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Showing below up to 500 results in range #251 to #750.
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- (hist) Exponential cdf [294 bytes]
- (hist) Secant zeta function [295 bytes]
- (hist) Relationship between cosine, Gudermannian, and sech [295 bytes]
- (hist) Derivative of Riemann zeta [295 bytes]
- (hist) Bessel J in terms of Bessel-Clifford [295 bytes]
- (hist) Prime number theorem, pi and x/log(x) [296 bytes]
- (hist) Antiderivative of arctanh [296 bytes]
- (hist) Unsigned Lah numbers [297 bytes]
- (hist) Relationship between sech, inverse Gudermannian, and cos [298 bytes]
- (hist) Paper:John H. Halton/On a General Fibonacci Identity [298 bytes]
- (hist) Square of i [299 bytes]
- (hist) Prime number theorem, logarithmic integral [300 bytes]
- (hist) Stirling polynomial [300 bytes]
- (hist) Binomial series [300 bytes]
- (hist) Relationship between csch, inverse Gudermannian, and cot [303 bytes]
- (hist) Relationship between coth, inverse Gudermannian, and csc [303 bytes]
- (hist) Signed Lah numbers [304 bytes]
- (hist) Dickman–de Bruijn function [305 bytes]
- (hist) Value of Ai(0) [306 bytes]
- (hist) Limiting value of Fresnel C [306 bytes]
- (hist) Relationship between Meixner polynomials and Charlier polynomials [307 bytes]
- (hist) Relationship between tanh, inverse Gudermannian, and sin [308 bytes]
- (hist) Limiting value of Fresnel S [308 bytes]
- (hist) Relationship between Hurwitz zeta and gamma function [310 bytes]
- (hist) Relationship between Chebyshev U and Gegenbauer C [310 bytes]
- (hist) Anger three-term recurrence [310 bytes]
- (hist) L n'(0)=-n [310 bytes]
- (hist) Hahn-Exton q-Bessel [311 bytes]
- (hist) Fundamental pair of periods [311 bytes]
- (hist) Integral representation of polygamma 2 [312 bytes]
- (hist) Functional equation for Riemann xi [312 bytes]
- (hist) Tanhc [314 bytes]
- (hist) Exponential pdf [314 bytes]
- (hist) Relationship between logarithm and Mangoldt [315 bytes]
- (hist) Norton's constant [316 bytes]
- (hist) Hypergeometric 1F0 [316 bytes]
- (hist) Hermite (physicist) polynomial at negative argument [316 bytes]
- (hist) Q-number of a negative [317 bytes]
- (hist) Sum of reciprocal Pochhammer symbols of a fixed exponent [318 bytes]
- (hist) Incomplete beta function [319 bytes]
- (hist) Incomplete Elliptic E [319 bytes]
- (hist) L n(0)=1 [319 bytes]
- (hist) Relationship between exponential integral Ei, cosine integral, and sine integral [320 bytes]
- (hist) Relationship between arcsin and arccsc [321 bytes]
- (hist) Paper:Charles Watkins Merrifield/The Sums of the Series of the Reciprocals of the Prime Numbers and of Their Powers [321 bytes]
- (hist) Ramanujan's sum [321 bytes]
- (hist) Q-Euler formula for e sub q [322 bytes]
- (hist) Relationship between coth and csch [322 bytes]
- (hist) Mills' constant [323 bytes]
- (hist) Gompertz constant [323 bytes]
- (hist) Euler totient is multiplicative [323 bytes]
- (hist) Airy zeta function at 2 [323 bytes]
- (hist) Kronecker delta [323 bytes]
- (hist) Epstein zeta function [324 bytes]
- (hist) Taylor series for Gudermannian [324 bytes]
- (hist) Relationship between Bessel I sub 1/2 and sinh [325 bytes]
- (hist) Derivative of Legendre chi 2 [326 bytes]
- (hist) Böhmer S [326 bytes]
- (hist) Sierpiński constant [327 bytes]
- (hist) Laurent series of the Riemann zeta function [328 bytes]
