship between Weber function 2 and Struve function 2
  ...f{E}_2$ denotes a [[Weber function]] and $\mathbf{H}_2$ denotes a [[Struve function]].
  223 bytes (31 words) - 04:11, 6 June 2016
• Modified Struve function
  where $\mathbf{H}_{\nu}$ denotes a [[Struve function]].
  341 bytes (47 words) - 18:32, 24 May 2016
• Relationship between Struve function and hypergeometric pFq
  ...a [[Struve function]], $\pi$ denotes [[pi]], $\Gamma$ denotes the [[gamma function]], and ${}_2F_1$ denotes the [[hypergeometric pFq]].
  443 bytes (55 words) - 13:18, 25 June 2016
• Inverse error function
  The inverse error function is the [[inverse function]] of the [[error function]]. We denote it by writing $\mathrm{erf}^{-1}$. [[Derivative of inverse error function]]<br />
  496 bytes (68 words) - 04:56, 16 September 2016
• Antiderivative of inverse error function
  where $\mathrm{erf}^{-1}$ denotes the [[inverse error function]], $\exp$ denotes the [[exponential]], and $\pi$ denotes [[pi]].
  369 bytes (50 words) - 03:48, 3 October 2016
• Dickman–de Bruijn function
  The Dickman–de Bruijn function $\rho(u)$ solves the [[initial value problem]]
  305 bytes (46 words) - 18:31, 24 May 2016
• Buchstab function
  The Buchstab function is a [[continuous]] function $\omega \colon [1,\infty) \rightarrow (0,\infty)$ defined by the [[initial
  453 bytes (67 words) - 18:31, 24 May 2016
• Debye function
  275 bytes (42 words) - 15:56, 10 July 2017
• Incomplete beta function
  The incomplete beta function is defined by
  319 bytes (42 words) - 01:56, 23 December 2016
• Fibonacci zeta function
  The Fibonacci zeta function is defined by * {{PaperReference|The Fibonacci Zeta Function|1976|Maruti Ram Murty|prev=Fibonacci numbers|next=Binet's formula}}
  704 bytes (99 words) - 00:25, 24 May 2017
• Riemann Siegel theta function
  The Riemann Siegel $\vartheta$ function is defined for $t \in \mathbb{R}$ by where $\log$ denotes the [[logarithm]] and $\Gamma$ denotes the [[gamma function]].
  548 bytes (76 words) - 18:33, 24 May 2016
• Error function is odd
  ...thrm{erf}$ denotes the [[error function]] (i.e. $\mathrm{erf}$ is an [[odd function]]).
  800 bytes (116 words) - 03:41, 28 March 2017
• Derivative of the exponential function
  where $e^z$ denotes the [[exponential function]].
  806 bytes (115 words) - 00:10, 23 December 2016
• Faddeeva function
  The Faddeeva function (also called the Kramp function) is defined by ...e [[error function]] and $\mathrm{erfc}$ denotes the [[complementary error function]].
  478 bytes (66 words) - 18:31, 24 May 2016
• Taylor series of the exponential function
  where $e^z$ is the [[exponential function]].
  255 bytes (35 words) - 04:03, 3 October 2016
• Imaginary error function
  18 bytes (2 words) - 22:19, 23 May 2016
• Generating function for partition function
  where $p(k)$ denotes the [[partition]] function. ...witz|author2=Irene A. Stegun|prev=Partition|next=Closed form for partition function with sinh}}: $24.2.1 \mathrm{I}.B.$
  541 bytes (72 words) - 20:34, 26 June 2016
• Bolzano function is continuous
  The [[Bolzano function]] is [[continuous]].
  126 bytes (12 words) - 17:10, 23 June 2016
• Cellérier function is continuous
  The [[Cellérier function]] is [[continuous]].
  129 bytes (13 words) - 17:12, 23 June 2016
• Darboux function is continuous
  The [[Darboux function]] is [[continuous]] on $\mathbb{R}$.
  142 bytes (15 words) - 17:14, 23 June 2016
• Faber function F1
  22 bytes (3 words) - 17:15, 23 June 2016
• Faber function F2
  22 bytes (3 words) - 17:16, 23 June 2016
• Knopp function is continuous
  The [[Knopp function]] $K_{a,b}$ is [[continuous]] on $\mathbb{R}$ for $a \in (0,1)$ and $ab>1$.
  179 bytes (23 words) - 03:31, 27 October 2016
• McCarthy function is continuous
  The [[McCarthy function]] is [[continuous]] on $\mathbb{R}$.
  143 bytes (15 words) - 13:40, 17 November 2016
• The Petr function is nowhere differentiable
  The [[Petr function]] is [[nowhere differentiable]] on $(0,1)$.
  146 bytes (14 words) - 20:34, 25 June 2017
• Riemann function is continuous
  The [[Riemann function]] is [[continuous]].
  126 bytes (12 words) - 03:27, 6 July 2016
• Schwarz function is continuous
  Let $M>0$. The [[Schwarz function]] is [[continuous]] on $(0,M)$.
  148 bytes (16 words) - 17:46, 20 September 2016
• Van der Waerden function is continuous
  The [[van der Waerden function]] is [[continuous]].
  134 bytes (14 words) - 03:17, 6 July 2016
• Derivative of inverse error function
  where $\mathrm{erf}^{-1}$ denotes the [[inverse error function]], $\pi$ denotes [[pi]], and $\exp$ denotes the [[exponential]].
  364 bytes (48 words) - 04:38, 16 September 2016
• Airy zeta function at 2
  ...m{Ai}}$ denotes the [[Airy zeta function]], $\Gamma$ denotes the [[gamma]] function, and $\pi$ denotes [[pi]].
  323 bytes (37 words) - 02:20, 2 November 2016
• Book:Aleksandar Ivić/The Riemann Zeta-Function
  {{Book|The Riemann Zeta-Function: Theory and Applications|1985|Dover Publications, Inc|0-486-42813-3|Aleksan :: 12.4. The Generalised von Mangoldt Function and the Möbius Function
  5 KB (554 words) - 16:39, 21 June 2016
• Logarithm (multivalued) of a positive integer power function
  67 bytes (7 words) - 06:28, 4 June 2016
• Gamma function written as a limit of a factorial, exponential, and a rising factorial
  where $\Gamma$ denotes the [[gamma]] function. ...(z) as integral of a power of log(1/t) for Re(z) greater than 0|next=Gamma function written as infinite product}}: §1.1 (2)
  603 bytes (87 words) - 20:56, 3 March 2018
• Gamma function written as infinite product
  where $\Gamma$ denotes the [[gamma]] function. ...Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Gamma function written as a limit of a factorial, exponential, and a rising factorial|next
  791 bytes (102 words) - 20:56, 3 March 2018
• Relationship between Anger function and Bessel J sub nu
  #REDIRECT [[Relationship between Anger function and Bessel J]]
  62 bytes (8 words) - 05:51, 6 June 2016
• Gauss' formula for gamma function
  where $\Gamma$ denotes the [[gamma]] function and $n!$ denotes the [[factorial]].
  462 bytes (62 words) - 07:03, 8 June 2016
• Book:Edward Charles Titchmarch/The Zeta-Function of Riemann
  #REDIRECT [[Book:Edward Charles Titchmarsh/The Zeta-Function of Riemann]]
  73 bytes (9 words) - 19:39, 9 June 2016
• Paper:Carl-Erik Fröberg/On the prime zeta function
  {{Book|On the prime zeta function|1968|BIT Numerical Mathematics||Carl-Erik Fröberg}}
  209 bytes (26 words) - 19:27, 15 June 2016
• Absolute convergence of secant zeta function
  The series defining the [[secant zeta function]] $\psi_s(z)$ converges absolutely in the following cases:
  430 bytes (56 words) - 06:09, 16 June 2016
• Möbius function is multiplicative
  The [[Möbius function]] is a [[multiplicative function]].
  332 bytes (41 words) - 01:32, 22 June 2016
• Cellérier function is nowhere differentiable
  The [[Cellérier function]] is [[nowhere differentiable]].
  141 bytes (14 words) - 17:12, 23 June 2016
• Darboux function is nowhere differentiable
  The [[Darboux function]] is [[nowhere differentiable]] on $\mathbb{R}$.
  154 bytes (16 words) - 17:14, 23 June 2016
• Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1
  ...e $\zeta$ denotes the [[Riemann zeta function]] and $\mu$ is the [[Möbius function]]. ...ical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Möbius function|next=Identity written as a sum of Möbius functions}}: $24.3.1. \mathrm{I}.
  506 bytes (69 words) - 01:34, 22 June 2016
• Complementary error function
  18 bytes (2 words) - 22:11, 23 May 2016
• Taylor series for error function
  where $\mathrm{erf}$ denotes the [[error function]] and $\pi$ denotes [[pi]], and $k!$ denotes the [[factorial]]. From the [[Taylor series of the exponential function]],
  1 KB (155 words) - 04:16, 3 October 2016
• Bolzano function
  A Bolzano function $B \colon [a,b] \rightarrow [c,d]$ is is defined as the limit of a sequence The Bolzano function is defined by the pointwise limit
  1 KB (239 words) - 17:10, 23 June 2016
• Bolzano function is nowhere differentiable
  The [[Bolzano function]] is [[nowhere differentiable]].
  138 bytes (13 words) - 17:11, 23 June 2016
• Cellérier function
  Let $a>1000$. The Cellérier function is defined as [[Cellérier function is continuous]]<br />
  359 bytes (51 words) - 17:11, 23 June 2016
• Riemann function
  The Riemann function is the function $R \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by [[Riemann function is continuous]]<br />
  612 bytes (89 words) - 03:26, 6 July 2016
• Darboux function
  The Darboux function is defined by where $\sin$ denotes the [[sine]] function.
  597 bytes (73 words) - 18:02, 25 June 2017
• Schwarz function
  ...r \cdot \rfloor$ denotes the [[floor]] function and let $M>0$. The Schwarz function $S \colon (0,M) \rightarrow \mathbb{R}$ is defined by [[Schwarz function is continuous]]<br />
  590 bytes (78 words) - 18:03, 25 June 2017
• Blancmange function
  #REDIRECT [[Takagi function]]
  29 bytes (3 words) - 16:50, 22 January 2016
• Takagi function is continuous
  The [[Takagi function]] is [[continuous]].
  125 bytes (12 words) - 03:15, 6 July 2016
• Van der Waerden function
  The van der Waerden function $V \colon \mathbb{R} \rightarrow \mathbb{R}$ is defined by the formula where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function.
  726 bytes (98 words) - 03:33, 6 July 2016
• Knopp function
  Let $a \in (0,1)$ $ab > 1$. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function.
  559 bytes (79 words) - 03:31, 27 October 2016
• Knopp function is nowhere differentiable
  The [[Knopp function]] $K_{a,b}$ is [[nowhere differentiable]] on $\mathbb{R}$ for $a \in (0,1)$
  192 bytes (24 words) - 03:32, 27 October 2016
• Petr function
  ...1}^{\infty} \dfrac{a_k}{10^k}$, where $a_k \in \{0,1,\ldots,9\}$. The Petr function $P_K \colon [0,1] \rightarrow \mathbb{R}$ is defined by [[The Petr function is continuous]]<br />
  673 bytes (100 words) - 20:34, 25 June 2017
• McCarthy function
  The McCarthy function $M$ is defined by [[McCarthy function is continuous]]<br />
  493 bytes (72 words) - 13:39, 17 November 2016
• Multiple gamma function
  where $\zeta_N$ denotes the [[Barnes zeta function]].
  211 bytes (35 words) - 18:32, 24 May 2016
• Elliptic gamma function
  The elliptic gamma function is defined by
  204 bytes (32 words) - 18:54, 24 May 2016
• Ramanujan theta function
  The Ramanujan theta function, $f$, is defined for $|ab|<1$ by
  379 bytes (61 words) - 16:02, 10 July 2017
• Sum of sum of divisors function equals product of Riemann zeta for Re(z) greater than k+1
  where $\sigma_n$ denotes the [[sum of divisors]] function and $\zeta$ denotes the [[Riemann zeta]].
  481 bytes (65 words) - 22:18, 25 June 2016
• Closed form for partition function
  #REDIRECT [[Closed form for partition function with sinh]]
  58 bytes (8 words) - 20:34, 26 June 2016
• Recurrence relation for partition function
  #REDIRECT [[Pure recurrence relation for partition function]]
  61 bytes (7 words) - 20:39, 26 June 2016
• Recurrence relation for partition function with sum of divisors
  ...the [[partition]] function and $\sigma_1$ denotes the [[sum of divisors]] function. ...tion function|next=Sum of divisors functions written in terms of partition function}}: $24.2.1 \mathrm{II}.A.$
  530 bytes (69 words) - 20:43, 26 June 2016
• Sum of divisors functions written in terms of partition function
  ...$ denotes the [[sum of divisors]] function and $p$ denotes the [[partition function]]. ...tition function with sum of divisors|next=Asymptotic formula for partition function}}: $24.2.1 \mathrm{II}.B.$
  604 bytes (83 words) - 20:47, 26 June 2016
• Asymptotic formula for partition function
  where $p$ denotes the [[partition]] function, $\exp$ denotes the [[exponential]], and $\pi$ denotes [[pi]]. ...ene A. Stegun|prev=Sum of divisors functions written in terms of partition function|next=findme}}: $24.2.1 \mathrm{III}$
  533 bytes (66 words) - 00:10, 28 June 2016
• Series for log(Riemann zeta) in terms of Mangoldt function
  * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Series for log(riemann zeta)
  515 bytes (68 words) - 05:42, 5 July 2016
• Logarithmic derivative of Riemann zeta in terms of Mangoldt function
  ...\zeta$ denotes the [[Riemann zeta]] and $\Lambda$ denotes the [[Mangoldt]] function. * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Logarithmic derivative of Ri
  546 bytes (72 words) - 09:05, 19 November 2016
• Van der Waerden function is nowhere differentiable
  The [[van der Waerden function]] is [[nowhere differentiable]].
  146 bytes (15 words) - 03:18, 6 July 2016
• Riemann function is almost nowhere differentiable
  The [[Riemann function]] is [[nowhere differentiable]] except at points of the form $\pi \dfrac{2p
  218 bytes (29 words) - 03:28, 6 July 2016
• Generating function for Hermite (physicist) polynomials
  497 bytes (60 words) - 22:55, 8 July 2016
• Paper:Maruti Ram Murty/The Fibonacci Zeta Function
  {{Book|The Fibonacci Zeta Function|1976|Journal of the London Mathematical Society||Maruti Ram Murty}} [[Fibonacci zeta function]] <br />
  387 bytes (52 words) - 00:28, 24 May 2017
• Schwarz function is nowhere differentiable on a dense subset
  Let $M>0$. The [[Schwarz function]] is [[nowhere differentiable]] on a [[dense]] subset of $(0,M)$.
  182 bytes (21 words) - 17:47, 20 September 2016
• K-function
  The $K$-function is defined by ...ient]], $\log$ denotes the [[logarithm]], and $\Gamma$ denotes the [[gamma function]].
  461 bytes (59 words) - 19:40, 25 September 2016
• Hyperfactorial in terms of K-function
  where $H$ denotes the [[hyperfactorial]] and $K$ denotes the [[K-function]].
  205 bytes (25 words) - 19:40, 25 September 2016
• Paper:Richard E. Crandall/On the quantum zeta function
  {{Book|On the quantum zeta function|1996|Journal of Physics A: Mathematical and General||Richard E. Crandall}}
  246 bytes (35 words) - 02:15, 2 November 2016
• Thomae function is continuous at irrationals
  The [[Thomae function]] is [[continuous]] at all [[irrational number|irrational numbers]].
