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  1. 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)
  2. 1F1(a;r;z)1F1(a;r;-z)=2F3(a,r-a;r,r/2,r/2+1/2;z^2/4)
  3. 1Phi0(a;;z)1Phi0(b;;az)=1Phi0(ab;;z)
  4. 1Phi0(a;;z) as infinite product
  5. 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)
  6. 2F1(1,1;2;z)=-log(1-z)/z
  7. 2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z
  8. 2F1(1/2,1;3/2;-z^2)=arctan(z)/z
  9. 2F1(1/2,1;3/2;z^2)=log((1+z)/(1-z))/(2z)
  10. 2F1(a,b;a+b+1/2;z)^2=3F2(2a,a+b,2b;a+b+1/2,2a+2b;z)
  11. 2Phi1(q,-1;-q;z)=1+2Sum z^k/(1+q^k)
  12. 2cos(mt)cos(nt)=cos((m+n)t)+cos((m-n)t)
  13. Abel p
  14. Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z))
  15. Abs(log(1+z)) less than or equal to -log(1-abs(z))
  16. Abs(log(1-x)) less than 3x/2
  17. Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1
  18. Absolute convergence of secant zeta function
  19. Airy Ai
  20. Airy Bi
  21. Airy functions footer
  22. Airy zeta function
  23. Airy zeta function at 2
  24. Alexander operator
  25. Algebraic
  26. Algebraic number
  27. Alternating sum over bottom of binomial coefficient with top fixed equals 0
  28. Anger derivative recurrence
  29. Anger function
  30. Anger of integer order is Bessel J
  31. Anger three-term recurrence
  32. Antiderivative of arccos
  33. Antiderivative of arccosh
  34. Antiderivative of arcsin
  35. Antiderivative of arcsinh
  36. Antiderivative of arctan
  37. Antiderivative of arctanh
  38. Antiderivative of cosine integral
  39. Antiderivative of coth
  40. Antiderivative of hyperbolic cosecant
  41. Antiderivative of inverse error function
  42. Antiderivative of sech
  43. Antiderivative of sine integral
  44. Antiderivative of tanh
  45. Antiderivative of the logarithm
  46. Antiderivative of versine
  47. Apéry's constant
  48. Apéry's constant is irrational
  49. Arakawa-Kaneko zeta function
  50. Arccos
  51. Arccos as inverse cosine
  52. Arccosh
  53. Arccot
  54. Arccoth
  55. Arccsc
  56. Arccsch
  57. Arcsec
  58. Arcsech
  59. Arcsin
  60. Arcsin as inverse sine
  61. Arcsin cdf
  62. Arcsin pdf
  63. Arcsinh
  64. Arctan
  65. Arctanh
  66. Arithmetic functions
  67. Arithmetic zeta function
  68. Artin-Mazur zeta function
  69. Artin constant
  70. Associated Laguerre L
  71. Asymptotic behavior of Sievert integral
  72. Asymptotic formula for partition function
  73. 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
  74. B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt
  75. B(x,y)B(x+y,z)=B(y,z)B(y+z,x)
  76. B(x,y)B(x+y,z)=B(z,x)B(x+z,y)
  77. 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)