- (hist) Relationship between the Fransén–Robinson constant, e, pi, and logarithm [328 bytes]
- (hist) Hypergeometric 0F0 [328 bytes]
- (hist) Relationship between Bessel I sub -1/2 and cosh [330 bytes]
- (hist) Series for erf with exponential factored out [330 bytes]
- (hist) Barnes G at positive integer [330 bytes]
- (hist) Cosh is even [331 bytes]
- (hist) Q-exponential e sub 1/q [332 bytes]
- (hist) Möbius function is multiplicative [332 bytes]
- (hist) Sinhc [334 bytes]
- (hist) Three-term recurrence for Bateman F [334 bytes]
- (hist) Böhmer C [334 bytes]
- (hist) T(n)=n(n+1)/2 [334 bytes]
- (hist) Glaisher–Kinkelin constant [335 bytes]
- (hist) Laplace pdf [335 bytes]
- (hist) 1/q-number as a q-number [336 bytes]
- (hist) Tanh is odd [336 bytes]
- (hist) Incomplete Elliptic K [338 bytes]
- (hist) Porter's constant [338 bytes]
- (hist) Bickley-Naylor [338 bytes]
- (hist) Fransén–Robinson constant [340 bytes]
- (hist) Harmonic number [340 bytes]
- (hist) Relationship between Chebyshev T and hypergeometric 2F1 [340 bytes]
- (hist) Relationship between cos and cosh [340 bytes]
- (hist) Derivative of erfi [340 bytes]
- (hist) Modified Struve function [341 bytes]
- (hist) Exton q-exponential [342 bytes]
- (hist) L(-n)=(-1)^nL(n) [342 bytes]
- (hist) Dirichlet eta [343 bytes]
- (hist) Relationship between Chebyshev U and hypergeometric 2F1 [344 bytes]
- (hist) Legendre chi in terms of polylogarithm [347 bytes]
- (hist) Antiderivative of sech [347 bytes]
- (hist) Sinh is odd [347 bytes]
- (hist) L n'(x)=-Sum L k(x) [348 bytes]
- (hist) Arcsin as inverse sine [349 bytes]
- (hist) Log(1+x) less than x [350 bytes]
- (hist) Cauchy pdf [350 bytes]
- (hist) Paper:Daniel S. Moak/The q-gamma function for q greater than 1 [351 bytes]
- (hist) Continuous uniform cdf [351 bytes]
- (hist) Logarithm of a complex number [353 bytes]
- (hist) Gamma recurrence relation [355 bytes]
- (hist) Shi [356 bytes]
- (hist) XL n'(x)=nL n(x)-n L (n-1)(x) [356 bytes]
- (hist) F(n+1)F(n-1)-F(n)^2=(-1)^n [357 bytes]
- (hist) Q-Euler formula for E sub q [358 bytes]
- (hist) F(2n)=F(n+1)^2-F(n-1)^2 [358 bytes]
- (hist) Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta [359 bytes]
- (hist) Relationship between Airy Ai and modified Bessel K [359 bytes]
- (hist) Sister Celine's polynomials [359 bytes]
- (hist) Cellérier function [359 bytes]
- (hist) Gamma(n+1)=n! [359 bytes]
- (hist) Log base a in terms of logarithm base b [359 bytes]
- (hist) Paper:Yilmaz Simsek/q-Dedekind type sums related to q-zeta function and basic L-series [360 bytes]
- (hist) Wallis product [361 bytes]
- (hist) 2cos(mt)cos(nt)=cos((m+n)t)+cos((m-n)t) [361 bytes]
- (hist) F(2n+1)=F(n+1)^2+F(n)^2 [362 bytes]
- (hist) Integral of log of inverse erf from 0 to 1 [363 bytes]
- (hist) X/(1+x) less than log(1+x) [363 bytes]
- (hist) Antiderivative of hyperbolic cosecant [364 bytes]
- (hist) Tetrahedral numbers [364 bytes]
- (hist) Exponential of logarithm [364 bytes]
- (hist) Derivative of inverse error function [364 bytes]
- (hist) Bessel polynomial in terms of Bessel functions [365 bytes]
- (hist) Relationship between sin and sinh [365 bytes]
- (hist) Paper:Matilde Lalín/Secant zeta functions [365 bytes]
- (hist) Relationship between integral of x*log(sin(x)), and Apéry's constant, pi, and logarithm [365 bytes]
