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  1. (hist) ‎Absolute convergence of secant zeta function ‎[430 bytes]
  2. (hist) ‎Sum rule for derivatives ‎[431 bytes]
  3. (hist) ‎Hypergeometric 2F0 ‎[431 bytes]
  4. (hist) ‎Jacobi nc ‎[432 bytes]
  5. (hist) ‎Stirling numbers of the second kind ‎[432 bytes]
  6. (hist) ‎Log(x) less than or equal to n(x^(1/n)-1) ‎[432 bytes]
  7. (hist) ‎E (0,1)(z)=1/(1-z) for abs(z) less than 1 ‎[432 bytes]
  8. (hist) ‎Symmetry relation of exponential integral E ‎[433 bytes]
  9. (hist) ‎Relationship between csch and csc ‎[434 bytes]
  10. (hist) ‎B(x,y+1)=(y/x)B(x+1,y) ‎[434 bytes]
  11. (hist) ‎Difference of cosh and sinh ‎[434 bytes]
  12. (hist) ‎Matrix exponential ‎[434 bytes]
  13. (hist) ‎Weierstrass elliptic ‎[435 bytes]
  14. (hist) ‎Legendre chi ‎[436 bytes]
  15. (hist) ‎Dawson D+ ‎[437 bytes]
  16. (hist) ‎L(n)^2-5F(n)^2=4(-1)^n ‎[437 bytes]
  17. (hist) ‎Taylor series ‎[438 bytes]
  18. (hist) ‎Doubling identity for sinh (2) ‎[438 bytes]
  19. (hist) ‎Q-derivative of q-Sine ‎[439 bytes]
  20. (hist) ‎Greatest prime factor ‎[440 bytes]
  21. (hist) ‎Period of cosh ‎[440 bytes]
  22. (hist) ‎Artin-Mazur zeta function ‎[441 bytes]
  23. (hist) ‎Jackson q-Bessel (2) ‎[441 bytes]
  24. (hist) ‎Halving identity for sinh ‎[441 bytes]
  25. (hist) ‎Jackson q-Bessel (1) ‎[442 bytes]
  26. (hist) ‎Product rule for derivatives ‎[442 bytes]
  27. (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]
  28. (hist) ‎Orthogonality of Bateman F on R ‎[442 bytes]
  29. (hist) ‎Relationship between Struve function and hypergeometric pFq ‎[443 bytes]
  30. (hist) ‎Jacobi nd ‎[444 bytes]
  31. (hist) ‎Two-dimensional Laplace transform ‎[444 bytes]
  32. (hist) ‎Unit step function ‎[444 bytes]
  33. (hist) ‎Normal cdf ‎[444 bytes]
  34. (hist) ‎Generating function for Laguerre L ‎[444 bytes]
  35. (hist) ‎Q-difference equation for q-exponential E sub q ‎[447 bytes]
  36. (hist) ‎Paper:D.S. McAnally/q-exponential and q-gamma functions. I. q-exponential functions ‎[447 bytes]
  37. (hist) ‎Limit of quotient of consecutive Fibonacci numbers ‎[448 bytes]
  38. (hist) ‎Arccsch ‎[449 bytes]
  39. (hist) ‎Jacobi ns ‎[449 bytes]
  40. (hist) ‎Reciprocal Riemann zeta in terms of Mobius ‎[449 bytes]
  41. (hist) ‎Q-derivative of q-Cosine ‎[450 bytes]
  42. (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]
  43. (hist) ‎K(m)=(pi/2)2F1(1/2,1/2;1;m) ‎[452 bytes]
  44. (hist) ‎Buchstab function ‎[453 bytes]
  45. (hist) ‎E(m)=(pi/2)2F1(-1/2,1/2;1;m) ‎[453 bytes]
  46. (hist) ‎Glyphs ‎[454 bytes]
  47. (hist) ‎Relationship between Li 2(1),Li 2(-1), and pi ‎[454 bytes]
  48. (hist) ‎Limit of q-exponential E sub 1/q for 0 less than q less than 1 ‎[454 bytes]
  49. (hist) ‎Hypergeometric 2F3 ‎[454 bytes]
  50. (hist) ‎Catalan's constant ‎[455 bytes]