  78. B(x,y+1)=(y/(x+y))B(x,y)
  79. B(x,y+1)=(y/x)B(x+1,y)
  80. Barnes G
  81. Barnes G at positive integer
  82. Barnes G at z+1 in terms of Barnes G and gamma
  83. Barnes zeta function
  84. Basic hypergeometric phi
  85. Basic hypergeometric series psi
  86. Bateman F
  87. Bell numbers
  88. Bell polynomial
  89. Bernardi operator
  90. Bernoulli-Euler Gamma function
  91. Bernoulli B
  92. Bernoulli numbers
  93. Bernoulli polynomial and Hurwitz zeta
  94. Bernstein B
  95. Bessel-Clifford
  96. Bessel J
  97. Bessel J in terms of Bessel-Clifford
  98. Bessel Y
  99. Bessel at -n-1/2 in terms of Bessel polynomial
  100. Bessel at n+1/2 in terms of Bessel polynomial
  101. Bessel functions footer
  102. Bessel polynomial
  103. Bessel polynomial generalized hypergeometric
  104. Bessel polynomial in terms of Bessel functions
  105. Beta
  106. Beta as improper integral
  107. Beta as product of gamma functions
  108. Beta in terms of gamma
  109. Beta in terms of power of t over power of (1+t)
  110. Beta in terms of sine and cosine
  111. Beta is symmetric
  112. Bickley-Naylor
  113. Binet's formula
  114. Binomial coefficient
  115. Binomial coefficient ((n+1) choose k) equals (n choose k) + (n choose (k-1))
  116. Binomial coefficient (n choose 0) equals 1
  117. Binomial coefficient (n choose k) equals (-1)^k ((k-n-1) choose k)
  118. Binomial coefficient (n choose k) equals (n choose (n-k))
  119. Binomial coefficient (n choose n) equals 1
  120. Binomial series
  121. Binomial theorem
  122. Bohr-Mollerup theorem
  123. Bolzano function
  124. Bolzano function is continuous
  125. Bolzano function is nowhere differentiable
  126. Book:Aleksandar Ivić/The Riemann Zeta-Function
  127. Book:Alfred George Greenhill/The applications of elliptic functions
  128. Book:Andrew Gray/A Treatise on Bessel Functions
  129. Book:Andrew Gray/A Treatise on Bessel Functions/Second Edition
  130. Book:Arthur Erdélyi/Higher Transcendental Functions Volume I
  131. Book:Arthur Erdélyi/Higher Transcendental Functions Volume II
  132. Book:Arthur Erdélyi/Higher Transcendental Functions Volume III
  133. Book:Bernard Dwork/Generalized hypergeometric functions
  134. Book:Charalambos Charalambides/Discrete q-Distributions
  135. Book:Earl David Rainville/Special Functions
  136. Book:Edmund Taylor Whittaker/A course of modern analysis/Third edition
  137. Book:Edward Charles Titchmarsh/The Zeta-Function of Riemann
  138. Book:Elena Deza/Figurate Numbers
  139. Book:F.E. Relton/Applied Bessel Functions
  140. Book:G.H. Hardy/The General Theory Of Dirichlet's Series
  141. Book:Gabor Szegő/Orthogonal Polynomials/Fourth Edition
  142. Book:George E. Andrews/Special Functions
  143. Book:George Eyre Andrews/Number Theory
  144. Book:Harris Hancock/Lectures on the theory of elliptic functions
  145. Book:Ian N. Sneddon/Special Functions of Mathematical Physics and Chemistry
  146. Book:Ioannis Dimitrios Avgoustis/Definite Integration using the Generalized Hypergeometric Functions
  147. Book:Johan Thim/Continuous Nowhere Differentiable Functions
  148. Book:Johann Heinrich Graf/Einleitung in die Theorie der Gammafunktion und der Euler'schen Integrale
  149. Book:Larry C. Andrews/Special Functions of Mathematics for Engineers
  150. Book:Leonard Lewin/Dilogarithms and Associated Functions
  151. Book:Leonard Lewin/Polylogarithms and Associated Functions/Second Edition
  152. Book:Leonard Lewin/Structural Properties of Polylogarithms
  153. Book:Michael Wilensky/Ueber Besselsche Funktionen
  154. Book:Milton Abramowitz/Handbook of mathematical functions
  155. Book:Nicholas Higham/Functions of Matrices: Theory and Computation
  156. Book:Norman L. Johnson/Continuous Univariate Distributions Volume 2/Second Edition
  157. Book:Richard Beals/Special functions, a graduate text
  158. Book:Richard Dedekind/Essays on the Theory of Numbers