- (hist) L(n+1)L(n-1)-L(n)^2=5(-1)^(n+1) [365 bytes]
- (hist) T(n+1)=T(n)+n+1 [365 bytes]
- (hist) Reciprocal of i [366 bytes]
- (hist) (n+1)L (n+1)(x) = (2n+1-x)L n(x)-nL (n-1)(x) [366 bytes]
- (hist) X less than -log(1-x) [367 bytes]
- (hist) Q-Polygamma function [368 bytes]
- (hist) Antiderivative of inverse error function [369 bytes]
- (hist) Complex number [370 bytes]
- (hist) Hypergeometric 0F3 [370 bytes]
- (hist) Binomial coefficient (n choose k) equals (n choose (n-k)) [371 bytes]
- (hist) T(n+1)^2-T(n)^2=(n+1)^3 [372 bytes]
- (hist) Log a(b)=1/log b(a) [372 bytes]
- (hist) Relationship between cosine, imaginary number, logarithm, and the golden ratio [373 bytes]
- (hist) N^2=T(n)+T(n-1) [373 bytes]
- (hist) Lattice generated by doubly periodic periods [374 bytes]
- (hist) Doubling identity for cosh (1) [374 bytes]
- (hist) Lerch zeta function [376 bytes]
- (hist) Erdős-Borwein Constant [377 bytes]
- (hist) Relationship between Chebyshev T and Gegenbauer C [377 bytes]
- (hist) Binomial coefficient (n choose n) equals 1 [377 bytes]
- (hist) Log(x) less than or equal to x-1 [377 bytes]
- (hist) Matrix e^A=limit of (I+A/s)^s [377 bytes]
- (hist) Dirichlet L-function [379 bytes]
- (hist) Ramanujan theta function [379 bytes]
- (hist) Sum of Fibonacci numbers [379 bytes]
- (hist) Falling factorial [380 bytes]
- (hist) T(n)^2=T(T(n))+T(T(n)-1) [381 bytes]
- (hist) -log(1-x) less than x/(1-x) [381 bytes]
- (hist) Trigamma [384 bytes]
- (hist) Nth derivative of logarithm [384 bytes]
- (hist) Logarithm at -i [385 bytes]
- (hist) Bell polynomial [386 bytes]
- (hist) Relationship between spherical Bessel y and cosine [387 bytes]
- (hist) Paper:Maruti Ram Murty/The Fibonacci Zeta Function [387 bytes]
- (hist) Q-derivative power rule [388 bytes]
- (hist) Anger of integer order is Bessel J [389 bytes]
- (hist) Sum of Lucas numbers [389 bytes]
- (hist) Identity [390 bytes]
- (hist) Relationship between spherical Bessel j and sine [391 bytes]
- (hist) Hypergeometric 1F2 [391 bytes]
- (hist) Q-number when a=n is a natural number [392 bytes]
- (hist) Orthogonality of Chebyshev U on (-1,1) [392 bytes]
- (hist) Cauchy cdf [393 bytes]
- (hist) Bessel polynomial generalized hypergeometric [394 bytes]
- (hist) Meissel-Mertens constant in terms of the Euler-Mascheroni constant [395 bytes]
- (hist) Catalan's constant using Hurwitz zeta [396 bytes]
- (hist) Neumann polynomial [396 bytes]
- (hist) Q-Gamma at z+1 [396 bytes]
- (hist) Erf of conjugate is conjugate of erf [396 bytes]
- (hist) Tanh of a sum [396 bytes]
- (hist) Partial derivative of beta function [397 bytes]
- (hist) Relationship between logarithm and positive integer exponents [397 bytes]
- (hist) Recurrence relation of exponential integral E [399 bytes]
- (hist) Log e(z)=log(z) [399 bytes]
- (hist) Log 10(z)=log(z)/log(10) [399 bytes]
- (hist) Relationship between Bessel J and hypergeometric 0F1 [400 bytes]
- (hist) Relationship between coth and cot [400 bytes]
- (hist) Distance to integers [400 bytes]
- (hist) Log(z)=log(10)log 10(z) [400 bytes]
- (hist) Coth of a sum [400 bytes]
- (hist) Relationship between cosh and cos [401 bytes]
- (hist) Haversine [401 bytes]
- (hist) Sum of squares of Fibonacci numbers [401 bytes]
- (hist) Hypergeometric 3F2 [401 bytes]
- (hist) Euler E n'(x)=nE n-1(x) [401 bytes]
- (hist) Normal pdf [401 bytes]
- (hist) F(n+m+1)=F(n+1)F(m+1)+F(n)F(m) [402 bytes]
