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- (hist) Coversine [680 bytes]
- (hist) Pythagorean identity for sin and cos [678 bytes]
- (hist) Continued fraction for 2e^(z^2) integral from z to infinity e^(-t^2) dt for positive Re(z) [677 bytes]
- (hist) Logarithm of a quotient is a difference of logarithms [676 bytes]
- (hist) Logarithm of 1 [675 bytes]
- (hist) Legendre P [674 bytes]
- (hist) Petr function [673 bytes]
- (hist) Derivative of sech [672 bytes]
- (hist) Integral of (t-b)^(x-1)(a-t)^(y-1)/(t-x)^(x+y) dt=(a-b)^(x+y-1)/((a-c)^x(b-c)^y) B(x,y) [671 bytes]
- (hist) Kelvin functions footer [669 bytes]
- (hist) Golden ratio [667 bytes]
- (hist) Ratio test [666 bytes]
- (hist) Derivative of Jacobi theta 1 at 0 [665 bytes]
- (hist) Squares of theta relation for Jacobi theta 3 and Jacobi theta 4 [664 bytes]
- (hist) Squares of theta relation for Jacobi theta 2 and Jacobi theta 4 [664 bytes]
- (hist) Hankel H (2) in terms of csc and Bessel J [664 bytes]
- (hist) Arccosh [661 bytes]
- (hist) 2F1(a,b;a+b+1/2;z)^2=3F2(2a,a+b,2b;a+b+1/2,2a+2b;z) [659 bytes]
- (hist) Logarithm (multivalued) of a quotient is a difference of logarithms (multivalued) [658 bytes]
- (hist) Ramanujan tau [658 bytes]
- (hist) Squares of theta relation for Jacobi theta 4 and Jacobi theta 4 [654 bytes]
- (hist) Paper:Harry Bateman/Some Properties of a certain Set of Polynomials [654 bytes]
- (hist) Logarithm base a [653 bytes]
- (hist) Hankel H (1) in terms of csc and Bessel J [646 bytes]
- (hist) Polylogarithm [646 bytes]
- (hist) Integral of t^(x-1)(1-t^z)^(y-1) dt=(1/z)B(x/z,y) [645 bytes]
- (hist) 1F1(a;r;z)1F1(a;r;-z)=2F3(a,r-a;r,r/2,r/2+1/2;z^2/4) [644 bytes]
- (hist) Riemann zeta as integral of monomial divided by an exponential [644 bytes]
- (hist) Meissel-Mertens constant [644 bytes]
- (hist) Derivative of cosh [640 bytes]
- (hist) Q-Pochhammer [640 bytes]
- (hist) Binomial coefficient (n choose 0) equals 1 [639 bytes]
- (hist) Riemann xi [637 bytes]
- (hist) E [637 bytes]
- (hist) Elliptic K [635 bytes]
- (hist) 2F0(a,b;;z)2F0(a,b;;-z)=4F1(a,b,a/2+b/2,a/2+b/2+1/2;a+b;4z^2) [634 bytes]
- (hist) Exponential integral Ei series [634 bytes]
- (hist) Integral representation of Struve function [633 bytes]
- (hist) 1F1(a;2a;z)1F1(b;2b;-z)=2F3(a/2+b/2,a/2+b/2+1/2;a+1/2,b+1/2,a+b;z^2/4) [633 bytes]
- (hist) Derivative of sinh [633 bytes]
- (hist) Z coth(z) = sum of 2^(2n)B (2n) z^(2n)/(2n)! [632 bytes]
- (hist) Chi [632 bytes]
- (hist) Euler's formula [631 bytes]
- (hist) Logarithm (multivalued) of product is a sum of logarithms (multivalued) [629 bytes]
- (hist) 0F1(;r;z)0F1(;r;-z)=0F3(r,r/2,r/2+1/2;-z^2/4) [626 bytes]
- (hist) Error functions footer [624 bytes]
- (hist) 0F1(;r;z)0F1(;s;z)=2F1(r/2+s/2, r/2+s/2-1/2;r,s,r+s-1;4z) [622 bytes]
- (hist) Logarithmic derivative of Jacobi theta 2 equals negative tangent + a sum of sines [621 bytes]
- (hist) Polar coordinates [620 bytes]
- (hist) Squares of theta relation for Jacobi theta 1 and Jacobi theta 4 [615 bytes]
- (hist) Legendre's constant [615 bytes]
- (hist) Integral (t-b)^(x-1)(a-t)^(y-1)dt=(a-b)^(x+y-1)B(x,y) [612 bytes]
- (hist) Riemann function [612 bytes]
- (hist) Q-derivative [612 bytes]
- (hist) Gamma(1)=1 [606 bytes]
- (hist) Paper:James Whitbread Lee Glaisher/On certain definite integrals involving the exponential-integral [605 bytes]
- (hist) Logarithm of a quotient of Jacobi theta 4 equals a sum of sines [605 bytes]
- (hist) Sum of divisors functions written in terms of partition function [604 bytes]