  173 bytes (18 words) - 00:34, 9 December 2016
• McCarthy function is nowhere differentiable
  The [[McCarthy function]] is [[nowhere differentiable]] on $\mathbb{R}$.
  155 bytes (16 words) - 13:40, 17 November 2016
• Thomae function is discontinuous at rationals
  The [[Thomae function]] is discontinuous at all [[rational number|rational numbers]].
  168 bytes (18 words) - 00:34, 9 December 2016
• Identity function
  22 bytes (2 words) - 23:01, 25 December 2016
• Paper:Thomas Clausen/Uber die function sin(φ)+sin(2φ)/2^2+sin(3φ)/3^2+etc.
  {{Book|Uber die function sin(φ)+sin(2φ)/2^2+sin(3φ)/3^2+etc.|1832|Journal für die reine und ange
  286 bytes (39 words) - 06:38, 10 January 2017
• Dirichlet function
  The Dirichlet function $D \colon \mathbb{R} \rightarrow \{0,1\}$ is defined by [[Dirichlet function is nowhere continuous]]<br />
  525 bytes (71 words) - 07:29, 10 January 2017
• Dirichlet function is nowhere continuous
  The [[Dirichlet function]] is nowhere [[continuous]].
  136 bytes (13 words) - 07:31, 10 January 2017
• Bessel-Clifford function
  29 bytes (2 words) - 10:44, 11 January 2017
• Paper:Daniel S. Moak/The q-gamma function for q greater than 1
  {{Book|The q-gamma function for q greater than 1|1980|Aequationes Mathematicae||Daniel S. Moak|}}
  351 bytes (44 words) - 00:18, 30 May 2017
• Weierstrass function is nowhere differentiable
  The [[Weierstrass nowhere differentiable function|Weierstrass function]] is nowhere differentiable.
  182 bytes (17 words) - 17:52, 25 June 2017
• Airy zeta function
  ...elds a sequence of zeros $\{a_i\}_{i=1}^{\infty}$. We define the Airy zeta function using these zeros in the following way: [[Airy zeta function at 2]]<br />
  468 bytes (67 words) - 02:16, 2 November 2016
• Epstein zeta function
  Let $Q(m,n)=cm^2+bmn+an^2$. The Epstein zeta function is ...d=2129&uid=2&uid=70&uid=4&uid=3739256&sid=21104980545913 On Epstein's Zeta Function]<br />
  324 bytes (45 words) - 18:52, 24 May 2016
• Goss zeta function
  ...://arxiv.org/pdf/math/9801158.pdf The Riemann Hypothesis for the Goss zeta function $\mathbb{F}_q[T]$]
  120 bytes (19 words) - 01:25, 21 October 2014
• Lerch zeta function
  The Lerch zeta function is defined by [http://arxiv.org/pdf/1506.06161v1.pdf The Lerch zeta function III. Polylogarithms and special values]
  376 bytes (52 words) - 17:58, 24 June 2016
• Matsumoto zeta function
  Let $P(z)$ be a [[polynomial]]. Define the Matsumoto zeta function by
  183 bytes (27 words) - 18:53, 24 May 2016
• Weierstrass function is continuous
  The [[Weierstrass nowhere differentiable function|Weierstrass function]] is continuous.
  170 bytes (16 words) - 17:51, 25 June 2017
• The Petr function is continuous
  The [[Petr function]] is [[continuous]] on $(0,1)$.
  134 bytes (13 words) - 20:35, 25 June 2017
• Unit step function
  The unit step function $U \colon \mathbb{R} \rightarrow \left\{ 0, \dfrac{1}{2}, 1 \right\}$ is de
  444 bytes (52 words) - 22:08, 26 August 2017
• Integral representation of Struve function
  ...[[Struve function]], $\pi$ denotes [[pi]], $\Gamma$ denotes the [[gamma]] function, and $\sin$ denotes [[sine]]. ...author2=Irene A. Stegun|prev=findme|next=Integral representation of Struve function (2)}}: $12.1.6$
  633 bytes (79 words) - 16:11, 4 November 2017
• Integral representation of Struve function (2)
  where $\mathbf{H}$ denotes the [[Struve function]], $\pi$ denotes [[pi]], $\Gamma$ denotes [[gamma]]$, $\sin$ denotes [[sine ...l representation of Struve function|next=Integral representation of Struve function (3)}}: $12.1.7$
  699 bytes (87 words) - 16:14, 4 November 2017
• Integral representation of Struve function (3)
  ...denotes the [[gamma]] function, and $e^{-zt}$ denotes the [[exponential]] function. ...Abramowitz|author2=Irene A. Stegun|prev=Integral representation of Struve function (2)|next=Recurrence relation for Struve fuction}}: $12.1.8$
  813 bytes (108 words) - 19:53, 4 November 2017
• Recurrence relation for Struve function (2)
  ...truve function]], $\pi$ denotes [[pi]], and $\Gamma$ denotes the [[gamma]] function.
  549 bytes (73 words) - 16:27, 4 November 2017
• Ein function
  41 bytes (4 words) - 00:47, 25 March 2018
• Paper:Yilmaz Simsek/q-Dedekind type sums related to q-zeta function and basic L-series
  {{Book|q-Dedekind type sums related to q-zeta function and basic L-series|2006|Journal of Mathematical Analysis and Applications||
  360 bytes (42 words) - 04:52, 12 February 2018
• Generating function for Laguerre L
  where $e^{\frac{-xt}{1-t}}$ denotes an [[exponential]] function and $L_k$ denotes [[Laguerre L]].
  444 bytes (70 words) - 14:08, 15 March 2018
• Paper:R. M. Corless/On the Lambert W function
  {{Book|On the Lambert W function|1996|Advances in Computational Mathematics||R. M. Corless|author2=G. H. Gon
  209 bytes (33 words) - 18:23, 16 June 2018
• Paper:Lucas Jódar/On the hypergeometric matrix function
  {{Book|On the hypergeometric matrix function|1998|Journal of Computational and Applied Mathematics||Lucas Jódar}} [[Beta function|(1)]] <br />
  517 bytes (73 words) - 01:30, 3 August 2019

Page text matches

• Main Page
  <div class="grid"><center>[[Identity function|<randomimagebycategory categories="identityglyph" width="45" />]]<br /> [[Identity function|Identity]]</center></div>
  26 KB (2,938 words) - 15:47, 26 August 2023
• Hypergeometric pFq
  The generalized hypergeometric function ${}_pF_q$ is defined by
  2 KB (274 words) - 14:42, 15 March 2018
• Series for log(riemann zeta) over primes
  ...ct for Riemann zeta|next=Series for log(Riemann zeta) in terms of Mangoldt function}}: § Introduction $(2')$
  532 bytes (68 words) - 03:09, 1 July 2017
• Hypergeometric 0F0
  The hypergeometric ${}_0F_0$ function is defined by the series It is a special case of the [[hypergeometric pFq]] function.
  328 bytes (43 words) - 06:04, 10 January 2017
• Bernoulli-Euler Gamma function
  ....de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html#id1 Is the Gamma function misdefined?]
  128 bytes (16 words) - 22:45, 13 January 2015
• Hadamard gamma
  The Hadamard gamma function is defined by the formula where $\Gamma$ denotes the [[gamma function]].
  2 KB (267 words) - 18:28, 24 May 2016
• Spherical Bessel j
  The spherical Bessel function of the first kind is defined by where $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]].
  480 bytes (66 words) - 22:44, 20 June 2016
• Spherical Bessel y
  The spherical Bessel function of the second kind is where $Y_{\nu}$ denotes the [[Bessel Y sub nu|Bessel function of the second kind]].
  466 bytes (63 words) - 01:14, 18 July 2016
• Q-Gaussian distribution
  The $q$-Gaussian distribution has [[probability density function]]
  498 bytes (74 words) - 18:55, 24 May 2016
• Heaviside step function
  The Heaviside step function is
  148 bytes (22 words) - 18:32, 24 May 2016
• Hankel H (1)
  ...function of the first kind]] and $Y_{\nu}$ is the [[Bessel Y sub nu|Bessel function of the second kind]]. Note the similarity of these functions to the [[Hanke
  953 bytes (142 words) - 23:59, 22 December 2016
• Hankel H (2)
  ...Bessel function of the first kind]] and $Y_{\nu}$ is the [[Bessel Y|Bessel function of the second kind]]. Note the similarity of these functions to the [[Hanke
  809 bytes (127 words) - 23:58, 22 December 2016
• Modified Bessel I
  The modified Bessel function of the first kind is defined by ...notes the [[imaginary number]] and $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]].
  954 bytes (145 words) - 23:53, 10 June 2016
• Modified Bessel K
  The modified Bessel function of the second kind is defined by where $I_{\nu}$ is the [[Modified Bessel I sub nu|modified Bessel function of the first kind]].
  775 bytes (110 words) - 23:46, 10 June 2016
• Difference equation of hypergeometric type
  where the function $\rho$ satisfies a [[Pearson difference equation]]
  862 bytes (126 words) - 08:58, 17 January 2015
• Dawson D-
  The Dawson function $D-$ is defined by [[Error function]]<br />
  289 bytes (41 words) - 00:12, 29 October 2017
• Dawson D+
  The Dawson function $D+$ (sometimes called the Dawson $F$ function) is defined by [[Error function]]<br />
  437 bytes (60 words) - 00:21, 29 October 2017
• Böhmer C
  The Böhmer $C$ function is defined by
  334 bytes (48 words) - 06:51, 10 January 2017
• Thomae function
  Thomae's function (sometimes called the popcorn function, raindrop function, Stars over Babylon) is given by the formula File:Thomae.png|Plot of the Thomae function.
  1 KB (150 words) - 00:36, 9 December 2016
• Devil's staircase
  The Devil's staircase (also known as the Cantor function) is a function $c \colon [0,1] \rightarrow [0,1]$ can be expressed by the following rules: File:Cantor function.gif|Construction of $c$ in steps.
  1 KB (146 words) - 16:03, 10 July 2017
• Full list
  <td><center>[[File:erfthumb.png|45px|link=Error function]]<br /> [[Error function]]</center></td>
  12 KB (1,622 words) - 00:11, 5 May 2015
• Upper incomplete gamma
  The upper incomplete gamma function $\Gamma$ is defined by [[:Relationship between the exponential integral and upper incomplete gamma function]]
  262 bytes (35 words) - 03:22, 1 July 2017
• Sum of reciprocal Pochhammer symbols of a fixed exponent
  ...k)_p$ denotes the [[Pochhammer]] symbol and $\Gamma$ denotes the [[gamma]] function.
  318 bytes (44 words) - 12:40, 17 September 2016
• Beta in terms of sine and cosine
  ...], $\sin$ denotes the [[sine]] function, and $\cos$ denotes the [[cosine]] function.
  1 KB (223 words) - 21:04, 3 March 2018
• Beta in terms of gamma
  where $B$ denotes the [[beta]] function and $\Gamma$ denotes the [[gamma]] function.
  422 bytes (57 words) - 15:10, 6 October 2016
• Taylor series
  A Taylor series is a way to express a function as an infinite series under suitable differentiability conditions. The Tayl [[Taylor series of the exponential function]]<br />
  438 bytes (66 words) - 03:48, 6 June 2016
• Weierstrass factorization theorem
  ...a zero of order $0$ means that $f(0) \neq 0$). Then there exists an entire function $g$ and a sequence of integers $\{p_n\}$ such that [[Gamma function Weierstrass product]]<br />
  923 bytes (136 words) - 19:12, 26 November 2016
• Riemann-Landau xi
  The Riemann-Landau $\Xi$ function is defined by * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Functional equation for Riem
  403 bytes (49 words) - 15:28, 18 March 2017
• Signum
  ...\mathbb{R} \rightarrow \{-1,0,1\}$ (also called the sign function) is the function The function is occasionally extended to a function $\mathrm{sgn} \colon \mathbb{C} \rightarrow \mathbb{C}$ by
  999 bytes (131 words) - 05:12, 11 February 2018
• Legendre's constant
  where $\log$ denotes the [[logarithm]] and $\pi$ denotes the [[prime counting function]].
  615 bytes (88 words) - 18:59, 24 May 2016
• Logarithmically convex
  ...$f$ is called logarithmically convex (sometimes called superconvex) if the function $g(x)=\log f(x)$ is [[convex]].
  126 bytes (20 words) - 04:14, 20 March 2015
• Mertens
  Let $n \in \mathbb{Z}^+$. Define the Mertens function where $\mu$ is the [[Möbius function]].
  555 bytes (78 words) - 17:41, 12 October 2016
• Secant zeta function
  [[Absolute convergence of secant zeta function]]
  295 bytes (40 words) - 06:10, 16 June 2016
• Book:Leonard Lewin/Dilogarithms and Associated Functions
  ::8. Legendre's Chi-function ::8. Function Equations Involving Several Variables
  4 KB (409 words) - 20:56, 27 June 2016
• Hyperfactorial
  [[Hyperfactorial in terms of K-function]]<br />
  426 bytes (58 words) - 19:39, 25 September 2016
• Struve function
  [[Relationship between Struve function and hypergeometric pFq]]<br /> [[Relationship between Weber function 0 and Struve function 0]]<br />
  1 KB (218 words) - 01:09, 21 December 2017
• Elliptic K
  The elliptic $K$ function (also known as the complete elliptic integral of the first kind) is defined
  635 bytes (84 words) - 04:48, 21 December 2017
• Gamma
  ...lon \mathbb{C} \setminus \{0,-1,-2,\ldots\} \rightarrow \mathbb{C}$ is the function initially defined for $\mathrm{Re}(z)>0$ by the integral by the formula The [[analytic continuation]] of $\Gamma$ leads to a [[meromorphic function]] with [[pole | poles]] at the negative integers.
  5 KB (678 words) - 18:12, 16 June 2018
• Riemann zeta
  The Riemann zeta function $\zeta$ is defined for $\mathrm{Re}(z)>1$ by [[Laurent series of the Riemann zeta function]]<br />
  3 KB (417 words) - 18:40, 12 May 2017
• Q-Gamma
  Let $0<q<1$. Define the $q$-gamma function by the formula ...The function $\Gamma_q$ is a [[q-analogue | $q$-analogue]] of the [[gamma function]].
  2 KB (252 words) - 00:13, 30 May 2017
• Error function
  The (normalized) error function $\mathrm{erf}$ is defined by where $\pi$ denotes [[pi]] and $e^{-\tau^2}$ denotes the [[exponential]] function.
  2 KB (271 words) - 00:43, 25 June 2017
• Sine
  The sine function $\sin \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by ...-theory-fall-2006/projects/chan.pdf The sine product formula and the gamma function]<br />
  2 KB (207 words) - 17:34, 1 July 2017
• Partition
  The partition function $p \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ is defined so that $p(n)$ [[Generating function for partition function]]<br />
  820 bytes (109 words) - 20:50, 26 June 2016
• Format notes
  =Defining a new function= The such-and-such function $\mathrm{mathname} \colon domain \rightarrow codomain$ is defined by
  8 KB (1,248 words) - 04:48, 26 November 2016
• Q-Binomial
  The $q$-Binomial function is
  1 KB (203 words) - 18:56, 24 May 2016
• Closed form for partition function with sinh
  a [[sawtooth function]], define where $p$ denotes the [[partition]] function, $\pi$ denotes [[pi]], and $\sinh$ denotes the [[sinh|hyperbolic sine]].