- (hist) Abs(log(1-x)) less than 3x/2 [402 bytes]
- (hist) Logarithm at i [403 bytes]
- (hist) Riemann-Landau xi [403 bytes]
- (hist) Doubling identity for cosh (3) [403 bytes]
- (hist) Li 2(1)=pi^2/6 [404 bytes]
- (hist) Bessel at -n-1/2 in terms of Bessel polynomial [406 bytes]
- (hist) Relationship between tanh and tan [406 bytes]
- (hist) Abs(log(1+z)) less than or equal to -log(1-abs(z)) [407 bytes]
- (hist) Derivative [408 bytes]
- (hist) Rodrigues formula for Meixner polynomial [408 bytes]
- (hist) Series for q-sin sub q [408 bytes]
- (hist) Klein invariant J [411 bytes]
- (hist) Period parallelogram [411 bytes]
- (hist) Loggamma [412 bytes]
- (hist) Hypergeometric 4F1 [412 bytes]
- (hist) Arcsech [413 bytes]
- (hist) Relationship between Airy Bi and modified Bessel I [413 bytes]
- (hist) Erfi [413 bytes]
- (hist) Logarithm of exponential [413 bytes]
- (hist) Binomial coefficient ((n+1) choose k) equals (n choose k) + (n choose (k-1)) [414 bytes]
- (hist) Halving identity for cosh [414 bytes]
- (hist) Differential equation for Jacobi P [416 bytes]
- (hist) Sinh of a sum [416 bytes]
- (hist) Arcsin cdf [416 bytes]
- (hist) Lagrange polynomial [417 bytes]
- (hist) F(2n)=F(n)L(n) [417 bytes]
- (hist) Chebyshev psi function [418 bytes]
- (hist) Meromorphic continuation of q-exponential E sub q [418 bytes]
- (hist) Dirichlet series [418 bytes]
- (hist) Doubling identity for cosh (2) [419 bytes]
- (hist) Fibonacci polynomial [421 bytes]
- (hist) Relationship between Scorer Gi and Airy functions [421 bytes]
- (hist) Relationship between Bessel Y sub n and Bessel Y sub -n [421 bytes]
- (hist) Log 10(z)=log 10(e)log(z) [421 bytes]
- (hist) Derivative of Struve H0 [421 bytes]
- (hist) Beta in terms of gamma [422 bytes]
- (hist) L(n)=F(n+1)+F(n-1) [422 bytes]
- (hist) Cosh of a sum [422 bytes]
- (hist) L n(x)=(e^x/n!)d^n/dx^n(x^n e^(-x)) [422 bytes]
- (hist) Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials [423 bytes]
- (hist) Paper:Edmund Landau/Sur la série des inverse de nombres de Fibonacci [423 bytes]
- (hist) E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1 [424 bytes]
- (hist) Minkowski question mark [425 bytes]
- (hist) Fresnel S in terms of erf [425 bytes]
- (hist) Fresnel C in terms of erf [425 bytes]
- (hist) Limit of x^a log(x)=0 [425 bytes]
- (hist) Doubling identity for sinh (1) [425 bytes]
- (hist) Hyperfactorial [426 bytes]
- (hist) Relationship between Legendre polynomial and hypergeometric 2F1 [426 bytes]
- (hist) Logarithm at minus 1 [427 bytes]
- (hist) E is limit of (1+1/n)^n [427 bytes]
- (hist) B(x,y+1)=(y/(x+y))B(x,y) [427 bytes]
- (hist) Relationship between sech and sec [428 bytes]
- (hist) B(x,y)B(x+y,z)=B(y,z)B(y+z,x) [428 bytes]
- (hist) Orthogonality relation for cosine on (0,pi) [429 bytes]
- (hist) Lower incomplete gamma [430 bytes]
- (hist) Relationship between Scorer Hi and Airy functions [430 bytes]
- (hist) Binomial coefficient (n choose k) equals (-1)^k ((k-n-1) choose k) [430 bytes]
- (hist) Absolute convergence of secant zeta function [430 bytes]
- (hist) Sum rule for derivatives [431 bytes]
- (hist) Hypergeometric 2F0 [431 bytes]
- (hist) Jacobi nc [432 bytes]
- (hist) Stirling numbers of the second kind [432 bytes]
- (hist) Log(x) less than or equal to n(x^(1/n)-1) [432 bytes]
- (hist) E (0,1)(z)=1/(1-z) for abs(z) less than 1 [432 bytes]