  51. (hist) ‎Relationship between Bessel J sub n and Bessel J sub -n ‎[455 bytes]
  52. (hist) ‎Relationship between Anger function and Weber function ‎[456 bytes]
  53. (hist) ‎E^x is greater than 1+x for nonzero real x ‎[456 bytes]
  54. (hist) ‎Sqrt(1-z^2)2F1(1,1;3/2;z^2)=arcsin(z)/z ‎[456 bytes]
  55. (hist) ‎Period of sinh ‎[456 bytes]
  56. (hist) ‎Pythagorean identity for coth and csch ‎[456 bytes]
  57. (hist) ‎T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n ‎[456 bytes]
  58. (hist) ‎Floor ‎[457 bytes]
  59. (hist) ‎Antiderivative of the logarithm ‎[458 bytes]
  60. (hist) ‎Series for polygamma in terms of Riemann zeta ‎[459 bytes]
  61. (hist) ‎Paper:V.E. Hoggatt, Jr/Triangular numbers ‎[460 bytes]
  62. (hist) ‎B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt ‎[460 bytes]
  63. (hist) ‎Digamma at 1/2 ‎[461 bytes]
  64. (hist) ‎K-function ‎[461 bytes]
  65. (hist) ‎Riemann-Siegel Z ‎[462 bytes]
  66. (hist) ‎Gauss' formula for gamma function ‎[462 bytes]
  67. (hist) ‎Halving identity for tangent (1) ‎[462 bytes]
  68. (hist) ‎Bessel polynomial ‎[464 bytes]
  69. (hist) ‎Weierstrass elementary factors ‎[464 bytes]
  70. (hist) ‎Polygamma recurrence relation ‎[464 bytes]
  71. (hist) ‎Period of tanh ‎[464 bytes]
  72. (hist) ‎(c-a-b)2F1-(c-a)2F1(a-1)+b(1-z)2F1(b+1)=0 ‎[464 bytes]
  73. (hist) ‎H (-(n+1/2))(z)=(-1)^n J (n+1/2)(z) for integer n geq 0 ‎[464 bytes]
  74. (hist) ‎E^x is less than 1/(1-x) for nonzero real x less than 1 ‎[465 bytes]
  75. (hist) ‎Spherical Bessel y ‎[466 bytes]
  76. (hist) ‎Relationship between sine, imaginary number, logarithm, and the golden ratio ‎[467 bytes]
  77. (hist) ‎Airy zeta function ‎[468 bytes]
  78. (hist) ‎F(-n)=(-1)^(n+1)F(n) ‎[469 bytes]
  79. (hist) ‎Orthogonality of Laguerre L ‎[469 bytes]
  80. (hist) ‎Sum of cosh and sinh ‎[470 bytes]
  81. (hist) ‎Series for log(z) for absolute value of (z-1) less than 1 ‎[471 bytes]
  82. (hist) ‎Clausen sine ‎[471 bytes]
  83. (hist) ‎Taylor series of log(1+z) ‎[472 bytes]
  84. (hist) ‎Digamma at z+n ‎[472 bytes]
  85. (hist) ‎Integral of Bessel J for nu=2n+1 ‎[472 bytes]
  86. (hist) ‎(c-2a-(b-a)z)2F1+a(1-z)2F1(a+1)-(c-a)2F1(a-1)=0 ‎[472 bytes]
  87. (hist) ‎Weierstrass nowhere differentiable function ‎[473 bytes]
  88. (hist) ‎Value of derivative of trigamma at positive integer plus 1/2 ‎[473 bytes]
  89. (hist) ‎Generating relation for Bateman F ‎[473 bytes]
  90. (hist) ‎Euler phi ‎[474 bytes]
  91. (hist) ‎Limit of log(x)/x^a=0 ‎[474 bytes]
  92. (hist) ‎Bessel at n+1/2 in terms of Bessel polynomial ‎[475 bytes]
  93. (hist) ‎Beta as product of gamma functions ‎[475 bytes]
  94. (hist) ‎X/(1+x) less than 1-e^(-x) less than x for nonzero real x greater than -1 ‎[476 bytes]
  95. (hist) ‎B(x,y)B(x+y,z)=B(z,x)B(x+z,y) ‎[476 bytes]