  1 KB (172 words) - 20:40, 26 June 2016
• Pure recurrence relation for partition function
  ...the [[partition]] function and $\sigma_1$ denotes the [[sum of divisors]] function. ...rm for partition function with sinh|next=Recurrence relation for partition function with sum of divisors}}: $24.2.1 \mathrm{II}.A.$
  581 bytes (80 words) - 20:41, 26 June 2016
• Ihara zeta function
  Let $X$ be a finite [[graph]]. The Ihara zeta function is given by the formula ...an analogue of the [[Euler product]] representation of the [[Riemann zeta function]].
  765 bytes (122 words) - 18:52, 24 May 2016
• Exponential
  The exponential function $\exp \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula [[Derivative of the exponential function]]<br />
  2 KB (327 words) - 23:37, 21 October 2017
• Airy Bi
  The Airy function $\mathrm{Bi}$ (sometimes called the "Bairy function") is a solution of the [[Airy differential equation]] which is [[linearly independent]] from the [[Airy Ai]] function.
  2 KB (216 words) - 16:07, 21 October 2017
• Lambert W
  The Lambert $W$ function is the (multi-valued) function that satisfies the equation ...om/watch?v=AJD8kh3DSAM 6: Recursion, Infinite Tetrations and the Lambert W Function (4 August 2014)]
  974 bytes (142 words) - 18:24, 16 June 2018
• Apéry's constant
  where $\zeta$ denotes the [[Riemann zeta function]]. This constant is notable because it is known in general that for integer
  682 bytes (105 words) - 17:17, 24 June 2016
• Prime counting
  The prime counting function $\pi \colon \mathbb{R} \rightarrow \mathbb{Z}^+$ is defined by the formula
  600 bytes (82 words) - 06:35, 22 June 2016
• Hermite (probabilist)
  ...a sequence of [[Orthogonal polynomial|orthogonal polynomials]] with weight function $e^{-\frac{x^2}{2}}$. <strong>Theorem:</strong> ([[Generating function]]) The Hermite polynomials obey
  2 KB (228 words) - 18:41, 24 May 2016
• Fransén–Robinson constant
  where $\dfrac{1}{\Gamma}$ denotes the [[reciprocal gamma function]].
  340 bytes (46 words) - 20:17, 20 June 2016
• Beta
  The beta function $B$ (note: $B$ is [https://en.wikipedia.org/wiki/Beta capital $\beta$] in G [[Partial derivative of beta function]]<br />
  3 KB (578 words) - 19:49, 15 March 2018
• Polylogarithm
  [https://www.youtube.com/watch?v=6v60ivoC2z8 polylogarithm function]
  646 bytes (91 words) - 20:28, 25 June 2017
• Cosine
  The cosine function, $\cos \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula where $e^z$ is the [[exponential function]].
  1 KB (177 words) - 22:09, 19 December 2017
• Euler-Mascheroni constant
  #the [[Barnes G]] function
  2 KB (206 words) - 20:57, 3 March 2018
• Book:Arthur Erdélyi/Higher Transcendental Functions Volume II
  ::::7.2.3. Kelvin's function and related functions
  5 KB (521 words) - 05:44, 4 March 2018
• Book:Edward Charles Titchmarsh/The Zeta-Function of Riemann
  {{Book|The Zeta-Function of Riemann|1930|Cambridge University Press||Edward Charles Titchmarch}} ...r primes|$(2')$]] (and [[Series for log(Riemann zeta) in terms of Mangoldt function|$(2')$]])
  1 KB (170 words) - 15:23, 18 March 2017
• Factorial
  [[Gamma function]]
  242 bytes (35 words) - 19:40, 9 October 2016
• Pochhammer
  ...ochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by ...findme|next=findme}}: $18. (1)$ (note: Rainville calls this the "factorial function" and expresses it slightly differently by defining it by the equivalent for
  1 KB (187 words) - 23:25, 3 March 2018
• Mills' constant
  ...6/S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf A prime-representing function by W.H. Mills]
  323 bytes (52 words) - 19:00, 24 May 2016
• Tangent
  The tangent function is defined as the ratio of the [[sine]] and [[cosine]] functions:
  916 bytes (117 words) - 03:38, 6 July 2016
• Euler totient
  Euler's totient function $\phi \colon \{1,2,3,\ldots\} \rightarrow \{1,2,3,\ldots\}$ (not to be conf [https://www.youtube.com/watch?v=QbsWEVcjJy0 Euler's Phi Function] (6 October 2010)<br />
  2 KB (344 words) - 01:58, 21 December 2017
• Mobius function
  #REDIRECT [[Möbius function]]
  30 bytes (4 words) - 16:13, 4 October 2014
• Jacobi sn
  #$\mathrm{sn \hspace{2pt}}$ is an odd function
  815 bytes (121 words) - 19:06, 5 July 2016
• Jacobi cn
  #$\mathrm{cn \hspace{2pt}}$ is an even function
  762 bytes (111 words) - 19:06, 5 July 2016
• Mertens function
  Merten's function is defined by the formula where $\mu$ is the [[Möbius function]].
  129 bytes (22 words) - 16:13, 4 October 2014
• Möbius
  The Möbius function is the function $\mu$ defined by the formula [[Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1]]<br />
  2 KB (278 words) - 23:55, 8 December 2016
• Liouville lambda
  The Liouville function is defined by the formula where $\zeta$ denotes the [[Riemann zeta function]].
  1 KB (139 words) - 06:35, 22 June 2016
• Mangoldt
  The Mangoldt function is defined by the formula
  770 bytes (106 words) - 02:31, 28 November 2016
• Hurwitz zeta
  The Hurwitz zeta function is a generalization of the [[Riemann zeta]] function defined initially for $\mathrm{Re}(s)>1$ and $\mathrm{Re}(a)>0$ by [[Relationship between Hurwitz zeta and gamma function]]<br />
  575 bytes (76 words) - 01:27, 21 December 2016
• Laguerre L
  [[Generating function for Laguerre L]]<br /> ...ns for Scientists and Engineers|1968|W.W. Bell|prev=findme|next=Generating function for Laguerre L}}: $(6.3)$
  1 KB (180 words) - 14:37, 15 March 2018
• Euler E generating function
  where $e^{xt}$ denotes the [[exponential function]] and $E_n$ denotes an [[Euler E]] polynomial.
  519 bytes (75 words) - 01:05, 4 March 2018
• Gegenbauer C
  where $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, $\Gamma$ denotes [[gamma]], and $k!$ denotes the [[factorial]].
  1 KB (181 words) - 01:29, 20 December 2017
• Jacobi P
  ...nomial $P_n^{(\alpha,\beta)}$ are [[orthogonal polynomials]] with [[weight function]] $w(x)=(1-x)^{\alpha}(1-x)^{\beta}$ on the interval $[-1,1]$ that obey $P_
  826 bytes (106 words) - 03:30, 11 June 2016
• Exponential integral E
  [[Relationship between the exponential integral and upper incomplete gamma function]]<br />
  1 KB (199 words) - 00:45, 24 March 2018
• Logarithmic integral
  [[Prime counting function]] <br />
  994 bytes (117 words) - 03:33, 17 March 2018
• Sine integral
  where $\mathrm{sinc}$ denotes the [[sinc]] function. ...Physics and Chemistry|1956|Ian N. Sneddon|prev=Cosine integral|next=Error function}}: $\S 5 (5.10)$
  1,012 bytes (130 words) - 00:42, 25 June 2017
• Bateman F
  where ${}_3F_2$ denotes the [[generalized hypergeometric function]]. The first few Bateman polynomials are
  1 KB (135 words) - 11:57, 10 October 2019
• Euler E
  [[Euler E generating function]]<br />
  754 bytes (117 words) - 01:05, 4 March 2018
• Catalan's constant
  ...stant can be expressed as $\beta(2)$ where $\beta$ is the [[Dirichlet beta function]].
  455 bytes (51 words) - 15:40, 25 February 2018
• Klein invariant J
  The Klein j-invariant $j(\tau)$ is ''the'' [[modular function]] which encodes arithmetic information about the [[singular moduli]] of [[i
  411 bytes (58 words) - 18:28, 24 May 2016
• Hermite (physicist)
  ...polynomials $H_n$ are a sequence of [[orthogonal polynomials]] with weight function $w(x)=e^{-x^2}$ over $(-\infty,\infty)$. [[Generating function for Hermite (physicist) polynomials]]<br />
  2 KB (256 words) - 00:49, 9 July 2016
• Dirichlet beta
  The Dirichlet $\beta$ function is defined by
  489 bytes (65 words) - 00:54, 11 December 2016
• Sinc
  ...athrm{sinc}$ function (sometimes called the "unnormalized" $\mathrm{sinc}$ function) is defined by It appears in the definition of the [[Sine integral]] function.
  1 KB (175 words) - 02:19, 16 September 2016
• Sum of divisors
  The sum of positive divisors function, $\sigma_x$, is defined by [[Sum of divisors functions written in terms of partition function]]<br />
  559 bytes (83 words) - 20:50, 26 June 2016
• Logarithm
  [[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
  3 KB (452 words) - 05:03, 21 December 2017
• Polygamma
  The polygamma function of order $m$, $\psi^{(m)}(z)$, is defined by the formula ...mma]] function $\psi$ is the function $\psi^{(0)}(z)$ and the [[trigamma]] function is $\psi^{(1)}(z)$.
  2 KB (222 words) - 22:47, 17 March 2017
• Q-Polygamma function
  ...a functions of order $m$, $\psi_q^{(m)}$, are analogues of the [[polygamma function]] defined by ...ma_q(z).$ Here the function $\Gamma_q$ is the [[q-Gamma function|$q$-Gamma function]].
  368 bytes (60 words) - 18:56, 24 May 2016
• Polygamma functions
  #REDIRECT [[Polygamma function]]
  32 bytes (3 words) - 15:18, 14 October 2014
• Laplace transform
  ...tarrow \mathbb{C}$ be a function, then the Laplace transform of $f$ is the function defined by |Original function $f(t)$
  4 KB (583 words) - 03:37, 21 December 2017
• Arccos
  The function $\mathrm{arccos} \colon \mathbb{C} \setminus \{(-\infty,-1) \bigcup (1,\inf
  916 bytes (116 words) - 20:04, 22 November 2016
• Arcsin
  The function $\mathrm{arcsin} \colon \mathbb{C} \setminus \left\{ (-\infty,-1) \bigcup ( ...ub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0015%7CLOG_0028 On the function arc sin(x+iy)-Cayley]<br />
  1 KB (214 words) - 23:45, 22 December 2016
• Arctan
  The $\mathrm{arctan}$ function is the inverse function of the [[tangent]] function.<br />
  778 bytes (94 words) - 02:46, 16 September 2016
• Prime zeta P
  The prime zeta function is defined by [[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
  2 KB (248 words) - 23:29, 17 March 2017
• Arccot
  ...\right] \setminus \{0\}$ is the [[inverse function]] of the [[cotangent]] function.
  685 bytes (81 words) - 03:44, 6 July 2016
• Arcsec
  ...t\{ \dfrac{\pi}{2} \right\}$ is the [[inverse function]] of the [[secant]] function.
  526 bytes (62 words) - 03:44, 6 July 2016
• Arccsc
  ...} \right] \setminus \{0\}$ is the [[inverse function]] of the [[cosecant]] function.
  549 bytes (65 words) - 14:51, 19 September 2016
• Cotangent
  The cotangent function is defined by the formula where $\tan$ denotes the [[tangent]] function.
  970 bytes (123 words) - 03:38, 6 July 2016
• Secant
  The secant function is defined by
  814 bytes (101 words) - 20:45, 26 February 2017
• Cosecant
  The cosecant function is defined by where $\sin$ denotes the [[sine]] function.
  1 KB (145 words) - 15:39, 10 July 2017
• Sinh
  The hyperbolic sine function $\mathrm{sinh} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by ...to-one]], its [[inverse function]] the [[arcsinh|inverse hyperbolic sine]] function is clear.
  2 KB (217 words) - 23:44, 21 October 2017
• Cosh
  The hyperbolic cosine function $\cosh \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by
  2 KB (204 words) - 23:44, 21 October 2017
• Coth
  where $\tanh$ denotes the [[Tanh|hyperbolic tangent]] function. ...be.com/watch?v=Pz7BDxef3HU Calculus I - Derivative of Hyperbolic Cotangent Function coth(x) - Proof]
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• Sech
  The hyperbolic secant function $\mathrm{sech} \colon \mathbb{R} \rightarrow (0,1]$ is defined by ...define the [[arcsech|inverse hyperbolic secant function]] as the [[inverse function]] of $\mathrm{sech}$ restricted to $[0,\infty)$.
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• Csch
  The hyperbolic cosecant function $\mathrm{csch} \colon \mathbb{R} \setminus \{0\} \rightarrow \mathbb{R} \se ...e]], its [[inverse function]], the [[arccsch|inverse hyperbolic cosecant]] function is clear.
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• Arcsinh
  ...}$ is function is the [[inverse function]] of the [[sinh|hyperbolic sine]] function. It may be defined by
  792 bytes (98 words) - 23:28, 11 December 2016
• Arccosh
  ...\mathrm{arccosh}$ is the [[inverse function]] of the [[hyperbolic cosine]] function. It may be defined by
  661 bytes (81 words) - 23:42, 11 December 2016
• Arctanh
  ...m{arctanh}$ is the [[inverse function]] of the [[tanh|hyperbolic tangent]] function. It may be defined by
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• Arccoth
  ...arccoth}$ is the [[inverse function]] of the [[coth|hyperbolic cotangent]] function. It may be defined by the following formula:
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• Arcsech
  ...on $\mathrm{arcsech} \colon (0,1] \rightarrow [0,\infty)$ is the [[inverse function]] of the [[sech|hyperbolic secant]].
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• Arccsch
  ...{R} \setminus \{0\} \rightarrow \mathbb{R} \setminus \{0\}$ is the inverse function of the [[csch|hyperbolic cosecant]].
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• Arithmetic functions
  An arithmetic function is a function $f \colon \{1,2,3,\ldots\} \rightarrow \mathbb{C}$.
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• Reciprocal gamma
  The reciprocal gamma function $\dfrac{1}{\Gamma}$ is defined by where $\Gamma$ denotes the [[gamma function]].
  893 bytes (108 words) - 10:50, 11 January 2017
• Jacobi ns
  The $\mathrm{ns}$ function is defined by where $\mathrm{sn}$ denotes the [[Jacobi sn]] function.
  449 bytes (61 words) - 19:07, 5 July 2016
• Jacobi sc
  The $\mathrm{sc}$ function is defined by ...{sn}$ is the [[Jacobi sn]] function and $\mathrm{cn}$ is the [[Jacobi cn]] function.
  505 bytes (72 words) - 19:07, 5 July 2016
• Jacobi nd
  The $\mathrm{nd}$ function is defined by where $\mathrm{dn}$ is the [[Jacobi dn]] function.
  444 bytes (61 words) - 19:07, 5 July 2016
• Jacobi ds
  The $\mathrm{ds}$ function is defined by ...{dn}$ is the [[Jacobi dn]] function and $\mathrm{sn}$ is the [[Jacobi sn]] function.
  505 bytes (72 words) - 19:07, 5 July 2016
• Jacobi nc
  The $\mathrm{nc}$ function is defined by where $\mathrm{cn}$ is the [[Jacobi cn]] function.