- (hist) Symmetry relation of exponential integral E [433 bytes]
- (hist) Relationship between csch and csc [434 bytes]
- (hist) B(x,y+1)=(y/x)B(x+1,y) [434 bytes]
- (hist) Difference of cosh and sinh [434 bytes]
- (hist) Matrix exponential [434 bytes]
- (hist) Weierstrass elliptic [435 bytes]
- (hist) Legendre chi [436 bytes]
- (hist) Dawson D+ [437 bytes]
- (hist) L(n)^2-5F(n)^2=4(-1)^n [437 bytes]
- (hist) Taylor series [438 bytes]
- (hist) Doubling identity for sinh (2) [438 bytes]
- (hist) Q-derivative of q-Sine [439 bytes]
- (hist) Greatest prime factor [440 bytes]
- (hist) Period of cosh [440 bytes]
- (hist) Artin-Mazur zeta function [441 bytes]
- (hist) Jackson q-Bessel (2) [441 bytes]
- (hist) Halving identity for sinh [441 bytes]
- (hist) Jackson q-Bessel (1) [442 bytes]
- (hist) Product rule for derivatives [442 bytes]
- (hist) Integral of (z^n)log(z)dz=(z^(n+1)/(n+1))log(z)-z^(n+1)/(n+1)^2 for integer n neq -1 [442 bytes]
- (hist) Orthogonality of Bateman F on R [442 bytes]
- (hist) Relationship between Struve function and hypergeometric pFq [443 bytes]
- (hist) Jacobi nd [444 bytes]
- (hist) Two-dimensional Laplace transform [444 bytes]
- (hist) Unit step function [444 bytes]
- (hist) Normal cdf [444 bytes]
- (hist) Generating function for Laguerre L [444 bytes]
- (hist) Q-difference equation for q-exponential E sub q [447 bytes]
- (hist) Paper:D.S. McAnally/q-exponential and q-gamma functions. I. q-exponential functions [447 bytes]
- (hist) Limit of quotient of consecutive Fibonacci numbers [448 bytes]
- (hist) Arccsch [449 bytes]
- (hist) Jacobi ns [449 bytes]
- (hist) Reciprocal Riemann zeta in terms of Mobius [449 bytes]
- (hist) Q-derivative of q-Cosine [450 bytes]
- (hist) Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z)) [452 bytes]
- (hist) K(m)=(pi/2)2F1(1/2,1/2;1;m) [452 bytes]
- (hist) Buchstab function [453 bytes]
- (hist) E(m)=(pi/2)2F1(-1/2,1/2;1;m) [453 bytes]
- (hist) Glyphs [454 bytes]
- (hist) Relationship between Li 2(1),Li 2(-1), and pi [454 bytes]
- (hist) Limit of q-exponential E sub 1/q for 0 less than q less than 1 [454 bytes]
- (hist) Hypergeometric 2F3 [454 bytes]
- (hist) Catalan's constant [455 bytes]
- (hist) Relationship between Bessel J sub n and Bessel J sub -n [455 bytes]
- (hist) Relationship between Anger function and Weber function [456 bytes]
- (hist) E^x is greater than 1+x for nonzero real x [456 bytes]
- (hist) Sqrt(1-z^2)2F1(1,1;3/2;z^2)=arcsin(z)/z [456 bytes]
- (hist) Period of sinh [456 bytes]
- (hist) Pythagorean identity for coth and csch [456 bytes]
- (hist) T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n [456 bytes]
- (hist) Floor [457 bytes]
- (hist) Antiderivative of the logarithm [458 bytes]
- (hist) Series for polygamma in terms of Riemann zeta [459 bytes]
- (hist) Paper:V.E. Hoggatt, Jr/Triangular numbers [460 bytes]
- (hist) B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt [460 bytes]
- (hist) Digamma at 1/2 [461 bytes]
- (hist) K-function [461 bytes]
- (hist) Riemann-Siegel Z [462 bytes]
- (hist) Gauss' formula for gamma function [462 bytes]
- (hist) Halving identity for tangent (1) [462 bytes]
- (hist) Bessel polynomial [464 bytes]
- (hist) Weierstrass elementary factors [464 bytes]
- (hist) Polygamma recurrence relation [464 bytes]
- (hist) Period of tanh [464 bytes]
- (hist) (c-a-b)2F1-(c-a)2F1(a-1)+b(1-z)2F1(b+1)=0 [464 bytes]