- (hist) Gamma function written as a limit of a factorial, exponential, and a rising factorial [603 bytes]
- (hist) Relationship between logarithm (multivalued) and positive integer exponents [603 bytes]
- (hist) Spherical Hankel h (2) [603 bytes]
- (hist) Spherical Hankel h (1) [603 bytes]
- (hist) Integral t^(x-1)(1-t)^(y-1)(1+bt)^(-x-y)dt = (1+b)^(-x)B(x,y) [600 bytes]
- (hist) Prime counting [600 bytes]
- (hist) Li 2(z)=-Li 2(1/z)-(1/2)(log z)^2 + i pi log(z) + pi^2/3 [599 bytes]
- (hist) Darboux function [597 bytes]
- (hist) Orthogonality of Chebyshev T on (-1,1) [595 bytes]
- (hist) Derivative of Li 2(-1/x) [593 bytes]
- (hist) Z coth(z) = 2 Sum of (-1)^(n+1) zeta(2n) z^(2n)/pi^(2n) [592 bytes]
- (hist) Logarithmic derivative of Jacobi theta 4 equals a sum of sines [592 bytes]
- (hist) Derivative of arccot [592 bytes]
- (hist) Integral t^(x-1)(1+bt)^(-x-y) dt = b^(-x) B(x,y) [591 bytes]
- (hist) Second q-shifted factorial [591 bytes]
- (hist) Recurrence relation for Struve fuction [590 bytes]
- (hist) Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m [590 bytes]
- (hist) Schwarz function [590 bytes]
- (hist) Q-calculus footer [590 bytes]
- (hist) Scorer Hi [589 bytes]
- (hist) Scorer Gi [589 bytes]
- (hist) Logarithmic derivative of Jacobi theta 3 equals a sum of sines [588 bytes]
- (hist) Logarithmic derivative of Jacobi theta 1 equals cotangent + a sum of sines [588 bytes]
- (hist) Series for log(z+a) for positive a and Re(z) greater than -a [587 bytes]
- (hist) Reciprocal zeta function [587 bytes]
- (hist) Logarithm and friends footer [586 bytes]
- (hist) B(x,y)=2^(1-x-y)integral (1+t)^(x-1)(1-t)^(y-1)+(1+t)^(y-1)(1-t)^(x-1) dt [586 bytes]
- (hist) Book:Michael Wilensky/Ueber Besselsche Funktionen [583 bytes]
- (hist) Reciprocal Fibonacci constant [582 bytes]
- (hist) Brun's constant [582 bytes]
- (hist) Pure recurrence relation for partition function [581 bytes]
- (hist) Polygamma multiplication formula [580 bytes]
- (hist) General Dirichlet series [579 bytes]
- (hist) Logarithmic derivative of Riemann zeta in terms of series over primes [578 bytes]
- (hist) Digamma at 1 [578 bytes]
- (hist) Alternating sum over bottom of binomial coefficient with top fixed equals 0 [578 bytes]
- (hist) Hurwitz zeta [575 bytes]
- (hist) Riemann zeta at even integers [574 bytes]
- (hist) Derivative of the logarithm [574 bytes]
- (hist) Value of polygamma at 1/2 [574 bytes]
- (hist) (z/(1-q))2Phi1(q,q;q^2;z)=Sum z^k/(1-q^k) [568 bytes]
- (hist) Asymptotic behavior of Sievert integral [568 bytes]
- (hist) Closed formula for physicist's Hermite polynomials [568 bytes]
- (hist) Hypergeometric 1F1 [568 bytes]
- (hist) Logarithm of product is a sum of logarithms [567 bytes]
- (hist) T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k) [566 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) Dedekind eta [565 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) Relationship between sinh and sin [563 bytes]
- (hist) 1/B(n,m)=n((n+m-1) choose (m-1)) [561 bytes]
- (hist) 1/B(n,m)=m((n+m-1) choose (n-1)) [560 bytes]
- (hist) Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0 [559 bytes]
- (hist) Bessel-Clifford [559 bytes]
- (hist) Knopp function [559 bytes]
- (hist) Sum of divisors [559 bytes]
- (hist) Bernstein B [558 bytes]
- (hist) Q-sin sub q [557 bytes]
- (hist) Log((1+z)/(1-z)) as continued fraction [556 bytes]
- (hist) Elliptic E [556 bytes]
- (hist) Taylor series of log(1-z) [555 bytes]
- (hist) Genocchi numbers [555 bytes]