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• Jacobi cd
  The $\mathrm{cd}$ function is defined by ...{cn}$ is the [[Jacobi cn]] function and $\mathrm{dn}$ is the [[Jacobi dn]] function.
  505 bytes (72 words) - 19:06, 5 July 2016
• Jacobi sd
  The $\mathrm{sd}$ function is defined by ...{sn}$ is the [[Jacobi sn]] function and $\mathrm{dn}$ is the [[Jacobi dn]] function.
  493 bytes (71 words) - 19:08, 5 July 2016
• Jacobi cs
  The $\mathrm{cs}$ function is defined by ...{cn}$ is the [[Jacobi cn]] function and $\mathrm{sn}$ is the [[Jacobi sn]] function.
  493 bytes (71 words) - 19:06, 5 July 2016
• Jacobi dc
  The $\mathrm{dc}$ function is defined by ...{dn}$ is the [[Jacobi dn]] function and $\mathrm{cn}$ is the [[Jacobi cn]] function.
  493 bytes (71 words) - 19:06, 5 July 2016
• Taylor series of sine
  where $\sin$ denotes the [[sine]] function. Using the [[Taylor series of the exponential function]] and the definition of $\sin$,
  1 KB (206 words) - 03:19, 1 July 2017
• Fresnel C
  The Fresnel C function is defined by the formula
  871 bytes (132 words) - 05:10, 21 December 2017
• Fresnel S
  The Fresnel $S$ function is defined by
  867 bytes (129 words) - 17:21, 5 October 2016
• Bessel J
  where $\Gamma$ denotes the [[gamma]] function. [[Relationship between Anger function and Bessel J]]<br />
  3 KB (476 words) - 05:41, 4 March 2018
• Derivative of cosine
  ...he [[chain rule]], the [[reciprocal of i]], and the definition of the sine function,
  1,007 bytes (152 words) - 01:27, 1 July 2017
• Taylor series of cosine
  where $\cos$ denotes the [[cosine]] function. Using the [[Taylor series of the exponential function]] and the definition of $\cos$,
  1 KB (201 words) - 03:18, 1 July 2017
• Derivative of tangent
  ...$\tan$ denotes the [[tangent]] function and $\sec$ denotes the [[secant]] function.
  870 bytes (124 words) - 00:35, 26 April 2017
• Derivative of cosecant
  ...sc$ denotes the [[cosecant]] function and $\cot$ denotes the [[cotangent]] function.
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• Derivative of sinh
  and so using the [[derivative of the exponential function]], the [[derivative is a linear operator|linear property of the derivative]
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• Derivative of cosh
  and so using the [[derivative of the exponential function]], the [[derivative is a linear operator|linear property of the derivative]
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• Derivative of arccos
  where $\mathrm{arccos}$ denotes the [[arccos|inverse cosine]] function.
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• Dedekind zeta function
  ...finite [[field extension]] of the [[rational numbers]]. The Dedekind zeta function of $F$ is
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• Dirichlet L-function
  Let $\chi$ be a [[Dirichlet character]]. The Dirichlet $L$-function associated with $\chi$ is ...ac.uk/people/staff/mrwatkin//zeta/devlin.pdf How Euler discovered the zeta function]
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• Ramanujan tau
  The Ramanujan tau function $\tau \colon \mathbb{N} \rightarrow \mathbb{Z}$ is defined by the formulas ...q=e^{2\pi i z}$ with $\mathrm{Re}(z)>0$, $\eta$ denotes the [[Dedekind eta function]], and $\Delta$ denotes the [[discriminant modular form]].
  658 bytes (95 words) - 00:53, 23 December 2016
• Riemann-Siegel Z
  The Riemann-Siegel $Z$ function is defined by ...e [[Riemann-Siegel theta function]] and $\zeta$ denotes the [[Riemann zeta function]].
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• Riemann-Siegel theta function
  The Riemann-Siegel theta function is defined by where $\Gamma$ denotes the [[Gamma function]] and $\log$ denotes the [[logarithm]].
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• Barnes G
  The Barnes $G$ function is defined by the following [[Weierstrass factorization]]: where $\exp$ denotes the [[exponential function]] and $\gamma$ denotes the [[Euler-Mascheroni constant]].
  838 bytes (119 words) - 05:48, 6 June 2016
• Barnes G at positive integer
  where $G$ denotes the [[Barnes G]] function and $i!$ denotes the [[factorial]].
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• Glaisher–Kinkelin constant
  where $G$ is the [[Barnes G|Barnes $G$]] function.
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• Derivative of zeta at -1
  where $\zeta$ denotes the [[Riemann zeta function]], $A$ denotes the [[Glaisher–Kinkelin constant]], and $\log$ denotes the
  291 bytes (35 words) - 20:20, 20 June 2016
• Catalan's constant using Dirichlet beta
  ...e $K$ is [[Catalan's constant]] and $\beta$ denotes the [[Dirichlet beta]] function.
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• Catalan's constant using Legendre chi
  where $K$ is [[Catalan's constant]] and $\chi$ denotes the [[Legendre chi]] function.
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• Catalan's constant using Hurwitz zeta
  ...ant]], and $\zeta'$ denotes the partial derivative of the [[Hurwitz zeta]] function with respect to the first argument.
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• Legendre chi
  The Legendre chi function $\chi_{\nu}$ is defined by
  436 bytes (59 words) - 17:48, 25 June 2017
• Legendre chi in terms of polylogarithm
  where $\chi$ denotes the [[Legendre chi]] function and $\mathrm{Li}_{\nu}$ denotes the [[polylogarithm]].
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• Derivative of Legendre chi 2
  ...and $\mathrm{arctanh}$ denotes the [[Arctanh|inverse hyperbolic tangent]] function.
  326 bytes (41 words) - 01:31, 1 July 2017
• Book:Earl David Rainville/Special Functions
  ::8. The Gamma function ::16. The Beta function
  7 KB (822 words) - 05:19, 21 December 2017
• Bernoulli polynomial and Hurwitz zeta
  ...ernoulli B|Bernoulli polynomial]] and $\zeta$ denotes the [[Hurwitz zeta]] function.
  251 bytes (34 words) - 07:13, 16 June 2016
• Bessel at -n-1/2 in terms of Bessel polynomial
  where $J_{-n-\frac{1}{2}}$ denotes a [[Bessel J|Bessel function of the first kind]] and $y_n$ denotes a [[Bessel polynomial]].
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• Genocchi numbers
  The Genocchi numbers $G_n$ are given by the generating function
  555 bytes (74 words) - 18:57, 24 May 2016
• Q-exponential e sub q
  where $(q;q)_k$ denotes the [[q-shifted factorial]]. Note that this function is different than the [[q-exponential e sub 1/q |$q$-exponential $e_{\frac{
  541 bytes (80 words) - 03:30, 21 December 2016
• Constant functions are elliptic functions
  If $f$ is a [[constant function]], then $f$ is an [[elliptic function]].
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• Reciprocal zeta function
  ...eta function is $\dfrac{1}{\zeta(s)}$, where $\zeta$ is the [[Riemann zeta function]]. ...eta$ denotes the [[Riemann zeta function]], and $\mu$ denotes the [[Mobius function]].
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• Relationship between Lerch transcendent and Lerch zeta
  ...$\Phi$ denotes the [[Lerch transcendent]] and $L$ denotes the [[Lerch zeta function]].
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• Lefschetz zeta function
  Let $f$ be a function. The Lefschetz zeta function is
  244 bytes (44 words) - 18:52, 24 May 2016
• P-adic zeta function
  The $p$-adic zeta function is
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• Prime number theorem, pi and x/log(x)
  The function $\pi(x)$ obeys the formula where $\pi$ denotes the [[Prime counting|prime counting function]] and $\log$ denotes the [[logarithm]].
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• Prime number theorem, logarithmic integral
  where $\pi$ denotes the [[Prime counting|prime counting function]] and $\mathrm{li}$ denotes the [[logarithmic integral]].
  300 bytes (37 words) - 08:09, 8 June 2016
• Relationship between the exponential integral and upper incomplete gamma function
  ...integral E]] and $\Gamma$ denotes the [[incomplete gamma|incomplete gamma function]].
  268 bytes (36 words) - 00:16, 8 August 2016
• Relationship between cosine, imaginary number, logarithm, and the golden ratio
  where $\cos$ denotes the [[cosine]] function, $i$ denotes the [[imaginary number]], $\log$ denotes the [[logarithm]], an
  373 bytes (46 words) - 23:31, 27 June 2016
• Continuous
  ...ta$. If $f$ is continuous at all $x \in X$ we say that $f$ is a continuous function.
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• Coshc
  The $\mathrm{coshc}$ function is defined by
  270 bytes (34 words) - 19:08, 12 June 2017
• Thomae
  #REDIRECT [[Thomae function]]
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• Partial derivative of beta function
  ...$\Gamma$ denotes the [[gamma function]], and $\psi$ denotes the [[digamma function]].
  397 bytes (62 words) - 15:32, 23 June 2016
• Tanhc
  The $\mathrm{tanhc}$ function is defined by
  314 bytes (40 words) - 23:06, 11 June 2016
• Q-Fibonacci polynomials
  where $t(\xi) \neq 0$ is a function of a (real) variable $s$ and $q \neq 0$ is a [[real number]].
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• Arithmetic zeta function
  Let $X$ be a [[scheme]]. The arithmetic zeta function over $X$ is defined by
  166 bytes (26 words) - 18:51, 24 May 2016
• Artin-Mazur zeta function
  ...e the [[cardinality]] of the set $\mathrm{Fix}(f^n)$. The Artin-Mazur zeta function is
  441 bytes (74 words) - 18:52, 24 May 2016
• Arakawa-Kaneko zeta function
  where $\Gamma$ denotes the [[gamma function]] and $\mathrm{Li}_k$ denotes the [[polylogarithm]]. ...$ and $\xi_k$ has [[analytic continuation]] to $\mathbb{C}$ as an [[entire function]].
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• Barnes zeta function
  The Barnes zeta function is
  184 bytes (32 words) - 18:52, 24 May 2016
• Takagi function
  The Takagi function (also called the blancmange function) is defined by where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function.
  740 bytes (97 words) - 03:33, 6 July 2016
• Relationship between the exponential integral and incomplete gamma function
  ...[[Relationship between the exponential integral and upper incomplete gamma function]]
  95 bytes (11 words) - 03:42, 23 April 2015
• Greatest prime factor
  Define the greatest prime factor function $\mathrm{gpf}\colon \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ by
  440 bytes (56 words) - 06:34, 22 June 2016
• Meijer G-function
  ....youtube.com/watch?v=rZngIxZsxeA Mellin Barnes Integral for an Exponential Function]<br />
  111 bytes (16 words) - 02:03, 30 April 2015
• Relationship between Hurwitz zeta and gamma function
  ...a$ denotes the [[gamma function]] and $\zeta$ denotes the [[Hurwitz zeta]] function.
  310 bytes (42 words) - 07:13, 16 June 2016
• Laurent series of the Riemann zeta function
  where $\zeta$ denotes the [[Riemann zeta]] function and $\lambda_k$ denotes the [[Stieltjes constants]].
  328 bytes (42 words) - 05:02, 16 September 2016
• Euler phi
  The Euler phi function (not to be confused with the [[Euler totient]]) is defined for $q \in \math
  474 bytes (63 words) - 06:33, 22 June 2016
• Q-sin sub q
  The function $\sin_q$ is defined for $|z|<1$ by
  557 bytes (83 words) - 15:39, 11 July 2016
• Q-Euler formula for e sub q
  ...sub q|$q$-$\cos$]] function and $\sin_q$ is the [[q-sin sub q|$q$-$\sin$]] function.
  322 bytes (57 words) - 15:36, 11 July 2016
• Q-cos sub q
  The function $\cos_q$ is defined for $|z|<1$ by
  513 bytes (74 words) - 15:37, 11 July 2016
• Q-Cos
  The function $\mathrm{Cos}_q$ is defined by
  498 bytes (69 words) - 23:28, 26 June 2016
• Q-Euler formula for E sub q
  ...{Cos}$]] function and $\mathrm{Sin}_q$ is the [[q-Sin|$q$-$\mathrm{Sin}$]] function.
  358 bytes (59 words) - 23:10, 26 June 2016
• Q-Sin
  The function $\mathrm{Sin}_q$ is defined by
  479 bytes (70 words) - 00:49, 15 September 2016
• Derivative
  ...numbers]], $z_0 \in D$, and let $f \colon D \rightarrow \mathbb{C}$ be a [[function]]. We say that $f$ is (complex-) differentiable at $z_0$ if the [[limit]]
  408 bytes (63 words) - 05:10, 26 November 2016
• Complex conjugate of argument of error function
  where $\mathrm{erf}$ denotes the [[error function]] and $\overline{z}$ denotes the [[complex conjugate]].
  278 bytes (33 words) - 00:22, 8 August 2016
• Bohr-Mollerup theorem
  The [[gamma function]] is the unique function $f$ such that $f(1)=1$, $f(x+1)=xf(x)$ for $x>0$, and $f$ is [[logarithmica
  217 bytes (29 words) - 00:49, 1 October 2016
• Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta
  ..., $\log$ denotes the [[logarithm]], and $\zeta$ denotes the [[Riemann zeta function]].
  359 bytes (48 words) - 19:31, 15 June 2016
• Airy Ai
  ...e [[Airy differential equation]] linearly independent from the [[Airy Bi]] function. File:Airyaiplot.png|Graph of the Airy $\mathrm{Ai}$ function.
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• Integral representation of Airy Ai
  The [[Airy Ai]] function ...force $f(t)e^{zt} \Bigg |_{C}=0$. We will do this by first determining the function $f$. Plugging this back into the formula $(*)$ yields
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• Takagi function is nowhere differentiable
  The [[Takagi function]] is [[nowhere differentiable]].
  137 bytes (13 words) - 03:16, 6 July 2016
• Jackson q-Bessel (1)
  The Jackson $q$-Bessel function $J_{\nu}^{(1)}$ is defined by
  442 bytes (61 words) - 21:38, 17 June 2017
• Jackson q-Bessel (2)
  The Jackson $q$-Bessel function $J_{\nu}^{(2)}$ is defined by
  441 bytes (59 words) - 21:38, 17 June 2017
• Hahn-Exton q-Bessel
  The Hahn-Exton $q$-Bessel function, also called the Jackson $q$-Bessel function $J_{\nu}^{(3)}$, is defined by
  311 bytes (52 words) - 18:54, 24 May 2016
• Relationship between Airy Ai and modified Bessel K
  where $\mathrm{Ai}$ is the [[Airy Ai]] function and $K_{\nu}$ denotes the [[Modified Bessel K sub nu|modified Bessel $K$]].
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• Relationship between Airy Bi and modified Bessel I
  where $\mathrm{Bi}$ denotes the [[Airy Bi]] function and $I_{\nu}$ denotes the [[Modified Bessel I sub nu|modified Bessel $I$]].
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• Sum of values of sinc
  where $\mathrm{sinc}$ denotes the [[sinc|$\mathrm{sinc}$]] function and $\pi$ denotes [[pi]].
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• Tanc
  The $\mathrm{tanc}$ function is defined by
  252 bytes (34 words) - 19:53, 30 September 2016
• Sinhc
  The $\mathrm{sinhc}$ function is defined by
  334 bytes (45 words) - 23:06, 11 June 2016
• Spherical Hankel h (1)
  The spherical Hankel function $h_{\nu}^{(1)}$ is defined by ...t kind]] and $y_{\nu}$ is the [[Spherical Bessel y sub nu|spherical Bessel function of the second kind]].