- (hist) H (-(n+1/2))(z)=(-1)^n J (n+1/2)(z) for integer n geq 0 [464 bytes]
- (hist) E^x is less than 1/(1-x) for nonzero real x less than 1 [465 bytes]
- (hist) Spherical Bessel y [466 bytes]
- (hist) Relationship between sine, imaginary number, logarithm, and the golden ratio [467 bytes]
- (hist) Airy zeta function [468 bytes]
- (hist) F(-n)=(-1)^(n+1)F(n) [469 bytes]
- (hist) Orthogonality of Laguerre L [469 bytes]
- (hist) Sum of cosh and sinh [470 bytes]
- (hist) Series for log(z) for absolute value of (z-1) less than 1 [471 bytes]
- (hist) Clausen sine [471 bytes]
- (hist) Taylor series of log(1+z) [472 bytes]
- (hist) Digamma at z+n [472 bytes]
- (hist) Integral of Bessel J for nu=2n+1 [472 bytes]
- (hist) (c-2a-(b-a)z)2F1+a(1-z)2F1(a+1)-(c-a)2F1(a-1)=0 [472 bytes]
- (hist) Weierstrass nowhere differentiable function [473 bytes]
- (hist) Value of derivative of trigamma at positive integer plus 1/2 [473 bytes]
- (hist) Generating relation for Bateman F [473 bytes]
- (hist) Euler phi [474 bytes]
- (hist) Limit of log(x)/x^a=0 [474 bytes]
- (hist) Bessel at n+1/2 in terms of Bessel polynomial [475 bytes]
- (hist) Beta as product of gamma functions [475 bytes]
- (hist) X/(1+x) less than 1-e^(-x) less than x for nonzero real x greater than -1 [476 bytes]
- (hist) B(x,y)B(x+y,z)=B(z,x)B(x+z,y) [476 bytes]
- (hist) Faddeeva function [478 bytes]
- (hist) Pythagorean identity for tanh and sech [478 bytes]
- (hist) H (nu)(x) geq 0 for x gt 0 and nu geq 1/2 [478 bytes]
- (hist) E(2,1)(z)=cosh(sqrt(z)) [478 bytes]
- (hist) Q-Sin [479 bytes]
- (hist) Jinc [479 bytes]
- (hist) E(2,1)(-z^2)=cos(z) [479 bytes]
- (hist) Spherical Bessel j [480 bytes]
- (hist) Contour integral representation of reciprocal gamma [480 bytes]
- (hist) Sum of sum of divisors function equals product of Riemann zeta for Re(z) greater than k+1 [481 bytes]
- (hist) 2F1(1,1;2;z)=-log(1-z)/z [481 bytes]
- (hist) E(1,1)(z)=exp(z) [481 bytes]
- (hist) Value of polygamma at 1 [482 bytes]
- (hist) Q-difference equation for q-exponential E sub 1/q [482 bytes]
- (hist) Gamma'(z)/Gamma(z)=-gamma-1/z+Sum z/(k(z+k)) [482 bytes]
- (hist) Pochhammer symbol with non-negative integer subscript [483 bytes]
- (hist) 1Phi0(a;;z)1Phi0(b;;az)=1Phi0(ab;;z) [483 bytes]
- (hist) D/dz(z^(nu)H (nu))=z^(nu)H (nu-1) [484 bytes]
- (hist) X less than e^x-1 less than x/(1-x) for nonzero real x less than 1 [486 bytes]
- (hist) Logarithm diverges to negative infinity at 0 from right [487 bytes]
- (hist) Identity written as a sum of Möbius functions [487 bytes]
- (hist) Q-Hurwitz zeta [487 bytes]
- (hist) Integral representation of polygamma for Re(z) greater than 0 [488 bytes]
- (hist) Real and imaginary parts of log [488 bytes]
- (hist) Dirichlet beta [489 bytes]
- (hist) Series for log(z) for Re(z) greater than 1/2 [489 bytes]
- (hist) Logarithm (multivalued) of the exponential [489 bytes]
- (hist) Integral of Bessel J for nu=n+1 [489 bytes]
- (hist) Continuous [490 bytes]
- (hist) Orthogonality of Gegenbauer C on (-1,1) [490 bytes]
- (hist) Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1 [491 bytes]
- (hist) Jacobi sd [493 bytes]
- (hist) Jacobi cs [493 bytes]
- (hist) Jacobi dc [493 bytes]
- (hist) McCarthy function [493 bytes]
- (hist) Chain rule for derivatives [493 bytes]
- (hist) Product representation of totient [493 bytes]