  159. Book:Roelof Koekoek/Hypergeometric Orthogonal Polynomials and Their q-Analogues
  160. Book:Sir Thomas L. Heath/Euclid: The Thirteen Books of The Elements: Volume 2/Second Edition
  161. Book:T.S. Chihara/An Introduction to Orthogonal Polynomials
  162. Book:Thomas Ernst/A Comprehensive Treatment of q-Calculus
  163. Book:Victor Kac/Quantum Calculus
  164. Book:W.N. Bailey/Generalized Hypergeometric Series
  165. Book:W.W. Bell/Special Functions for Scientists and Engineers
  166. Book:Wilhelm Magnus/Formulas and Theorems for the Special Functions of Mathematical Physics/Third Edition
  167. Book:Yudell L. Luke/The Special Functions And Their Approximations, Volume I
  168. Boole polynomials
  169. Brun's constant
  170. Buchstab function
  171. Böhmer C
  172. Böhmer S
  173. C(a-(c-b)z)2F1-ac(1-z)2F1(a+1)+(c-a)(c-b)z2F1(c+1)=0
  174. C n^(lambda)'(x)=2lambda C (n+1)^(lambda+1)(x)
  175. Cahen's constant
  176. Catalan's constant
  177. Catalan's constant using Dirichlet beta
  178. Catalan's constant using Hurwitz zeta
  179. Catalan's constant using Legendre chi
  180. Catalan's identity
  181. Cauchy cdf
  182. Cauchy pdf
  183. Ceiling
  184. Cell
  185. Cellérier function
  186. Cellérier function is continuous
  187. Cellérier function is nowhere differentiable
  188. Chain rule for derivatives
  189. Chaitin's constant
  190. Champernowne constant
  191. Champernowne constant is transcendental
  192. Charlier polynomial
  193. Chebyshev T
  194. Chebyshev U
  195. Chebyshev psi function
  196. Chebyshev theta function
  197. Chi
  198. Clausen cosine
  199. Clausen sine
  200. Closed form for partition function with sinh
  201. Closed formula for physicist's Hermite polynomials
  202. Complex conjugate of argument of error function
  203. Complex number
  204. Constant functions are elliptic functions
  205. Constant multiple rule for derivatives
  206. Continued fraction
  207. Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt
  208. Continued fraction for 2e^(z^2) integral from z to infinity e^(-t^2) dt for positive Re(z)
  209. Continuous
  210. Continuous nowhere differentiable functions footer
  211. Continuous q-Hermite polynomial
  212. Continuous uniform cdf
  213. Continuous uniform pdf
  214. Contour integral representation of reciprocal gamma
  215. Convergence of Hypergeometric pFq
  216. Copeland-Erdős constant
  217. Copeland-Erdős is irrational
  218. Copeland-Erdős is normal
  219. Cosecant
  220. Cosh
  221. Cosh is even
  222. Cosh of a sum
  223. Coshc
  224. Cosine
  225. Cosine integral
  226. Cotangent
  227. Cotangent zeta function
  228. Coth
  229. Coth of a sum
  230. Covercosine
  231. Coversine
  232. Csch
  233. Cyclotomic polynomials
  234. D/dz(z^(-nu)H (nu))=1/(sqrt(pi)2^(nu)Gamma(nu+3/2))-z^(-nu)H (nu+1)
  235. D/dz(z^(nu)H (nu))=z^(nu)H (nu-1)
  236. Darboux function
  237. Darboux function is continuous
  238. Darboux function is nowhere differentiable
  239. Dawson D+
  240. Dawson D-
  241. Debye function
  242. Dedekind eta
  243. Dedekind zeta function
  244. Denisyuk polynomials
  245. Depreciated trigonometric functions footer
  246. Derivative
  247. Derivative is a linear operator
  248. Derivative of Bessel-Clifford
  249. Derivative of Bessel J with respect to its order
  250. Derivative of Bessel Y with respect to its order
  251. Derivative of Gudermannian
  252. Derivative of Jacobi theta 1 at 0
  253. Derivative of Legendre chi 2
  254. Derivative of Li 2(-1/x)
  255. Derivative of Riemann zeta
  256. Derivative of Struve H0
  257. Derivative of arccos
  258. Derivative of arccosh
  259. Derivative of arccot
  260. Derivative of arccoth
  261. Derivative of arccsc
  262. Derivative of arcsec
  263. Derivative of arcsin
  264. Derivative of arcsinh
  265. Derivative of arctan
  266. Derivative of arctanh
  267. Derivative of cosecant