- (hist) Mertens [555 bytes]
- (hist) Binet's formula [551 bytes]
- (hist) Exsecant [550 bytes]
- (hist) Recurrence relation for Struve function (2) [549 bytes]
- (hist) Arccsc [549 bytes]
- (hist) E is irrational [548 bytes]
- (hist) Riemann Siegel theta function [548 bytes]
- (hist) Logarithmic derivative of Riemann zeta in terms of Mangoldt function [546 bytes]
- (hist) Digamma functional equation [546 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) H (1/2)(z)=sqrt(2/(pi z))(1-cos(z)) [543 bytes]
- (hist) Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3 [542 bytes]
- (hist) Erfc [542 bytes]
- (hist) Relationship between logarithmic integral and exponential integral [542 bytes]
- (hist) Generating function for partition function [541 bytes]
- (hist) Q-exponential e sub q [541 bytes]
- (hist) Series for log(z) for Re(z) greater than 0 [540 bytes]
- (hist) 2Phi1(q,-1;-q;z)=1+2Sum z^k/(1+q^k) [539 bytes]
- (hist) Li2(z)=zPhi(z,2,1) [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) Dickson polynomial [538 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) Functional equation for Riemann zeta [536 bytes]
- (hist) Product representation of q-exponential E sub 1/q [535 bytes]
- (hist) Value of polygamma at positive integer [534 bytes]
- (hist) Asymptotic formula for partition function [533 bytes]
- (hist) Series for log(riemann zeta) over primes [532 bytes]
- (hist) Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4 [531 bytes]
- (hist) Chebyshev theta function [531 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) Recurrence relation for partition function with sum of divisors [530 bytes]
- (hist) Modular form [528 bytes]
- (hist) Quotient rule for derivatives [526 bytes]
- (hist) Faber F1 [526 bytes]
- (hist) Arcsec [526 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) Paper:H.J. Haubold/Mittag-Leffler Functions and Their Applications [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) Dirichlet function [525 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) Digamma at n+1/2 [523 bytes]
- (hist) Faber F2 [523 bytes]
- (hist) Euler numbers [522 bytes]
- (hist) Integral of Bessel J for nu=2n [522 bytes]
- (hist) Clausen cosine [521 bytes]
- (hist) Derivative of Gudermannian [521 bytes]
- (hist) Constant multiple rule for derivatives [519 bytes]
- (hist) Euler E generating function [519 bytes]
- (hist) Bernoulli B [519 bytes]
- (hist) Sum of totient equals zeta(z-1)/zeta(z) for Re(z) greater than 2 [518 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) Paper:Lucas Jódar/On the hypergeometric matrix function [517 bytes]
- (hist) (c-a-1)2F1+a2F1(a+1)-(c-1)2F1(c-1)=0 [517 bytes]
- (hist) Beta as improper integral [517 bytes]
- (hist) Li 2(z)+Li 2(1-z)=pi^2/6-log(z)log(1-z) [516 bytes]
- (hist) Log(1+z) as continued fraction [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) Sum over bottom of binomial coefficient with top fixed equals 2^n [516 bytes]
- (hist) Z coth(z) = 2z/(e^(2z)-1) + z [515 bytes]
- (hist) Series for log(Riemann zeta) in terms of Mangoldt function [515 bytes]
- (hist) E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936 [515 bytes]
- (hist) Q-zeta [514 bytes]
- (hist) Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1 [513 bytes]
- (hist) Q-cos sub q [513 bytes]
- (hist) 1+x greater than exp(x/(1+x)) for nonzero real x greater than -1 [512 bytes]
- (hist) (c-a-b)2F1+a(1-z)2F1(a+1)-(c-b)2F1(b-1)=0 [511 bytes]
- (hist) Bernoulli numbers [511 bytes]