- (hist) 1Phi0(a;;z) as infinite product [493 bytes]
- (hist) Relationship between Weber function and Anger function [494 bytes]
- (hist) Polygamma reflection formula [495 bytes]
- (hist) Book:Sir Thomas L. Heath/Euclid: The Thirteen Books of The Elements: Volume 2/Second Edition [495 bytes]
- (hist) Sum of totient equals z/((1-z) squared) [495 bytes]
- (hist) 2F1(1/2,1;3/2;z^2)=log((1+z)/(1-z))/(2z) [495 bytes]
- (hist) Inverse error function [496 bytes]
- (hist) Generating function for Hermite (physicist) polynomials [497 bytes]
- (hist) Q-Cos [498 bytes]
- (hist) Q-Gaussian distribution [498 bytes]
- (hist) 2F1(1/2,1;3/2;-z^2)=arctan(z)/z [499 bytes]
- (hist) Laplace cdf [499 bytes]
- (hist) Jacobi sc [505 bytes]
- (hist) Jacobi ds [505 bytes]
- (hist) Jacobi cd [505 bytes]
- (hist) 2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z [505 bytes]
- (hist) Integral of Bessel J for Re(nu) greater than -1 [505 bytes]
- (hist) (b-a)2F1+a2F1(a+1)-b2F1(b+1)=0 [505 bytes]
- (hist) Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1 [506 bytes]
- (hist) Relationship between Li 2(-1/x),Li 2(-x),Li 2(-1), and log^2(x) [508 bytes]
- (hist) Functional equation for Riemann zeta with cosine [508 bytes]
- (hist) Halving identity for tangent (3) [508 bytes]
- (hist) Halving identity for tangent (2) [510 bytes]
- (hist) Bernoulli numbers [511 bytes]
- (hist) (c-a-b)2F1+a(1-z)2F1(a+1)-(c-b)2F1(b-1)=0 [511 bytes]
- (hist) 1+x greater than exp(x/(1+x)) for nonzero real x greater than -1 [512 bytes]
- (hist) Q-cos sub q [513 bytes]
- (hist) Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1 [513 bytes]
- (hist) Q-zeta [514 bytes]
- (hist) E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936 [515 bytes]
- (hist) Series for log(Riemann zeta) in terms of Mangoldt function [515 bytes]
- (hist) Z coth(z) = 2z/(e^(2z)-1) + z [515 bytes]
- (hist) Sum over bottom of binomial coefficient with top fixed equals 2^n [516 bytes]
- (hist) C(a-(c-b)z)2F1-ac(1-z)2F1(a+1)+(c-a)(c-b)z2F1(c+1)=0 [516 bytes]
- (hist) Log(1+z) as continued fraction [516 bytes]
- (hist) Li 2(z)+Li 2(1-z)=pi^2/6-log(z)log(1-z) [516 bytes]
- (hist) Beta as improper integral [517 bytes]
- (hist) (c-a-1)2F1+a2F1(a+1)-(c-1)2F1(c-1)=0 [517 bytes]
- (hist) Paper:Lucas Jódar/On the hypergeometric matrix function [517 bytes]
- (hist) E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0 [518 bytes]
- (hist) Sum of totient equals zeta(z-1)/zeta(z) for Re(z) greater than 2 [518 bytes]
- (hist) Bernoulli B [519 bytes]
- (hist) Euler E generating function [519 bytes]
- (hist) Constant multiple rule for derivatives [519 bytes]
- (hist) Derivative of Gudermannian [521 bytes]
- (hist) Clausen cosine [521 bytes]
- (hist) Integral of Bessel J for nu=2n [522 bytes]
- (hist) Euler numbers [522 bytes]
- (hist) Faber F2 [523 bytes]
- (hist) Digamma at n+1/2 [523 bytes]
- (hist) Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x [524 bytes]
- (hist) Dirichlet function [525 bytes]
- (hist) D/dz(z^(-nu)H (nu))=1/(sqrt(pi)2^(nu)Gamma(nu+3/2))-z^(-nu)H (nu+1) [525 bytes]
- (hist) Paper:H.J. Haubold/Mittag-Leffler Functions and Their Applications [525 bytes]
- (hist) U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n [525 bytes]
- (hist) Arcsec [526 bytes]
- (hist) Faber F1 [526 bytes]
- (hist) Quotient rule for derivatives [526 bytes]
- (hist) Modular form [528 bytes]