  268. Derivative of cosh
  269. Derivative of cosine
  270. Derivative of cosine integral
  271. Derivative of cotangent
  272. Derivative of coth
  273. Derivative of erfi
  274. Derivative of hyperbolic cosecant
  275. Derivative of inverse error function
  276. Derivative of prime zeta
  277. Derivative of secant
  278. Derivative of sech
  279. Derivative of sine
  280. Derivative of sine integral
  281. Derivative of sinh
  282. Derivative of tangent
  283. Derivative of tanh
  284. Derivative of the exponential function
  285. Derivative of the logarithm
  286. Derivative of versine
  287. Derivative of zeta at -1
  288. Derivatives of Hypergeometric pFq
  289. Devil's staircase
  290. Devil's staircase is continuous
  291. Devil's staircase is not absolutely continuous
  292. Dickman–de Bruijn function
  293. Dickson polynomial
  294. Difference equation of hypergeometric type
  295. Difference of cosh and sinh
  296. Differential equation for Hypergeometric pFq
  297. Differential equation for Jacobi P
  298. Digamma
  299. Digamma at 1
  300. Digamma at 1/2
  301. Digamma at n+1
  302. Digamma at n+1/2
  303. Digamma at z+n
  304. Digamma functional equation
  305. Dilogarithm
  306. Dirichlet L-function
  307. Dirichlet beta
  308. Dirichlet beta in terms of Lerch transcendent
  309. Dirichlet eta
  310. Dirichlet function
  311. Dirichlet function is nowhere continuous
  312. Dirichlet series
  313. Distance to integers
  314. Doubling identity for cosh (1)
  315. Doubling identity for cosh (2)
  316. Doubling identity for cosh (3)
  317. Doubling identity for sinh (1)
  318. Doubling identity for sinh (2)
  319. Doubly periodic function
  320. E
  321. E(1,1)(z)=exp(z)
  322. E(2,1)(-z^2)=cos(z)
  323. E(2,1)(z)=cosh(sqrt(z))
  324. E(m)=(pi/2)2F1(-1/2,1/2;1;m)
  325. E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936
  326. E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1
  327. E^x greater than (1+x/y)^y greater than exp(xy/(x+y) for x greater than 0 and y greater than 0)
  328. E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0
  329. E^x is greater than 1+x for nonzero real x
  330. E^x is less than 1/(1-x) for nonzero real x less than 1
  331. E (0,1)(z)=1/(1-z) for abs(z) less than 1
  332. E is irrational
  333. E is limit of (1+1/n)^n
  334. Ei(x)=-Integral from -x to infinity of e^(-t)/t dt
  335. Elliptic E
  336. Elliptic K
  337. Elliptic function
  338. Elliptic gamma function
  339. Entire
  340. Entire exponential integral
  341. Epstein zeta function
  342. Erdős-Borwein Constant
  343. Erdős-Borwein Constant is irrational
  344. Erf of conjugate is conjugate of erf
  345. Erfc
  346. Erfi
  347. Error function
  348. Error function is odd
  349. Error functions footer
  350. Euler's formula
  351. Euler-Jackson q-difference operator
  352. Euler-Mascheroni constant
  353. Euler E
  354. Euler E generating function
  355. Euler E n'(x)=nE n-1(x)
  356. Euler numbers
  357. Euler phi
  358. Euler product
  359. Euler product for Riemann zeta
  360. Euler totient
  361. Euler totient is multiplicative
  362. Exponential
  363. Exponential cdf
  364. Exponential e in terms of basic hypergeometric phi
  365. Exponential integral E
  366. Exponential integral Ei
  367. Exponential integral Ei series
  368. Exponential of logarithm
  369. Exponential pdf
  370. Exsecant
  371. Exton q-exponential
  372. F(-n)=(-1)^(n+1)F(n)
  373. F(2n)=F(n)L(n)
  374. F(2n)=F(n+1)^2-F(n-1)^2
  375. F(2n+1)=F(n+1)^2+F(n)^2
  376. F(n+1)F(n-1)-F(n)^2=(-1)^n
  377. F(n+m+1)=F(n+1)F(m+1)+F(n)F(m)
  378. Faber F1
  379. Faber F1 is continuous
  380. Faber F1 is nowhere differentiable
  381. Faber F2
  382. Faber F2 is continuous
  383. Faber F2 is nowhere differentiable
  384. Factorial
  385. Faddeeva function
  386. Falling factorial
  387. Fermat numbers
  388. Fibonacci numbers