  96. (hist) ‎Faddeeva function ‎[478 bytes]
  97. (hist) ‎Pythagorean identity for tanh and sech ‎[478 bytes]
  98. (hist) ‎H (nu)(x) geq 0 for x gt 0 and nu geq 1/2 ‎[478 bytes]
  99. (hist) ‎E(2,1)(z)=cosh(sqrt(z)) ‎[478 bytes]
  100. (hist) ‎Q-Sin ‎[479 bytes]
  101. (hist) ‎Jinc ‎[479 bytes]
  102. (hist) ‎E(2,1)(-z^2)=cos(z) ‎[479 bytes]
  103. (hist) ‎Spherical Bessel j ‎[480 bytes]
  104. (hist) ‎Contour integral representation of reciprocal gamma ‎[480 bytes]
  105. (hist) ‎Sum of sum of divisors function equals product of Riemann zeta for Re(z) greater than k+1 ‎[481 bytes]
  106. (hist) ‎2F1(1,1;2;z)=-log(1-z)/z ‎[481 bytes]
  107. (hist) ‎E(1,1)(z)=exp(z) ‎[481 bytes]
  108. (hist) ‎Value of polygamma at 1 ‎[482 bytes]
  109. (hist) ‎Q-difference equation for q-exponential E sub 1/q ‎[482 bytes]
  110. (hist) ‎Gamma'(z)/Gamma(z)=-gamma-1/z+Sum z/(k(z+k)) ‎[482 bytes]
  111. (hist) ‎Pochhammer symbol with non-negative integer subscript ‎[483 bytes]
  112. (hist) ‎1Phi0(a;;z)1Phi0(b;;az)=1Phi0(ab;;z) ‎[483 bytes]
  113. (hist) ‎D/dz(z^(nu)H (nu))=z^(nu)H (nu-1) ‎[484 bytes]
  114. (hist) ‎X less than e^x-1 less than x/(1-x) for nonzero real x less than 1 ‎[486 bytes]
  115. (hist) ‎Logarithm diverges to negative infinity at 0 from right ‎[487 bytes]
  116. (hist) ‎Identity written as a sum of Möbius functions ‎[487 bytes]
  117. (hist) ‎Q-Hurwitz zeta ‎[487 bytes]
  118. (hist) ‎Integral representation of polygamma for Re(z) greater than 0 ‎[488 bytes]
  119. (hist) ‎Real and imaginary parts of log ‎[488 bytes]
  120. (hist) ‎Dirichlet beta ‎[489 bytes]
  121. (hist) ‎Series for log(z) for Re(z) greater than 1/2 ‎[489 bytes]
  122. (hist) ‎Logarithm (multivalued) of the exponential ‎[489 bytes]
  123. (hist) ‎Integral of Bessel J for nu=n+1 ‎[489 bytes]
  124. (hist) ‎Continuous ‎[490 bytes]
  125. (hist) ‎Orthogonality of Gegenbauer C on (-1,1) ‎[490 bytes]
  126. (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]
  127. (hist) ‎Jacobi sd ‎[493 bytes]
  128. (hist) ‎Jacobi cs ‎[493 bytes]
  129. (hist) ‎Jacobi dc ‎[493 bytes]
  130. (hist) ‎McCarthy function ‎[493 bytes]
  131. (hist) ‎Chain rule for derivatives ‎[493 bytes]
  132. (hist) ‎Product representation of totient ‎[493 bytes]
  133. (hist) ‎1Phi0(a;;z) as infinite product ‎[493 bytes]
  134. (hist) ‎Relationship between Weber function and Anger function ‎[494 bytes]
  135. (hist) ‎Polygamma reflection formula ‎[495 bytes]
  136. (hist) ‎Book:Sir Thomas L. Heath/Euclid: The Thirteen Books of The Elements: Volume 2/Second Edition ‎[495 bytes]
  137. (hist) ‎Sum of totient equals z/((1-z) squared) ‎[495 bytes]
  138. (hist) ‎2F1(1/2,1;3/2;z^2)=log((1+z)/(1-z))/(2z) ‎[495 bytes]
  139. (hist) ‎Inverse error function ‎[496 bytes]
  140. (hist) ‎Generating function for Hermite (physicist) polynomials ‎[497 bytes]