- (hist) Halving identity for tangent (2) [510 bytes]
- (hist) Halving identity for tangent (3) [508 bytes]
- (hist) Functional equation for Riemann zeta with cosine [508 bytes]
- (hist) Relationship between Li 2(-1/x),Li 2(-x),Li 2(-1), and log^2(x) [508 bytes]
- (hist) Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1 [506 bytes]
- (hist) (b-a)2F1+a2F1(a+1)-b2F1(b+1)=0 [505 bytes]
- (hist) Integral of Bessel J for Re(nu) greater than -1 [505 bytes]
- (hist) 2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z [505 bytes]
- (hist) Jacobi cd [505 bytes]
- (hist) Jacobi ds [505 bytes]
- (hist) Jacobi sc [505 bytes]
- (hist) Laplace cdf [499 bytes]
- (hist) 2F1(1/2,1;3/2;-z^2)=arctan(z)/z [499 bytes]
- (hist) Q-Gaussian distribution [498 bytes]
- (hist) Q-Cos [498 bytes]
- (hist) Generating function for Hermite (physicist) polynomials [497 bytes]
- (hist) Inverse error function [496 bytes]
- (hist) 2F1(1/2,1;3/2;z^2)=log((1+z)/(1-z))/(2z) [495 bytes]
- (hist) Sum of totient equals z/((1-z) squared) [495 bytes]
- (hist) Book:Sir Thomas L. Heath/Euclid: The Thirteen Books of The Elements: Volume 2/Second Edition [495 bytes]
- (hist) Polygamma reflection formula [495 bytes]
- (hist) Relationship between Weber function and Anger function [494 bytes]
- (hist) 1Phi0(a;;z) as infinite product [493 bytes]
- (hist) Product representation of totient [493 bytes]
- (hist) Chain rule for derivatives [493 bytes]
- (hist) McCarthy function [493 bytes]
- (hist) Jacobi dc [493 bytes]
- (hist) Jacobi cs [493 bytes]
- (hist) Jacobi sd [493 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) Orthogonality of Gegenbauer C on (-1,1) [490 bytes]
- (hist) Continuous [490 bytes]
- (hist) Integral of Bessel J for nu=n+1 [489 bytes]
- (hist) Logarithm (multivalued) of the exponential [489 bytes]
- (hist) Series for log(z) for Re(z) greater than 1/2 [489 bytes]
- (hist) Dirichlet beta [489 bytes]
- (hist) Real and imaginary parts of log [488 bytes]
- (hist) Integral representation of polygamma for Re(z) greater than 0 [488 bytes]
- (hist) Q-Hurwitz zeta [487 bytes]
- (hist) Identity written as a sum of Möbius functions [487 bytes]
- (hist) Logarithm diverges to negative infinity at 0 from right [487 bytes]
- (hist) X less than e^x-1 less than x/(1-x) for nonzero real x less than 1 [486 bytes]
- (hist) D/dz(z^(nu)H (nu))=z^(nu)H (nu-1) [484 bytes]
- (hist) 1Phi0(a;;z)1Phi0(b;;az)=1Phi0(ab;;z) [483 bytes]
- (hist) Pochhammer symbol with non-negative integer subscript [483 bytes]
- (hist) Gamma'(z)/Gamma(z)=-gamma-1/z+Sum z/(k(z+k)) [482 bytes]
- (hist) Q-difference equation for q-exponential E sub 1/q [482 bytes]
- (hist) Value of polygamma at 1 [482 bytes]
- (hist) E(1,1)(z)=exp(z) [481 bytes]
- (hist) 2F1(1,1;2;z)=-log(1-z)/z [481 bytes]
- (hist) Sum of sum of divisors function equals product of Riemann zeta for Re(z) greater than k+1 [481 bytes]
- (hist) Contour integral representation of reciprocal gamma [480 bytes]
- (hist) Spherical Bessel j [480 bytes]
- (hist) E(2,1)(-z^2)=cos(z) [479 bytes]
- (hist) Jinc [479 bytes]
- (hist) Q-Sin [479 bytes]
- (hist) E(2,1)(z)=cosh(sqrt(z)) [478 bytes]
- (hist) H (nu)(x) geq 0 for x gt 0 and nu geq 1/2 [478 bytes]
- (hist) Pythagorean identity for tanh and sech [478 bytes]
- (hist) Faddeeva function [478 bytes]
- (hist) B(x,y)B(x+y,z)=B(z,x)B(x+z,y) [476 bytes]
- (hist) X/(1+x) less than 1-e^(-x) less than x for nonzero real x greater than -1 [476 bytes]
- (hist) Beta as product of gamma functions [475 bytes]