  603 bytes (91 words) - 23:58, 22 December 2016
• Spherical Hankel h (2)
  The spherical Hankel function $h_{\nu}^{(2)}$ is defined by ...t kind]] and $y_{\nu}$ is the [[Spherical Bessel y sub nu|spherical Bessel function of the second kind]].
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• Bessel functions footer
  <div class="grid2"><center>[[Anger function|<randomimagebycategory categories="Angerglyph" width="45" />]]<br /> [[Anger function]]</center></div>
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• Relationship between Bessel I sub 1/2 and sinh
  ...e $I_{\frac{1}{2}}$ denotes the [[Modified Bessel I sub nu|modified Bessel function of the first kind]] and $\sinh$ denotes the [[Sinh|hyperbolic sine]].
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• Dilogarithm
  The dilogarithm function $\mathrm{Li}_2$ is defined for $|z| \leq 1$ by [http://maths.dur.ac.uk/~dma0hg/dilog.pdf The Dilogarithm function]<br />
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• Bernoulli numbers
  ...Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Gamma function written as infinite product|next=Euler-Mascheroni constant}}: §1.13 (1)
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• Q-Gamma at z+1
  where $\Gamma_q$ denotes the [[q-Gamma|$q$-gamma]] function and $[z]_q$ denotes the [[q-number|$q$-number]] of $z$. * {{PaperReference|The q-gamma function for q greater than 1|1980|Daniel S. Moak|prev=Q-shifted factorial|next=find
  396 bytes (56 words) - 00:17, 30 May 2017
• Relationship between sine and hypergeometric 0F1
  where $\sin$ denotes the [[sine]] function and ${}_0F_1$ denotes the [[hypergeometric pFq]].
  275 bytes (33 words) - 07:34, 8 June 2016
• Q-derivative of q-Sine
  ...[Q-Sin|$q$-Sine function]], and $\mathrm{Cos}_q$ is the [[Q-Cos|$q$-cosine function]].
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• Relationship between Bessel J and hypergeometric 0F1
  ...ssel J|Bessel function of the first kind]], $\Gamma$ denotes the [[gamma]] function and ${}_0F_1$ denotes the [[hypergeometric 0F1]].
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• Doubly periodic function
  ...\rightarrow \mathbb{C}$ is called doubly periodic if it has two [[periodic function|periods]] $\omega_1$ and $\omega_2$ such that $\dfrac{\omega_1}{\omega_2}$
  207 bytes (32 words) - 21:04, 6 June 2015
• Periodic function
  ...omega \in X$. This concept is related to the notion of a [[doubly periodic function]].
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• Functions named after Pafnuty Chebyshev
  *[[Chebyshev psi function|Chebyshev $\psi$]] *[[Chebyshev theta function|Chebyshev $\vartheta$]]
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• Weierstrass elliptic
  The Weierstrass elliptic function is [https://www.youtube.com/watch?v=WnaUZrPnZ30 Weierstrass Elliptic Function -- adding terms]<br />
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• Weierstrass nowhere differentiable function
  The Weierstrass function is [[Weierstrass function is continuous]]<br />
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• Weierstrass sigma
  Let $\Lambda \subset \mathbb{C}$ be a [[lattice]]. The Weierstrass $\sigma$ function is defined by
  279 bytes (41 words) - 18:38, 24 May 2016
• Weierstrass zeta
  Let $\Lambda \subset \mathbb{C}$ be a [[lattice]]. The Weierstrass zeta function is defined by
  272 bytes (37 words) - 18:51, 24 May 2016
• Kelvin ber
  The $\mathrm{ber}_{\nu}$ function is defined as ...part]] of a [[complex number]] and $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]].
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• Kelvin bei
  The $\mathrm{bei}_{\nu}$ function is defined as ...part]] of a [[complex number]] and $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]].
  902 bytes (126 words) - 05:42, 4 March 2018
• Kelvin ker
  The $\mathrm{ker}_{\nu}$ function is defined as ...ber]] and $K_{\nu}$ denotes the [[Modified Bessel K sub nu|modified Bessel function $K_{\nu}$]].
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• Kelvin kei
  The $\mathrm{kei}_{\nu}$ function is defined as
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• Weierstrass function
  #REDIRECT [[Weierstrass nowhere differentiable function]]
  57 bytes (5 words) - 02:30, 22 May 2015
• Chebyshev theta function
  The Chebyshev $\vartheta$ function is
  531 bytes (72 words) - 02:18, 28 November 2016
• Chebyshev psi function
  The Chebyshev $\psi$ function is where $\Lambda$ denotes the [[Mangoldt function]].
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• Doubly periodic
  #REDIRECT [[Doubly periodic function]]
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• Fundamental pair of periods
  ...$(\omega_1,\omega_2)$ is a fundamental pair of periods if every [[periodic function|period]] of $f$ is of the form $m\omega_1+n\omega_2$ for all [[integers]] $
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• Period parallelogram
  Let $(\omega_1,\omega_2)$ form a [[fundamental pair of periods]] for some function. The points $m\omega_1+n\omega_2$ form the vertices of the period parallelg
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• Lattice generated by doubly periodic periods
  ...al (i.e. they are the [[periodic function|periods]] of a [[doubly periodic function]]). The set $\Omega(\omega_1,\omega_2)=\left\{ m \omega_1 +n\omega_2 \colon
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• Elliptic function
  A function $f$ is called elliptic if it is a [[doubly periodic function]] and it is [[meromorphic]]. <strong>Theorem:</strong> A nonconstant [[elliptic function]] has a [[fundamental pair of periods]].
  2 KB (259 words) - 00:00, 23 December 2016
• Elliptic functions
  #REDIRECT [[Elliptic function]]
  31 bytes (3 words) - 21:20, 6 June 2015
• Relationship between spherical Bessel j and sine
  ...erical Bessel function of the first kind]] and $\sin$ denotes the [[sine]] function.
  391 bytes (59 words) - 07:34, 8 June 2016
• Relationship between spherical Bessel y and cosine
  ...cal Bessel function of the second kind]] and $\cos$ denotes the [[cosine]] function.
  387 bytes (59 words) - 01:15, 18 July 2016
• Book:Milton Abramowitz/Handbook of mathematical functions
  ::4.1. Logarithmic Function ::4.2. Exponential Function
  22 KB (2,666 words) - 05:08, 21 December 2017
• Book:Johan Thim/Continuous Nowhere Differentiable Functions
  ::3.1 Bolzano function (≈1830) ::3.2 Cellérier function (≈1860)
  1 KB (115 words) - 18:04, 25 June 2017
• Hahn polynomial
  where ${}_3F_2$ denotes the [[generalized hypergeometric function]].
  784 bytes (115 words) - 18:42, 24 May 2016
• Relationship between Bessel I sub -1/2 and cosh
  ...$I_{-\frac{1}{2}}$ denotes [[Modified Bessel I sub nu|the modified Bessel function of the first kind]] and $\cosh$ denotes the [[Cosh|hyperbolic cosine]].
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• Relationship between Bessel I and Bessel J
  ...sel I sub nu|modified Bessel $I$]] and $J_n$ denotes the [[Bessel J|Bessel function of the first kind]].
  280 bytes (45 words) - 20:27, 27 June 2016
• Anger function
  Let $\nu \in \mathbb{C}$. The Anger function $\mathbf{J}_{\nu}$ is defined by [[Relationship between Anger function and Bessel J sub nu]]<br />
  1 KB (193 words) - 04:05, 6 June 2016
• Relationship between Anger function and Bessel J
  ...{J}_n$ denotes an [[Anger function]] and $J_n$ denotes a [[Bessel J|Bessel function of the first kind]].
  278 bytes (42 words) - 20:27, 27 June 2016
• Weber function
  The Weber function is defined by [[Relationship between Weber function and Anger function]]<br />
  781 bytes (98 words) - 04:13, 6 June 2016
• Relationship between Weber function and Anger function
  ...u}$ denotes a [[Weber function]] and $\mathbf{J}_{\nu}$ denotes an [[Anger function]]. ...unction and Weber function|next=Relation between Weber function and Struve function}}: 12.3.5
  494 bytes (70 words) - 04:16, 6 June 2016
• Relationship between Anger function and Weber function
  ...u}$ denotes an [[Anger function]] and $\mathbf{E}_{\nu}$ denotes a [[Weber function]]. ...gun|prev=Weber function|next=Relationship between Weber function and Anger function}}: 12.3.4
  456 bytes (65 words) - 04:15, 6 June 2016
• Struve functions
  #REDIRECT [[Struve function]]
  29 bytes (3 words) - 18:14, 28 June 2015
• Relationship between Weber function 0 and Struve function 0
  ...f{E}_0$ denotes a [[Weber function]] and $\mathbf{H}_0$ denotes a [[Struve function]].
  249 bytes (32 words) - 13:18, 25 June 2016
• Relationship between Weber function 1 and Struve function 1
  ...f{E}_1$ denotes a [[Weber function]] and $\mathbf{H}_1$ denotes a [[Struve function]].
  263 bytes (34 words) - 13:19, 25 June 2016
• Relationship between Weber function and Struve function
  #REDIRECT [[Relationship between Weber function 0 and Struve function 0]]
  73 bytes (8 words) - 18:20, 28 June 2015
• Relationship between Weber function 2 and Struve function 2
  ...f{E}_2$ denotes a [[Weber function]] and $\mathbf{H}_2$ denotes a [[Struve function]].
  223 bytes (31 words) - 04:11, 6 June 2016
• Modified Struve function
  where $\mathbf{H}_{\nu}$ denotes a [[Struve function]].
  341 bytes (47 words) - 18:32, 24 May 2016
• Jacobi theta 1
  Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_1$ function is defined by where $\sin$ denotes the [[sine]] function.
  1 KB (176 words) - 20:23, 25 June 2017
• Relationship between Struve function and hypergeometric pFq
  ...a [[Struve function]], $\pi$ denotes [[pi]], $\Gamma$ denotes the [[gamma function]], and ${}_2F_1$ denotes the [[hypergeometric pFq]].
  443 bytes (55 words) - 13:18, 25 June 2016
• Lower incomplete gamma
  The lower incomplete gamma function is defined for $\mathrm{Re}(a)>0$ by A single-valued analytic function of $a$ and $x$ can be defined as
  430 bytes (64 words) - 18:32, 24 May 2016
• Fibonacci numbers
  [[Fibonacci zeta function]]<br /> ...ence|The Fibonacci Zeta Function|1976|Maruti Ram Murty|next=Fibonacci zeta function}}
  2 KB (362 words) - 13:12, 1 August 2018
• Inverse error function
  The inverse error function is the [[inverse function]] of the [[error function]]. We denote it by writing $\mathrm{erf}^{-1}$. [[Derivative of inverse error function]]<br />
  496 bytes (68 words) - 04:56, 16 September 2016
• Integral of log of inverse erf from 0 to 1
  where $\mathrm{erf}^{-1}$ denotes the [[inverse error function]], $\log$ denotes the [[logarithm]], and $\gamma$ denotes the [[Euler-Masch
  363 bytes (43 words) - 01:54, 1 July 2017
• Integral of inverse erf from 0 to 1
  where $\mathrm{erf}^{-1}$ denotes the [[inverse error function]] and $\pi$ denotes [[pi]].
  284 bytes (36 words) - 04:46, 16 September 2016
• Antiderivative of inverse error function
  where $\mathrm{erf}^{-1}$ denotes the [[inverse error function]], $\exp$ denotes the [[exponential]], and $\pi$ denotes [[pi]].
  369 bytes (50 words) - 03:48, 3 October 2016
• Dickman–de Bruijn function
  The Dickman–de Bruijn function $\rho(u)$ solves the [[initial value problem]]
  305 bytes (46 words) - 18:31, 24 May 2016
• Buchstab function
  The Buchstab function is a [[continuous]] function $\omega \colon [1,\infty) \rightarrow (0,\infty)$ defined by the [[initial
  453 bytes (67 words) - 18:31, 24 May 2016
• Incomplete beta function
  The incomplete beta function is defined by
  319 bytes (42 words) - 01:56, 23 December 2016
• Relationship between incomplete beta and hypergeometric 2F1
  where $B_x$ denotes the [[incomplete beta function]] and ${}_2F_1$ denotes the [[hypergeometric pFq]].
  264 bytes (38 words) - 20:26, 5 July 2016
• Fibonacci zeta function
  The Fibonacci zeta function is defined by * {{PaperReference|The Fibonacci Zeta Function|1976|Maruti Ram Murty|prev=Fibonacci numbers|next=Binet's formula}}
  704 bytes (99 words) - 00:25, 24 May 2017
• Functions named after Peter Gustav Lejeune Dirichlet
  *[[Dirichlet L-function]]
  64 bytes (6 words) - 19:33, 25 July 2015
• Riemann Siegel theta function
  The Riemann Siegel $\vartheta$ function is defined for $t \in \mathbb{R}$ by where $\log$ denotes the [[logarithm]] and $\Gamma$ denotes the [[gamma function]].
  548 bytes (76 words) - 18:33, 24 May 2016
• Error function is odd
  ...thrm{erf}$ denotes the [[error function]] (i.e. $\mathrm{erf}$ is an [[odd function]]).
  800 bytes (116 words) - 03:41, 28 March 2017
• Erfc
  The complementary error function $\mathrm{erfc}$ is defined by the formula where $\mathrm{erf}$ denotes the [[error function]].
  542 bytes (68 words) - 21:56, 6 July 2016
• Identity
  Let $X$ be any set. The function $\mathrm{id} \colon X \rightarrow X$ defined by $\mathrm{id}(x)=x$ is calle
  390 bytes (58 words) - 23:02, 25 December 2016
• Reciprocal Fibonacci constant
  [[Fibonacci zeta function]]<br />
  582 bytes (75 words) - 03:40, 25 June 2017
• Derivative of the exponential function
  where $e^z$ denotes the [[exponential function]].
  806 bytes (115 words) - 00:10, 23 December 2016
• Faddeeva function
  The Faddeeva function (also called the Kramp function) is defined by ...e [[error function]] and $\mathrm{erfc}$ denotes the [[complementary error function]].
  478 bytes (66 words) - 18:31, 24 May 2016
• Taylor series of the exponential function
  where $e^z$ is the [[exponential function]].
  255 bytes (35 words) - 04:03, 3 October 2016
• Bessel-Clifford
  The Bessel-Clifford function $\mathcal{C}_n$ is defined by where $\dfrac{1}{\Gamma}$ denotes the [[reciprocal gamma]] function.
  559 bytes (71 words) - 16:03, 29 April 2017
• Floor
  The floor function $\mathrm{floor} \colon \mathbb{R} \rightarrow \mathbb{Z}$ (sometimes writte
  457 bytes (70 words) - 00:41, 23 December 2016
• Ceiling
  The ceiling function $\mathrm{ceil} \colon \mathbb{R} \rightarrow \mathbb{Z}$ (sometimes written
  887 bytes (129 words) - 12:44, 14 July 2023
• Falling factorial
  where $\Gamma$ denotes the [[gamma function]].
  380 bytes (54 words) - 12:27, 11 August 2016
• Book:Thomas Ernst/A Comprehensive Treatment of q-Calculus
  ::2.16 Weierstraß, formal power series and the $\Gamma$ function :::3.7.1 The $\Gamma$ function
  11 KB (1,149 words) - 15:21, 18 December 2023
• Value of Ai(0)
  ...m{Ai}$ denotes the [[Airy Ai]] function and $\Gamma$ denotes the [[gamma]] function.