- (hist) Recurrence relation for partition function with sum of divisors [530 bytes]
- (hist) U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1) [530 bytes]
- (hist) Chebyshev theta function [531 bytes]
- (hist) Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4 [531 bytes]
- (hist) Series for log(riemann zeta) over primes [532 bytes]
- (hist) Asymptotic formula for partition function [533 bytes]
- (hist) Value of polygamma at positive integer [534 bytes]
- (hist) Product representation of q-exponential E sub 1/q [535 bytes]
- (hist) Functional equation for Riemann zeta [536 bytes]
- (hist) B(x,y)B(x+y,z)B(x+y+z,u)=Gamma(x)Gamma(y)Gamma(z)Gamma(u)/Gamma(x+y+z+u) [537 bytes]
- (hist) Dickson polynomial [538 bytes]
- (hist) Z/(1-sqrt(q))2Phi1(q,sqrt(q);sqrt(q^3);z)=Sum z^k/(1-q^(k-1/2)) [538 bytes]
- (hist) Li2(z)=zPhi(z,2,1) [538 bytes]
- (hist) 2Phi1(q,-1;-q;z)=1+2Sum z^k/(1+q^k) [539 bytes]
- (hist) Series for log(z) for Re(z) greater than 0 [540 bytes]
- (hist) Q-exponential e sub q [541 bytes]
- (hist) Generating function for partition function [541 bytes]
- (hist) Relationship between logarithmic integral and exponential integral [542 bytes]
- (hist) Erfc [542 bytes]
- (hist) Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3 [542 bytes]
- (hist) H (1/2)(z)=sqrt(2/(pi z))(1-cos(z)) [543 bytes]
- (hist) E^x greater than (1+x/y)^y greater than exp(xy/(x+y) for x greater than 0 and y greater than 0) [546 bytes]
- (hist) Digamma functional equation [546 bytes]
- (hist) Logarithmic derivative of Riemann zeta in terms of Mangoldt function [546 bytes]
- (hist) Riemann Siegel theta function [548 bytes]
- (hist) E is irrational [548 bytes]
- (hist) Arccsc [549 bytes]
- (hist) Recurrence relation for Struve function (2) [549 bytes]
- (hist) Exsecant [550 bytes]
- (hist) Binet's formula [551 bytes]
- (hist) Mertens [555 bytes]
- (hist) Genocchi numbers [555 bytes]
- (hist) Taylor series of log(1-z) [555 bytes]
- (hist) Elliptic E [556 bytes]
- (hist) Log((1+z)/(1-z)) as continued fraction [556 bytes]
- (hist) Q-sin sub q [557 bytes]
- (hist) Bernstein B [558 bytes]
- (hist) Sum of divisors [559 bytes]
- (hist) Knopp function [559 bytes]
- (hist) Bessel-Clifford [559 bytes]
- (hist) Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0 [559 bytes]
- (hist) 1/B(n,m)=m((n+m-1) choose (n-1)) [560 bytes]
- (hist) 1/B(n,m)=n((n+m-1) choose (m-1)) [561 bytes]
- (hist) Relationship between sinh and sin [563 bytes]
- (hist) H (3/2)(z)=sqrt(z/(2pi))(1+2/z^2)-sqrt(2/(pi z))(sin(z)+cos(z)/z) [563 bytes]
- (hist) Dedekind eta [565 bytes]
- (hist) Integral of (1+t)^(2x-1)(1-t)^(2y-1)(1+t^2)^(-x-y)dt=2^(x+y-2)B(x,y) [566 bytes]
- (hist) T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k) [566 bytes]
- (hist) Logarithm of product is a sum of logarithms [567 bytes]
- (hist) Hypergeometric 1F1 [568 bytes]
- (hist) Closed formula for physicist's Hermite polynomials [568 bytes]
- (hist) Asymptotic behavior of Sievert integral [568 bytes]
- (hist) (z/(1-q))2Phi1(q,q;q^2;z)=Sum z^k/(1-q^k) [568 bytes]
- (hist) Value of polygamma at 1/2 [574 bytes]
- (hist) Derivative of the logarithm [574 bytes]
- (hist) Riemann zeta at even integers [574 bytes]
- (hist) Hurwitz zeta [575 bytes]
- (hist) Alternating sum over bottom of binomial coefficient with top fixed equals 0 [578 bytes]
- (hist) Digamma at 1 [578 bytes]