  389. Fibonacci polynomial
  390. Fibonacci zeta at 1 is irrational
  391. Fibonacci zeta function
  392. Fibonacci zeta in terms of a sum of binomial coefficients
  393. Floor
  394. Format notes
  395. Fransén–Robinson constant
  396. Fresnel C
  397. Fresnel C in terms of erf
  398. Fresnel C is odd
  399. Fresnel S
  400. Fresnel S in terms of erf
  401. Fresnel S is odd
  402. Full list
  403. Functional equation for Riemann xi
  404. Functional equation for Riemann zeta
  405. Functional equation for Riemann zeta with cosine
  406. Functions named after Carl Gustav Jacob Jacobi
  407. Functions named after Pafnuty Chebyshev
  408. Functions named after Peter Gustav Lejeune Dirichlet
  409. Fundamental pair of periods
  410. Gamma
  411. Gamma'(z)/Gamma(z)=-gamma-1/z+Sum z/(k(z+k))
  412. Gamma(1)=1
  413. Gamma(n+1)=n!
  414. Gamma(z)Gamma(1-z)=pi/sin(pi z)
  415. Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0
  416. Gamma(z+1)=zGamma(z)
  417. Gamma function written as a limit of a factorial, exponential, and a rising factorial
  418. Gamma function written as infinite product
  419. Gamma recurrence relation
  420. Gauss' formula for gamma function
  421. Gegenbauer C
  422. Gelfond-Schneider constant is transcendental
  423. Gelfond constant
  424. Gelfond constant is transcendental
  425. Gelfond–Schneider constant
  426. General Dirichlet series
  427. Generalized q-Bessel
  428. Generating function for Hermite (physicist) polynomials
  429. Generating function for Laguerre L
  430. Generating function for partition function
  431. Generating relation for Bateman F
  432. Genocchi numbers
  433. Gieseking constant
  434. Glaisher–Kinkelin constant
  435. Glyphs
  436. Goh-Schmutz constant
  437. Golden ratio
  438. Gompertz constant
  439. Goss zeta function
  440. Graph
  441. Greatest prime factor
  442. Gudermannian
  443. H (-(n+1/2))(z)=(-1)^n J (n+1/2)(z) for integer n geq 0
  444. H (1/2)(z)=sqrt(2/(pi z))(1-cos(z))
  445. H (3/2)(z)=sqrt(z/(2pi))(1+2/z^2)-sqrt(2/(pi z))(sin(z)+cos(z)/z)
  446. H (nu)(x) geq 0 for x gt 0 and nu geq 1/2
  447. Hacovercosine
  448. Hacoversine
  449. Hadamard gamma
  450. Hahn-Exton q-Bessel
  451. Hahn polynomial
  452. Halving identity for cosh
  453. Halving identity for sinh
  454. Halving identity for tangent (1)
  455. Halving identity for tangent (2)
  456. Halving identity for tangent (3)
  457. Hankel H (1)
  458. Hankel H (1) in terms of csc and Bessel J
  459. Hankel H (2)
  460. Hankel H (2) in terms of csc and Bessel J
  461. Hankel functions footer
  462. Harmonic number
  463. Havercosine
  464. Haversine
  465. Heaviside step function
  466. Hermite (physicist)
  467. Hermite (physicist) polynomial at negative argument
  468. Hermite (probabilist)
  469. Humbert polynomials
  470. Hurwitz zeta
  471. Hurwitz zeta absolute convergence
  472. Hyperbolic trigonometric functions footer
  473. Hyperfactorial
  474. Hyperfactorial in terms of K-function
  475. Hypergeometric 0F0
  476. Hypergeometric 0F1
  477. Hypergeometric 0F3
  478. Hypergeometric 1F0
  479. Hypergeometric 1F1
  480. Hypergeometric 1F2
  481. Hypergeometric 2F0
  482. Hypergeometric 2F1
  483. Hypergeometric 2F3
  484. Hypergeometric 3F2
  485. Hypergeometric 4F1
  486. Hypergeometric functions footer
  487. Hypergeometric pFq
  488. Identity
  489. Identity written as a sum of Möbius functions
  490. Ihara zeta function
  491. Imaginary number
  492. Incomplete Elliptic E
  493. Incomplete Elliptic K
  494. Incomplete beta function
  495. Integer
  496. Integers
  497. Integral (t-b)^(x-1)(a-t)^(y-1)dt=(a-b)^(x+y-1)B(x,y)
  498. Integral from a to a
  499. Integral of (1+bt^z)^(-y)t^x dt = (1/z)*b^(-(x+1)/z) B((x+1)/z,y-(x+1)/z)
  500. Integral of (1+t)^(2x-1)(1-t)^(2y-1)(1+t^2)^(-x-y)dt=2^(x+y-2)B(x,y)

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