  306 bytes (34 words) - 23:06, 9 June 2016
• Value of Ai'(0)
  ...m{Ai}$ denotes the [[Airy Ai]] function and $\Gamma$ denotes the [[gamma]] function.
  287 bytes (35 words) - 23:06, 9 June 2016
• Porter's constant
  ...he [[Euler-Mascheroni constant]], and $\zeta$ denotes the [[Riemann zeta]] function.
  338 bytes (43 words) - 19:00, 24 May 2016
• Norton's constant
  ...[[logarithm]], $\pi$ denotes [[pi]], $\zeta$ denotes the [[Riemann zeta]] function, and $C$ denotes [[Porter's constant]].
  316 bytes (40 words) - 19:00, 24 May 2016
• Sierpiński constant
  ...otes the [[Euler-Mascheroni constant]], and $\Gamma$ denotes the [[gamma]] function.
  327 bytes (42 words) - 19:00, 24 May 2016
• Relationship between sine, imaginary number, logarithm, and the golden ratio
  where $\sin$ denotes the [[sine]] function, $i$ denotes the [[imaginary number]], $\log$ denotes the [[logarithm]], an
  467 bytes (60 words) - 23:31, 27 June 2016
• Derivative of arctan
  where $\mathrm{arctan}$ denotes the [[arctan|inverse tangent]] function.
  720 bytes (99 words) - 13:05, 22 September 2016
• Value of Anger at 0
  where $\textbf{J}_{\nu}$ denotes an [[Anger function]], $\sin$ denotes the [[sine]], and $\pi$ denotes [[pi]].
  256 bytes (34 words) - 05:49, 6 June 2016
• Anger three-term recurrence
  where $\textbf{J}_{\nu}$ denote the [[Anger function]], $\pi$ denotes [[pi]], and $\sin$ denotes [[sine]].
  310 bytes (46 words) - 05:50, 6 June 2016
• Anger derivative recurrence
  where $\mathbf{J}_{\nu}$ denotes the [[Anger function]].
  218 bytes (31 words) - 05:50, 6 June 2016
• Loggamma
  The loggamma function $\log \Gamma$ is defined by the [[principal branch]] of where $\log$ denotes the [[logarithm]] and $\Gamma$ denotes the [[gamma]] function.
  412 bytes (55 words) - 18:32, 24 May 2016
• Contour integral representation of reciprocal gamma
  ...] function, $\pi$ denotes [[pi]], and $e^{-t}$ denotes the [[exponential]] function.
  480 bytes (63 words) - 07:17, 16 June 2016
• Barnes G at z+1 in terms of Barnes G and gamma
  ...e $G$ denotes the [[Barnes G]] function and $\Gamma$ denotes the [[gamma]] function.
  226 bytes (30 words) - 12:52, 17 September 2016
• Ramanujan tau is multiplicative
  [[Ramanujan tau]] is a [[multiplicative function]].
  134 bytes (13 words) - 00:53, 23 December 2016
• Functional equation for Riemann xi
  * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann xi|next=Riemann-Land
  312 bytes (36 words) - 15:27, 18 March 2017
• Generating function for partition function
  where $p(k)$ denotes the [[partition]] function. ...witz|author2=Irene A. Stegun|prev=Partition|next=Closed form for partition function with sinh}}: $24.2.1 \mathrm{I}.B.$
  541 bytes (72 words) - 20:34, 26 June 2016
• Bolzano function is continuous
  The [[Bolzano function]] is [[continuous]].
  126 bytes (12 words) - 17:10, 23 June 2016
• Cellérier function is continuous
  The [[Cellérier function]] is [[continuous]].
  129 bytes (13 words) - 17:12, 23 June 2016
• Darboux function is continuous
  The [[Darboux function]] is [[continuous]] on $\mathbb{R}$.
  142 bytes (15 words) - 17:14, 23 June 2016
• Knopp function is continuous
  The [[Knopp function]] $K_{a,b}$ is [[continuous]] on $\mathbb{R}$ for $a \in (0,1)$ and $ab>1$.
  179 bytes (23 words) - 03:31, 27 October 2016
• McCarthy function is continuous
  The [[McCarthy function]] is [[continuous]] on $\mathbb{R}$.
  143 bytes (15 words) - 13:40, 17 November 2016
• The Petr function is nowhere differentiable
  The [[Petr function]] is [[nowhere differentiable]] on $(0,1)$.
  146 bytes (14 words) - 20:34, 25 June 2017
• Riemann function is continuous
  The [[Riemann function]] is [[continuous]].
  126 bytes (12 words) - 03:27, 6 July 2016
• Schwarz function is continuous
  Let $M>0$. The [[Schwarz function]] is [[continuous]] on $(0,M)$.
  148 bytes (16 words) - 17:46, 20 September 2016
• Van der Waerden function is continuous
  The [[van der Waerden function]] is [[continuous]].
  134 bytes (14 words) - 03:17, 6 July 2016
• Derivative of inverse error function
  where $\mathrm{erf}^{-1}$ denotes the [[inverse error function]], $\pi$ denotes [[pi]], and $\exp$ denotes the [[exponential]].
  364 bytes (48 words) - 04:38, 16 September 2016
• Relationship between logarithm and Mangoldt
  ...s that $d$ is a [[divisor]] of $n$, and $\Lambda$ denotes the [[Mangoldt]] function.
  315 bytes (42 words) - 16:31, 16 June 2016
• Airy zeta function at 2
  ...m{Ai}}$ denotes the [[Airy zeta function]], $\Gamma$ denotes the [[gamma]] function, and $\pi$ denotes [[pi]].
  323 bytes (37 words) - 02:20, 2 November 2016
• Fibonacci zeta at 1 is irrational
  ...ant]]) is an [[irrational number]], where $F$ denotes the [[Fibonacci zeta function]].
  234 bytes (28 words) - 17:37, 19 August 2016
• Q-derivative of q-Cosine
  ...Cos|$q$-Cosine function]], and $\mathrm{Sin}$ denotes the [[Q-Sin|$q$-Sine function]].
  450 bytes (59 words) - 23:29, 26 June 2016
• Q-Gamma at 1
  where $\Gamma_q$ denotes the [[Q-Gamma|$q$-gamma]] function.
  191 bytes (22 words) - 17:53, 17 October 2016
• Relation between polygamma and Hurwitz zeta
  ...^{(m)}$ denotes the [[polygamma]] and $\zeta$ denotes the [[Hurwitz zeta]] function.
  253 bytes (33 words) - 06:34, 11 June 2016
• Hurwitz zeta absolute convergence
  The [[Hurwitz zeta]] function $\zeta(s,a)$ is [[absolutely convergent]] for all $s$ with $\mathrm{Re}(s)>
  223 bytes (30 words) - 07:12, 16 June 2016
• Trigamma
  The trigamma function $\psi^{(1)}$ is defined by where $\log \Gamma$ denotes the [[loggamma]] function.
  384 bytes (49 words) - 03:11, 21 December 2016
• Euler-Jackson q-difference operator
  ...} \setminus \{1\}$. The Euler-Jackson $q$-difference operator applied to a function $f$ is given by the formula
  174 bytes (32 words) - 20:04, 3 June 2016
• Book:Aleksandar Ivić/The Riemann Zeta-Function
  {{Book|The Riemann Zeta-Function: Theory and Applications|1985|Dover Publications, Inc|0-486-42813-3|Aleksan :: 12.4. The Generalised von Mangoldt Function and the Möbius Function
  5 KB (554 words) - 16:39, 21 June 2016
• Book:Andrew Gray/A Treatise on Bessel Functions
  ...ter IX: Relations between Bessel Functions and Legendre Functions. Green's Function : Appendix I: Formulae for the Gamma Function and the Hypergeometric Function
  2 KB (220 words) - 19:38, 21 June 2017
• Book:Andrew Gray/A Treatise on Bessel Functions/Second Edition
  * [[Definition:Bessel Function|Bessel Functions]] ...ter IX: Relations between Bessel Functions and Legendre Functions. Green's Function
  2 KB (212 words) - 16:39, 21 June 2016
• Book:F.E. Relton/Applied Bessel Functions
  * [[Definition:Bessel Function|Bessel Functions]] : I. The Error Function; Beta and Gamma Functions
  908 bytes (110 words) - 16:40, 21 June 2016
• Book:Ian N. Sneddon/Special Functions of Mathematical Physics and Chemistry
  ::5. The gamma function and related functions :::[[Error function|$(5.11)$]]
  3 KB (300 words) - 03:07, 25 June 2017
• Book:Larry C. Andrews/Special Functions of Mathematics for Engineers
  * [[Definition:Special Function|Special Functions]] : Chapter 2. The Gamma Function and Related Functions
  7 KB (681 words) - 21:57, 3 June 2016
• Chain rule for derivatives
  ...nd $g$ be differentiable functions for which we may define the [[composite function]] $f \circ g$. Then the following formula holds:
  493 bytes (71 words) - 17:22, 27 June 2016
• Polar coordinates
  ...efinition of [[tangent]] and application of the [[arctan|inverse tangent]] function.
  620 bytes (88 words) - 17:23, 27 June 2016
• Logarithm (multivalued)
  The (multivalued) logarithm function $\mathrm{Log} \colon \mathbb{C} \rightarrow \mathscr{P}\left( \mathbb{C} \r
  1 KB (139 words) - 19:33, 25 June 2017
• Series for log(z+a) for positive a and Re(z) greater than -a
  ...greater than 1|next=Limit at infinity of logarithm times reciprocal power function is zero}}: $4.1.29$
  587 bytes (82 words) - 17:28, 27 June 2016
• Book:Arthur Erdélyi/Higher Transcendental Functions Volume I
  :1.1. Definition of the gamma function ...t of a factorial, exponential, and a rising factorial|$(2)$]] (and [[Gamma function written as infinite product|$(2)$]]
  10 KB (1,331 words) - 06:07, 4 March 2018
• Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0
  where $\Gamma$ denotes the [[gamma]] function and $\log$ denotes the [[logarithm]].
  559 bytes (71 words) - 20:56, 3 March 2018
• Gamma function written as a limit of a factorial, exponential, and a rising factorial
  where $\Gamma$ denotes the [[gamma]] function. ...(z) as integral of a power of log(1/t) for Re(z) greater than 0|next=Gamma function written as infinite product}}: §1.1 (2)
  603 bytes (87 words) - 20:56, 3 March 2018
• Gamma function written as infinite product
  where $\Gamma$ denotes the [[gamma]] function. ...Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Gamma function written as a limit of a factorial, exponential, and a rising factorial|next
  791 bytes (102 words) - 20:56, 3 March 2018
• Reciprocal gamma written as an infinite product
  where $\dfrac{1}{\Gamma}$ is the [[reciprocal gamma]] function, and $\gamma$ is the [[Euler-Mascheroni constant]]. ...on|author2=George Neville Watson|prev=Euler-Mascheroni constant|next=Gamma function written as infinite product}}: $\S 12 \cdot 11$
  1 KB (138 words) - 20:56, 3 March 2018
• Erf of conjugate is conjugate of erf
  where $\mathrm{erf}$ denotes the [[error function]] and $\overline{z}$ denotes the [[complex conjugate]]. ...atical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Error function is odd|next=findme}}: 7.1.10
  396 bytes (49 words) - 05:03, 5 June 2016
• Book:Arthur Erdélyi/Higher Transcendental Functions Volume III
  ::::14.6.2. The modular function $\mathcal{J}(z)$ :::17.1. Elementary functions of number theory generated by Riemann's zeta function
  5 KB (544 words) - 05:44, 4 March 2018
• Book:Gabor Szegő/Orthogonal Polynomials/Fourth Edition
  ::1.4. Orthogonality; weight function; vectors in function spaces ::1.7. The Gamma function
  3 KB (352 words) - 02:58, 15 July 2018
• Book:G.H. Hardy/The General Theory Of Dirichlet's Series
  ...FOR THE SUM OF THE COEFFICIENTS OF A DIRICHLET'S SERIES: THE ORDER OF THE FUNCTION REPRESENTED BY THE SERIES
  819 bytes (110 words) - 23:27, 17 March 2017
• Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt
  ...zeros of the [[Hermite polynomials]], and $H_k^{(n)}$ denote the [[weight function]] for the Hermite polynomials.
  910 bytes (135 words) - 21:14, 6 June 2016
• Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4
  where $\mathrm{erf}$ denotes the [[error function]].
  531 bytes (70 words) - 02:18, 6 June 2016
• Anger of integer order is Bessel J
  ...{J}_n$ denotes an [[Anger function]] and $J_n$ denotes a [[Bessel J|Bessel function of the first kind]]. ...4|Milton Abramowitz|author2=Irene A. Stegun|prev=Anger function|next=Weber function}}: 12.3.2
  389 bytes (57 words) - 04:07, 6 June 2016
• Versine
  The versine function $\mathrm{versin} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by th where $ \cos$ denotes the [[cosine]] function.
  734 bytes (92 words) - 02:33, 5 January 2017
• Coversine
  The coversine function $\mathrm{coversin} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by where $\sin$ denotes the [[sine]] function.
  680 bytes (84 words) - 04:27, 3 October 2016
• Haversine
  The haversine function $\mathrm{haversin} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by
  401 bytes (46 words) - 03:45, 28 March 2017
• Exsecant
  The exsecant function $\mathrm{exsec}$ is defined by
  550 bytes (68 words) - 04:11, 28 March 2017
• Relationship between Anger function and Bessel J sub nu
  #REDIRECT [[Relationship between Anger function and Bessel J]]
  62 bytes (8 words) - 05:51, 6 June 2016
• Euler's formula
  where $e^{iz}$ denotes the [[exponential function]], $\cos$ denotes the [[cosine]], $i$ denotes the [[imaginary number]], and
  631 bytes (99 words) - 00:20, 23 December 2016
• E^x is less than 1/(1-x) for nonzero real x less than 1
  where $e^x$ denotes the [[exponential function]].
  465 bytes (69 words) - 00:32, 23 December 2016
• Gauss' formula for gamma function
  where $\Gamma$ denotes the [[gamma]] function and $n!$ denotes the [[factorial]].
  462 bytes (62 words) - 07:03, 8 June 2016
• Book:Edward Charles Titchmarch/The Zeta-Function of Riemann
  #REDIRECT [[Book:Edward Charles Titchmarsh/The Zeta-Function of Riemann]]
  73 bytes (9 words) - 19:39, 9 June 2016
• Number theory functions footer
  <div class="grid"><center>[[File:Chebthetathumb.png|45px|link=Chebyshev theta function]]<br /> [[Chebyshev theta function|Chebyshev $\vartheta$]]</center></div>
  2 KB (233 words) - 11:15, 11 January 2017
• Paper:Harry Bateman/Some Properties of a certain Set of Polynomials
  :1. The generating function. :2. Transformation of the generating function.
  654 bytes (90 words) - 01:59, 22 June 2016
• Hankel H (1) in terms of csc and Bessel J
  ...otes the [[imaginary number]], and $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]].
  646 bytes (95 words) - 04:09, 11 June 2016
• Hankel H (2) in terms of csc and Bessel J
  ...$\csc$ denotes the [[cosecant]], $J_{-\nu}$ denotes the [[Bessel J|Bessel function of the first kind]], and $e^{\nu \pi i}$ denotes the [[exponential]].
