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  • 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]
    1 KB (191 words) - 05:53, 4 March 2018
  • 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 $[0,\infty)$.
    1 KB (134 words) - 23:35, 21 October 2017
  • Csch
    The hyperbolic cosecant function $\mathrm{csch} \colon \mathbb{R} \setminus \{0\} \rightarrow \mathbb{R} \se ...e]], its [[inverse function]], the [[arccsch|inverse hyperbolic cosecant]] function is clear.
    1 KB (137 words) - 23:35, 21 October 2017
  • Arcsinh
    ...}$ 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
    718 bytes (89 words) - 23:47, 11 December 2016
  • Arccoth
    ...arccoth}$ is the [[inverse function]] of the [[coth|hyperbolic cotangent]] function. It may be defined by the following formula:
    722 bytes (89 words) - 01:40, 16 September 2016
  • Arcsech
    ...on $\mathrm{arcsech} \colon (0,1] \rightarrow [0,\infty)$ is the [[inverse function]] of the [[sech|hyperbolic secant]].
    413 bytes (47 words) - 03:46, 6 July 2016
  • Arccsch
    ...{R} \setminus \{0\} \rightarrow \mathbb{R} \setminus \{0\}$ is the inverse function of the [[csch|hyperbolic cosecant]].
    449 bytes (52 words) - 03:46, 6 July 2016
  • Arithmetic functions
    An arithmetic function is a function $f \colon \{1,2,3,\ldots\} \rightarrow \mathbb{C}$.
    219 bytes (29 words) - 00:04, 11 December 2016
  • 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.
    432 bytes (60 words) - 19:07, 5 July 2016
  • 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.
    740 bytes (106 words) - 02:48, 5 January 2017
  • Derivative of sinh
    and so using the [[derivative of the exponential function]], the [[derivative is a linear operator|linear property of the derivative]
    633 bytes (94 words) - 07:52, 8 June 2016
  • Derivative of cosh
    and so using the [[derivative of the exponential function]], the [[derivative is a linear operator|linear property of the derivative]
    640 bytes (97 words) - 23:59, 16 June 2016
  • Derivative of arccos
    where $\mathrm{arccos}$ denotes the [[arccos|inverse cosine]] function.
    736 bytes (106 words) - 07:29, 8 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
  • 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]].
    462 bytes (67 words) - 18:30, 24 May 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
  • 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]].
    330 bytes (45 words) - 12:52, 17 September 2016
  • Glaisher–Kinkelin constant
    where $G$ is the [[Barnes G|Barnes $G$]] function.
    335 bytes (49 words) - 16:01, 16 June 2016
  • 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.
    216 bytes (25 words) - 08:00, 8 June 2016
  • Catalan's constant using Legendre chi
    where $K$ is [[Catalan's constant]] and $\chi$ denotes the [[Legendre chi]] function.
    687 bytes (103 words) - 15:46, 25 February 2018
  • Catalan's constant using Hurwitz zeta
    ...ant]], and $\zeta'$ denotes the partial derivative of the [[Hurwitz zeta]] function with respect to the first argument.
    396 bytes (53 words) - 08:01, 8 June 2016
  • 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]].
    347 bytes (49 words) - 07:59, 8 June 2016
  • 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]].
    406 bytes (64 words) - 20:27, 27 June 2016
  • 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]].
    155 bytes (19 words) - 00:01, 23 December 2016
  • 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
  • Relationship between Lerch transcendent and Lerch zeta
    ...$\Phi$ denotes the [[Lerch transcendent]] and $L$ denotes the [[Lerch zeta function]].
    253 bytes (35 words) - 16:34, 20 June 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
  • 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]].
    296 bytes (39 words) - 20:25, 27 June 2016
  • 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.
    490 bytes (88 words) - 20:40, 11 April 2015
  • Coshc
    The $\mathrm{coshc}$ function is defined by
    270 bytes (34 words) - 19:08, 12 June 2017
  • Thomae
    #REDIRECT [[Thomae function]]
    29 bytes (3 words) - 20:48, 11 April 2015
  • 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]].
    773 bytes (129 words) - 07:25, 16 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
  • 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
  • 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.
    2 KB (209 words) - 02:00, 18 December 2016
  • 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
    5 KB (873 words) - 00:50, 13 September 2023
  • 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 $K$]].
    359 bytes (48 words) - 23:07, 9 June 2016

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