  664 bytes (100 words) - 04:50, 11 June 2016
• Relationship between Bessel J sub n and Bessel J sub -n
  where $J_{-n}$ denotes the [[Bessel J|Bessel function of the first kind]].
  455 bytes (70 words) - 04:52, 11 June 2016
• Relationship between Bessel Y sub n and Bessel Y sub -n
  where $Y_{-n}$ denotes the [[Bessel Y|Bessel function of the second kind]].
  421 bytes (62 words) - 04:54, 11 June 2016
• Value of polygamma at 1
  ..., $m!$ denotes the [[factorial]], and $\zeta$ denotes the [[Riemann zeta]] function.
  482 bytes (62 words) - 22:45, 17 March 2017
• Polygamma multiplication formula
  where $\psi^{(m)}$ denotes the [[polygamma]] function and $\log$ denotes the [[logarithm]].
  580 bytes (77 words) - 22:51, 17 March 2017
• Book:Edmund Taylor Whittaker/A course of modern analysis/Third edition
  ::Chapter XII The Gamma Function ::Chapter XIII The Zeta Function of Riemann
  2 KB (229 words) - 16:39, 21 June 2016
• Gamma recurrence relation
  where $\Gamma$ denotes the [[gamma function]].
  355 bytes (46 words) - 02:28, 12 June 2016
• The Euler-Mascheroni constant exists
  ...e\sum_{k=1}^{\infty} \dfrac{1}{k^2}$ converges (it is the [[Riemann zeta]] function evaluated at $z=2$), we can conclude that $\displaystyle\sum_{k=1}^{\infty}
  2 KB (279 words) - 20:31, 20 June 2016
• Paper:Carl-Erik Fröberg/On the prime zeta function
  {{Book|On the prime zeta function|1968|BIT Numerical Mathematics||Carl-Erik Fröberg}}
  209 bytes (26 words) - 19:27, 15 June 2016
• Paper:Matilde Lalín/Secant zeta functions
  ::[[Absolute convergence of secant zeta function|Theorem 1]]<br />
  365 bytes (44 words) - 20:25, 20 June 2016
• Absolute convergence of secant zeta function
  The series defining the [[secant zeta function]] $\psi_s(z)$ converges absolutely in the following cases:
  430 bytes (56 words) - 06:09, 16 June 2016
• Derivative of Bessel J with respect to its order
  ...t kind]], $\log$ denotes the [[logarithm]], $\psi$ denotes the [[digamma]] function, and $k!$ denotes the [[factorial]].
  685 bytes (98 words) - 22:45, 19 June 2016
• Derivative of Bessel Y with respect to its order
  ...$\cot$ denotes the [[cotangent]], $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]], $\pi$ denotes [[pi]], and $\csc$ denotes the [[cosecan
  774 bytes (118 words) - 00:50, 28 August 2016
• Identity written as a sum of Möbius functions
  where $\mu$ denotes the [[Möbius function]]. ...nn zeta as a sum of Möbius function for Re(z) greater than 1|next=Möbius function is multiplicative}}: $24.3.1 \mathrm{I}.B.$
  487 bytes (69 words) - 01:33, 22 June 2016
• Möbius function is multiplicative
  The [[Möbius function]] is a [[multiplicative function]].
  332 bytes (41 words) - 01:32, 22 June 2016
• Sum of totient equals zeta(z-1)/zeta(z) for Re(z) greater than 2
  ...tes the [[Euler totient|totient]] and $\zeta$ denotes the [[Riemann zeta]] function.
  518 bytes (68 words) - 04:49, 22 June 2016
• Euler totient is multiplicative
  The [[Euler totient|totient]] is a [[multiplicative]] function.
  323 bytes (36 words) - 04:50, 22 June 2016
• Beta as improper integral
  where $B$ denotes the [[beta]] function.
  517 bytes (72 words) - 20:57, 3 March 2018
• Beta is symmetric
  where $B$ denotes the [[beta]] function.
  765 bytes (117 words) - 20:57, 3 March 2018
• Beta as product of gamma functions
  where $B$ denotes the [[beta]] function and $\Gamma$ denotes the [[gamma]] function.
  475 bytes (67 words) - 20:58, 3 March 2018
• Digamma at 1
  where $\psi$ denotes the [[digamma]] function and $\gamma$ denotes the [[Euler-Mascheroni constant]].
  578 bytes (67 words) - 23:22, 3 March 2018
• Digamma at z+n
  where $\psi$ denotes the [[digamma]] function.
  472 bytes (59 words) - 23:22, 3 March 2018
• Cellérier function is nowhere differentiable
  The [[Cellérier function]] is [[nowhere differentiable]].
  141 bytes (14 words) - 17:12, 23 June 2016
• Darboux function is nowhere differentiable
  The [[Darboux function]] is [[nowhere differentiable]] on $\mathbb{R}$.
  154 bytes (16 words) - 17:14, 23 June 2016
• Faber F1 is continuous
  The [[Faber F1]] function is [[continuous]].
  127 bytes (13 words) - 17:16, 23 June 2016
• Euler product for Riemann zeta
  * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann zeta|next=Series for
  727 bytes (88 words) - 05:00, 16 September 2016
• Faber F1 is nowhere differentiable
  The [[Faber F1]] function is [[nowhere differentiable]].
  139 bytes (14 words) - 17:18, 23 June 2016
• Faber F2 is nowhere differentiable
  The [[Faber F2]] function is [[nowhere differentiable]].
  139 bytes (14 words) - 17:17, 23 June 2016
• Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1
  ...e $\zeta$ denotes the [[Riemann zeta function]] and $\mu$ is the [[Möbius function]]. ...ical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Möbius function|next=Identity written as a sum of Möbius functions}}: $24.3.1. \mathrm{I}.
  506 bytes (69 words) - 01:34, 22 June 2016
• Book:Victor Kac/Quantum Calculus
  :12 Classical Partition Function and Euler's Product Formula
  2 KB (195 words) - 05:16, 21 January 2017
• Error functions footer
  <div class="grid"><center>[[Error function|<randomimagebycategory categories="Errorglyph" width="45" />]]<br /> [[Error function]]</center></div>
  624 bytes (80 words) - 21:57, 6 July 2016
• Erfi
  The imaginary error function $\mathrm{erfi}$ is defined by where $\mathrm{erf}$ denotes the [[error function]].
  413 bytes (54 words) - 23:10, 23 October 2016
• Jinc
  The $\mathrm{jinc}$ function is defined by where $J_1$ denotes the [[Bessel J|Bessel function of the first kind]].
  479 bytes (66 words) - 04:03, 22 September 2016
• Inverse Gudermannian
  ...ian $\mathrm{gd}^{-1}$ is the [[inverse function]] of the [[Gudermannian]] function. It may be defined by the following formula for $x \in \mathbb{R}$:
  966 bytes (122 words) - 23:10, 11 June 2016
• Q-exponential e sub 1/q
  The $q$-exponential $e_{\frac{1}{q}}$ is an [[entire]] function and is defined for $0 < |q| < 1$ by
  332 bytes (48 words) - 04:05, 21 December 2016
• Q-exponential E sub 1/q
  The $E_{\frac{1}{q}}$ function is defined by the formula
  728 bytes (100 words) - 04:30, 26 December 2016
• Derivative of arcsin
  where $\arcsin$ denotes the [[arcsin|inverse sine]] function.
  845 bytes (121 words) - 23:33, 8 December 2016
• Derivative of arcsec
  where $\mathrm{arcsec}$ is the [[arcsec|inverse secant]] function.
  955 bytes (143 words) - 23:51, 8 December 2016
• Derivative of arccot
  where $\mathrm{arccot}$ denotes the [[arccot|inverse cotangent]] function.
  592 bytes (88 words) - 07:31, 8 June 2016
• Pythagorean identity for sin and cos
  where $\sin$ denotes the [[sine]] function and $\cos$ denotes the [[cosine]] function.
  678 bytes (109 words) - 18:51, 15 December 2016
• Gompertz constant
  where $e$ denotes the base of the [[exponential function]] and $\mathrm{Ei}$ denotes the [[exponential integral Ei]].
  323 bytes (48 words) - 18:59, 24 May 2016
• Series for erf with exponential factored out
  where $\mathrm{erf}$ denotes the [[error function]] and $\pi$ denotes [[pi]].
  330 bytes (43 words) - 03:54, 3 October 2016
• Taylor series for error function
  where $\mathrm{erf}$ denotes the [[error function]] and $\pi$ denotes [[pi]], and $k!$ denotes the [[factorial]]. From the [[Taylor series of the exponential function]],
  1 KB (155 words) - 04:16, 3 October 2016
• Scorer Gi
  The Scorer $\mathrm{Gi}$ function is a solution of the [[differential equation]] $y''(x)-x y(x)=\dfrac{1}{\pi
  589 bytes (86 words) - 23:03, 9 June 2016
• Relationship between Scorer Gi and Airy functions
  ...enotes the [[Airy Ai]] function, and $\mathrm{Bi}$ denotes the [[Airy Bi]] function.
  421 bytes (62 words) - 22:53, 9 June 2016
• Scorer Hi
  The Scorer $\mathrm{Hi}$ function is a solution of the [[differential equation]] $y''(x)-x y(x)=\dfrac{1}{\pi
  589 bytes (87 words) - 23:00, 9 June 2016
• Relationship between Scorer Hi and Airy functions
  ...enotes the [[Airy Ai]] function, and $\mathrm{Bi}$ denotes the [[Airy Bi]] function.
  430 bytes (66 words) - 15:28, 6 October 2016
• Bolzano function
  A Bolzano function $B \colon [a,b] \rightarrow [c,d]$ is is defined as the limit of a sequence The Bolzano function is defined by the pointwise limit
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• Bolzano function is nowhere differentiable
  The [[Bolzano function]] is [[nowhere differentiable]].
  138 bytes (13 words) - 17:11, 23 June 2016
• Cellérier function
  Let $a>1000$. The Cellérier function is defined as [[Cellérier function is continuous]]<br />
  359 bytes (51 words) - 17:11, 23 June 2016
• Riemann function
  The Riemann function is the function $R \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by [[Riemann function is continuous]]<br />
  612 bytes (89 words) - 03:26, 6 July 2016
• Darboux function
  The Darboux function is defined by where $\sin$ denotes the [[sine]] function.
  597 bytes (73 words) - 18:02, 25 June 2017
• Schwarz function
  ...r \cdot \rfloor$ denotes the [[floor]] function and let $M>0$. The Schwarz function $S \colon (0,M) \rightarrow \mathbb{R}$ is defined by [[Schwarz function is continuous]]<br />
  590 bytes (78 words) - 18:03, 25 June 2017
• Blancmange function
  #REDIRECT [[Takagi function]]
  29 bytes (3 words) - 16:50, 22 January 2016
• Takagi function is continuous
  The [[Takagi function]] is [[continuous]].
  125 bytes (12 words) - 03:15, 6 July 2016
• Van der Waerden function
  The van der Waerden function $V \colon \mathbb{R} \rightarrow \mathbb{R}$ is defined by the formula where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function.
  726 bytes (98 words) - 03:33, 6 July 2016
• Faber F1
  The Faber function $F_1$ is defined by
  526 bytes (72 words) - 03:32, 6 July 2016
• Faber F2
  The Faber function $F_2$ is defined by
  523 bytes (71 words) - 03:32, 6 July 2016
• Faber functions
  #REDIRECT [[Faber function F1]]
  31 bytes (4 words) - 12:03, 5 January 2016
• Knopp function
  Let $a \in (0,1)$ $ab > 1$. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function.
  559 bytes (79 words) - 03:31, 27 October 2016
• Knopp function is nowhere differentiable
  The [[Knopp function]] $K_{a,b}$ is [[nowhere differentiable]] on $\mathbb{R}$ for $a \in (0,1)$
  192 bytes (24 words) - 03:32, 27 October 2016
• Petr function
  ...1}^{\infty} \dfrac{a_k}{10^k}$, where $a_k \in \{0,1,\ldots,9\}$. The Petr function $P_K \colon [0,1] \rightarrow \mathbb{R}$ is defined by [[The Petr function is continuous]]<br />
  673 bytes (100 words) - 20:34, 25 June 2017
• McCarthy function
  The McCarthy function $M$ is defined by [[McCarthy function is continuous]]<br />
  493 bytes (72 words) - 13:39, 17 November 2016
• Multiple gamma function
  where $\zeta_N$ denotes the [[Barnes zeta function]].
  211 bytes (35 words) - 18:32, 24 May 2016
• Elliptic gamma function
  The elliptic gamma function is defined by
  204 bytes (32 words) - 18:54, 24 May 2016
• Gamma(1)=1
  where $\Gamma$ denotes the [[gamma]] function.
  606 bytes (80 words) - 19:47, 15 March 2018
• Ramanujan theta function
  The Ramanujan theta function, $f$, is defined for $|ab|<1$ by
  379 bytes (61 words) - 16:02, 10 July 2017
• Distance to integers
  Define the function $\mathrm{dist}_{\mathbb{Z}} \colon \mathbb{R} \rightarrow \mathbb{R}$ by where $\inf$ denotes the [[infimum]]. This function can be computed using the [[floor]] and [[ceiling]] functions:
  400 bytes (61 words) - 03:12, 6 July 2016
• Faber F2 is continuous
  The [[Faber F2]] function is [[continuous]].
  127 bytes (13 words) - 17:19, 23 June 2016
• Stieltjes constants
  [[Laurent series of the Riemann zeta function]]<br />
  283 bytes (41 words) - 00:53, 9 December 2016
• Book:Harris Hancock/Lectures on the theory of elliptic functions
  :::1. One-valued function. Regular function. Zeros :::7. A one-valued function that is regular at all points of the plane is a constant
  21 KB (2,630 words) - 06:16, 25 June 2016
• Jacobi theta 2
  Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_2$ function is defined by where $\cos$ denotes the [[cosine]] function.
  1 KB (187 words) - 20:23, 25 June 2017
• Jacobi theta 3
  Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_3$ function is defined by where $\cos$ denotes the [[cosine]] function.
  1 KB (175 words) - 20:23, 25 June 2017
• Jacobi theta 4
  Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_4$ function is defined by where $\cos$ denotes the [[cosine]] function.
  1 KB (185 words) - 20:24, 25 June 2017
• Sum of sum of divisors function equals product of Riemann zeta for Re(z) greater than k+1
  where $\sigma_n$ denotes the [[sum of divisors]] function and $\zeta$ denotes the [[Riemann zeta]].
  481 bytes (65 words) - 22:18, 25 June 2016
• Closed form for partition function
  #REDIRECT [[Closed form for partition function with sinh]]
  58 bytes (8 words) - 20:34, 26 June 2016
• Recurrence relation for partition function
  #REDIRECT [[Pure recurrence relation for partition function]]
  61 bytes (7 words) - 20:39, 26 June 2016
• Recurrence relation for partition function with sum of divisors
  ...the [[partition]] function and $\sigma_1$ denotes the [[sum of divisors]] function. ...tion function|next=Sum of divisors functions written in terms of partition function}}: $24.2.1 \mathrm{II}.A.$
  530 bytes (69 words) - 20:43, 26 June 2016
• Sum of divisors functions written in terms of partition function
  ...$ denotes the [[sum of divisors]] function and $p$ denotes the [[partition function]]. ...tition function with sum of divisors|next=Asymptotic formula for partition function}}: $24.2.1 \mathrm{II}.B.$
  604 bytes (83 words) - 20:47, 26 June 2016
• Asymptotic formula for partition function
  where $p$ denotes the [[partition]] function, $\exp$ denotes the [[exponential]], and $\pi$ denotes [[pi]]. ...ene A. Stegun|prev=Sum of divisors functions written in terms of partition function|next=findme}}: $24.2.1 \mathrm{III}$
  533 bytes (66 words) - 00:10, 28 June 2016
• Hypergeometric 1F0
  The hypergeometric ${}_1F_0$ function is defined by the series
  316 bytes (41 words) - 06:04, 10 January 2017
• Hypergeometric 2F1
  The (Gauss) hypergeometric ${}_2F_1$ function (often written simply as $F$) is defined by the series ...neq 0, -1, -2, \ldots$. It is a special case of the [[hypergeometric pFq]] function.
  3 KB (501 words) - 23:23, 3 March 2018
• Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m
  where $\Gamma$ denotes the [[gamma]] function, $(a)_{m+1}$ denotes the [[Pochhammer]] symbol, and ${}_2F_1$ denotes the [
  590 bytes (79 words) - 23:15, 12 July 2016
• Riemann zeta as contour integral
  * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann zeta as integral of
  764 bytes (111 words) - 23:42, 17 March 2017
• Integral of monomial times Bessel J
  ...l J|Bessel function of the first kind]] and $\Gamma$ denotes the [[gamma]] function.
  760 bytes (110 words) - 16:53, 27 June 2016
• Integral of Bessel J for Re(nu) greater than -1
  where $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]].
  505 bytes (72 words) - 16:56, 27 June 2016
• Integral of Bessel J for nu=2n
  where $J_{2n}$ denotes the [[Bessel J|Bessel function of the first kind]].
  522 bytes (79 words) - 16:57, 27 June 2016
• Integral of Bessel J for nu=2n+1
  where $J_{2n+1}$ denotes the [[Bessel J|Bessel function of the first kind]].
  472 bytes (70 words) - 16:59, 27 June 2016
• Integral of Bessel J for nu=n+1
  where $J_{n+1}$ denotes the [[Bessel J|Bessel function of the first kind]].
  489 bytes (74 words) - 17:01, 27 June 2016
• Integral of Bessel J for nu=1
  where $J_1$ denotes the [[Bessel J|Bessel function of the first kind]].
  1 KB (193 words) - 23:18, 20 February 2018
• Riccati-Bessel S
  The Riccati-Bessel function $S_n$ is defined by the formula ...$\pi$ denotes [[pi]] and $J_{n+\frac{1}{2}}$ denotes the [[Bessel J|Bessel function of the first kind]].
  287 bytes (43 words) - 21:02, 27 June 2016
• Hypergeometric 1F1
  ...$ (sometimes denoted by $M$, sometimes called the confluent hypergeometric function of the first kind) is defined by the series
  568 bytes (93 words) - 20:31, 17 June 2017
• Hypergeometric 1F2
  [[Relationship between Struve function and hypergeometric pFq]]<br />
  391 bytes (53 words) - 06:06, 10 January 2017
• Series for log(Riemann zeta) in terms of Mangoldt function
  * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Series for log(riemann zeta)
  515 bytes (68 words) - 05:42, 5 July 2016
• Logarithmic derivative of Riemann zeta in terms of series over primes
  ...function|next=Logarithmic derivative of Riemann zeta in terms of Mangoldt function}}: § Introduction $(2{'}{'})$
  578 bytes (77 words) - 02:17, 1 July 2017
• Logarithmic derivative of Riemann zeta in terms of Mangoldt function
  ...\zeta$ denotes the [[Riemann zeta]] and $\Lambda$ denotes the [[Mangoldt]] function. * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Logarithmic derivative of Ri
  546 bytes (72 words) - 09:05, 19 November 2016
• Van der Waerden function is nowhere differentiable
  The [[van der Waerden function]] is [[nowhere differentiable]].
  146 bytes (15 words) - 03:18, 6 July 2016
• Riemann function is almost nowhere differentiable
  The [[Riemann function]] is [[nowhere differentiable]] except at points of the form $\pi \dfrac{2p
  218 bytes (29 words) - 03:28, 6 July 2016
• Continuous nowhere differentiable functions footer
  <div class="grid2"><center>[[Darboux function|<randomimagebycategory categories="darbouxglyph" width=45 />]] <br /> [[Darboux function]]</center></div>
  925 bytes (102 words) - 03:36, 6 July 2016
• Closed formula for physicist's Hermite polynomials
  ...ynomials]], $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, and $k!$ denotes the [[factorial]]. * {{BookReference|Special Functions|1960|Earl David Rainville|prev=Generating function for Hermite (physicist) polynomials|next=findme}}: $103. (2)$
  568 bytes (73 words) - 23:33, 8 July 2016
• Series for q-sin sub q
  where $\sin_q$ denotes the [[q-sin sub q|$\sin_q$]] function.
  408 bytes (56 words) - 15:39, 11 July 2016
• Omega constant
  This means it equals $W(1)$, where $W$ denotes the [[Lambert W]] function.
  226 bytes (33 words) - 00:41, 5 August 2016
• Digamma at 1/2
  where $\psi$ denotes the [[digamma]] function, $\gamma$ denotes the [[Euler-Mascheroni constant]], and $\log$ denotes the
  461 bytes (54 words) - 00:50, 9 August 2016
• Digamma at n+1/2
  where $\psi$ denotes the [[digamma]] function and $\gamma$ denotes the [[Euler-Mascheroni constant]].
  523 bytes (59 words) - 00:52, 9 August 2016
• Binet's formula
  ...ence|The Fibonacci Zeta Function|1976|Maruti Ram Murty|prev=Fibonacci zeta function|next=Fibonacci zeta in terms of a sum of binomial coefficients}}
  551 bytes (78 words) - 22:32, 25 May 2017
• Fibonacci zeta in terms of a sum of binomial coefficients
  where $F(z)$ denotes the [[Fibonacci zeta function]], $\displaystyle{{-z} \choose j}$ denotes a [[binomial coefficient]], and * {{PaperReference|The Fibonacci Zeta Function|1976|Maruti Ram Murty|prev=Binet's formula|next=findme}}
  2 KB (344 words) - 12:59, 11 August 2016
• Paper:Maruti Ram Murty/The Fibonacci Zeta Function
  {{Book|The Fibonacci Zeta Function|1976|Journal of the London Mathematical Society||Maruti Ram Murty}} [[Fibonacci zeta function]] <br />
  387 bytes (52 words) - 00:28, 24 May 2017
• Normalized sinc
  The normalized sinc function $\mathrm{sinc}_{\pi}$ is defined by
  267 bytes (37 words) - 02:22, 16 September 2016
• Reciprocal Riemann zeta
  ...fined by $\dfrac{1}{\zeta(z)},$ where $\zeta$ denotes the [[Riemann zeta]] function.
  266 bytes (34 words) - 03:54, 16 September 2016
• Reciprocal Riemann zeta in terms of Mobius
  where $\zeta$ denotes the [[Riemann zeta]] and $\mu$ denotes the [[Möbius]] function.
  449 bytes (57 words) - 19:31, 21 June 2017
• Schwarz function is nowhere differentiable on a dense subset
  Let $M>0$. The [[Schwarz function]] is [[nowhere differentiable]] on a [[dense]] subset of $(0,M)$.
  182 bytes (21 words) - 17:47, 20 September 2016
• Derivative of prime zeta
  where $P$ denotes the [[prime zeta P|prime zeta]] function and $\log$ denotes the [[logarithm]]. Recall the definition of the prime zeta function:
  1 KB (164 words) - 17:46, 19 October 2016
• K-function
  The $K$-function is defined by ...ient]], $\log$ denotes the [[logarithm]], and $\Gamma$ denotes the [[gamma function]].
  461 bytes (59 words) - 19:40, 25 September 2016
• Hyperfactorial in terms of K-function
  where $H$ denotes the [[hyperfactorial]] and $K$ denotes the [[K-function]].
  205 bytes (25 words) - 19:40, 25 September 2016
• Fresnel S in terms of erf
  ...$ denotes the [[imaginary number]], and $\mathrm{erf}$ denotes the [[error function]].
  425 bytes (58 words) - 17:17, 5 October 2016
• Fresnel S is odd
  The [[Fresnel S]] function is an [[odd function]].
  133 bytes (15 words) - 17:13, 5 October 2016
• Fresnel C is odd
  [[Fresnel C]] is an [[odd function]].
  120 bytes (13 words) - 17:23, 5 October 2016
• Fresnel C in terms of erf
  ...$ denotes the [[imaginary number]], and $\mathrm{erf}$ denotes the [[error function]].
  425 bytes (57 words) - 17:26, 5 October 2016
• Beta in terms of power of t over power of (1+t)
  where $B$ denotes the [[beta]] function.
  1 KB (170 words) - 01:16, 1 July 2017
• Derivative of erfi
  ...unction]], $\pi$ denotes [[pi]], and $e^{z^2}$ denotes the [[exponential]] function.
  340 bytes (44 words) - 23:12, 23 October 2016
• Paper:Richard E. Crandall/On the quantum zeta function
  {{Book|On the quantum zeta function|1996|Journal of Physics A: Mathematical and General||Richard E. Crandall}}
  246 bytes (35 words) - 02:15, 2 November 2016
• Thomae function is continuous at irrationals
  The [[Thomae function]] is [[continuous]] at all [[irrational number|irrational numbers]].
  173 bytes (18 words) - 00:34, 9 December 2016
• McCarthy function is nowhere differentiable
  The [[McCarthy function]] is [[nowhere differentiable]] on $\mathbb{R}$.
  155 bytes (16 words) - 13:40, 17 November 2016
• Riemann zeta as integral of monomial divided by an exponential
  where $\zeta$ denotes the [[Riemann zeta]] function and $e^t$ denotes the [[exponential]]. ...itchmarsh|prev=Logarithmic derivative of Riemann zeta in terms of Mangoldt function|next=Riemann zeta as contour integral}}: § Introduction $(3)$
  644 bytes (87 words) - 23:38, 17 March 2017
• Arcsin as inverse sine
  ...ac{\pi}{2} \right]$ is the [[inverse function]] of the restricted [[sine]] function $\sin \colon \left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right] \rightarrow [-1
  349 bytes (43 words) - 19:59, 22 November 2016
• Arccos as inverse cosine
  ...ghtarrow [0,\pi]$ is the [[inverse function]] of the restricted [[cosine]] function $\cos \colon \left[0,\pi\right] \rightarrow [-1,1]$.
  284 bytes (33 words) - 20:06, 22 November 2016
• Product of Weierstrass elementary factors is entire
  ...s elementary factors|Weierstrass elementary factor]] defines an [[entire]] function which has a zero at each $a_k$ and no other zeros. If a complex number $w$
  845 bytes (138 words) - 19:29, 26 November 2016
• Thomae function is discontinuous at rationals
  The [[Thomae function]] is discontinuous at all [[rational number|rational numbers]].
  168 bytes (18 words) - 00:34, 9 December 2016
• Q-shifted factorial
  * {{PaperReference|The q-gamma function for q greater than 1|1980|Daniel S. Moak|prev=Q-Gamma|next=Q-Gamma at z+1}}
  997 bytes (133 words) - 01:14, 25 December 2018
• Book:George E. Andrews/Special Functions
  ::1.11 A Probabilistic Evaluation of the Beta Function ::1.12 The $p$-adic Gamma Function
  6 KB (697 words) - 20:59, 18 December 2016
• Asymptotic behavior of Sievert integral
  ...^{-x}$ denotes the [[exponential]], and $\mathrm{erf}$ denotes the [[error function]].
  568 bytes (72 words) - 02:08, 21 December 2016
• Bessel J in terms of Bessel-Clifford
  ...$ denotes [[Bessel J]] and $\mathcal{C}_n$ denotes the [[Bessel-Clifford]] function.
  295 bytes (41 words) - 10:44, 11 January 2017
• Böhmer S
  The Böhmer $S$ function is defined by
  326 bytes (47 words) - 06:51, 10 January 2017
• Clausen sine
  Let $s \in \mathbb{C}$. The Clausen sine function $\mathrm{Cl}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined as the
  471 bytes (66 words) - 19:44, 7 September 2020
• Clausen cosine
  Let $s \in \mathbb{C}$. The Clausen cosine function $\tilde{\mathrm{Cl}}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined
  521 bytes (72 words) - 18:56, 7 September 2020
• Paper:Thomas Clausen/Uber die function sin(φ)+sin(2φ)/2^2+sin(3φ)/3^2+etc.
  {{Book|Uber die function sin(φ)+sin(2φ)/2^2+sin(3φ)/3^2+etc.|1832|Journal für die reine und ange
  286 bytes (39 words) - 06:38, 10 January 2017
• Dirichlet function
  The Dirichlet function $D \colon \mathbb{R} \rightarrow \{0,1\}$ is defined by [[Dirichlet function is nowhere continuous]]<br />
  525 bytes (71 words) - 07:29, 10 January 2017
• Dirichlet function is nowhere continuous
  The [[Dirichlet function]] is nowhere [[continuous]].
  136 bytes (13 words) - 07:31, 10 January 2017
• Relationship between Bessel-Clifford and hypergeometric 0F1
  where $\mathcal{C}_n$ denotes the [[Bessel-Clifford]] function and ${}_0F_1$ denotes [[hypergeometric 0F1]].
  272 bytes (33 words) - 10:46, 11 January 2017
• Derivative of Bessel-Clifford
  where $\mathcal{C}_n$ denotes the [[Bessel-Clifford]] function.
  242 bytes (32 words) - 10:48, 11 January 2017
• Reciprocal gamma is entire
  The [[reciprocal gamma]] function is [[entire]].
  131 bytes (13 words) - 10:50, 11 January 2017
• Riemann zeta at even integers
  * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann zeta as contour inte
  574 bytes (77 words) - 23:57, 17 March 2017
• Functional equation for Riemann zeta
  * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann zeta at even integer
  536 bytes (70 words) - 00:00, 18 March 2017
• Functional equation for Riemann zeta with cosine
  * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Functional equation for Riem
  508 bytes (66 words) - 15:18, 18 March 2017
• Riemann-Landau xi is even
  [[Riemann-Landau xi]] is an [[even function]]. * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann-Landau xi|next=findm
  268 bytes (28 words) - 15:29, 18 March 2017
• Havercosine
  The havercosine function $\mathrm{havercos} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by
  239 bytes (28 words) - 03:51, 28 March 2017
• Covercosine
  The coversine function $\mathrm{coversin} \colon \mathbb{C} \rightarrow \mathbb{C}$ defined by
  221 bytes (27 words) - 03:47, 28 March 2017
• Vercosine
  The vercosine function $\mathrm{vercos} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by
  221 bytes (27 words) - 03:51, 28 March 2017
• Hacovercosine
  The hacovercosine function $\mathrm{hacovercos} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined b
  241 bytes (28 words) - 03:50, 28 March 2017
• Hacoversine
  The hacoversine function $\mathrm{hacoversin} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined b
  242 bytes (29 words) - 03:52, 28 March 2017
• Paper:Daniel S. Moak/The q-gamma function for q greater than 1
  {{Book|The q-gamma function for q greater than 1|1980|Aequationes Mathematicae||Daniel S. Moak|}}
  351 bytes (44 words) - 00:18, 30 May 2017
• Book:Yudell L. Luke/The Special Functions And Their Approximations, Volume I
  :II. The Gamma function and Related Functions ::2.4. The Logarithmic Derivative of the Gamma Function
  2 KB (244 words) - 02:02, 30 May 2017

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