  141. (hist) ‎Q-Cos ‎[498 bytes]
  142. (hist) ‎Q-Gaussian distribution ‎[498 bytes]
  143. (hist) ‎2F1(1/2,1;3/2;-z^2)=arctan(z)/z ‎[499 bytes]
  144. (hist) ‎Laplace cdf ‎[499 bytes]
  145. (hist) ‎Jacobi sc ‎[505 bytes]
  146. (hist) ‎Jacobi ds ‎[505 bytes]
  147. (hist) ‎Jacobi cd ‎[505 bytes]
  148. (hist) ‎2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z ‎[505 bytes]
  149. (hist) ‎Integral of Bessel J for Re(nu) greater than -1 ‎[505 bytes]
  150. (hist) ‎(b-a)2F1+a2F1(a+1)-b2F1(b+1)=0 ‎[505 bytes]
  151. (hist) ‎Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1 ‎[506 bytes]
  152. (hist) ‎Relationship between Li 2(-1/x),Li 2(-x),Li 2(-1), and log^2(x) ‎[508 bytes]
  153. (hist) ‎Functional equation for Riemann zeta with cosine ‎[508 bytes]
  154. (hist) ‎Halving identity for tangent (3) ‎[508 bytes]
  155. (hist) ‎Halving identity for tangent (2) ‎[510 bytes]
  156. (hist) ‎Bernoulli numbers ‎[511 bytes]
  157. (hist) ‎(c-a-b)2F1+a(1-z)2F1(a+1)-(c-b)2F1(b-1)=0 ‎[511 bytes]
  158. (hist) ‎1+x greater than exp(x/(1+x)) for nonzero real x greater than -1 ‎[512 bytes]
  159. (hist) ‎Q-cos sub q ‎[513 bytes]
  160. (hist) ‎Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1 ‎[513 bytes]
  161. (hist) ‎Q-zeta ‎[514 bytes]
  162. (hist) ‎E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936 ‎[515 bytes]
  163. (hist) ‎Series for log(Riemann zeta) in terms of Mangoldt function ‎[515 bytes]
  164. (hist) ‎Z coth(z) = 2z/(e^(2z)-1) + z ‎[515 bytes]
  165. (hist) ‎Sum over bottom of binomial coefficient with top fixed equals 2^n ‎[516 bytes]
  166. (hist) ‎C(a-(c-b)z)2F1-ac(1-z)2F1(a+1)+(c-a)(c-b)z2F1(c+1)=0 ‎[516 bytes]
  167. (hist) ‎Log(1+z) as continued fraction ‎[516 bytes]
  168. (hist) ‎Li 2(z)+Li 2(1-z)=pi^2/6-log(z)log(1-z) ‎[516 bytes]
  169. (hist) ‎Beta as improper integral ‎[517 bytes]
  170. (hist) ‎(c-a-1)2F1+a2F1(a+1)-(c-1)2F1(c-1)=0 ‎[517 bytes]
  171. (hist) ‎Paper:Lucas Jódar/On the hypergeometric matrix function ‎[517 bytes]
  172. (hist) ‎E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0 ‎[518 bytes]
  173. (hist) ‎Sum of totient equals zeta(z-1)/zeta(z) for Re(z) greater than 2 ‎[518 bytes]
  174. (hist) ‎Bernoulli B ‎[519 bytes]
  175. (hist) ‎Euler E generating function ‎[519 bytes]
  176. (hist) ‎Constant multiple rule for derivatives ‎[519 bytes]
  177. (hist) ‎Derivative of Gudermannian ‎[521 bytes]
  178. (hist) ‎Clausen cosine ‎[521 bytes]
  179. (hist) ‎Integral of Bessel J for nu=2n ‎[522 bytes]
  180. (hist) ‎Euler numbers ‎[522 bytes]
  181. (hist) ‎Faber F2 ‎[523 bytes]
  182. (hist) ‎Digamma at n+1/2 ‎[523 bytes]
  183. (hist) ‎Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x ‎[524 bytes]
  184. (hist) ‎Dirichlet function ‎[525 bytes]
  185. (hist) ‎D/dz(z^(-nu)H (nu))=1/(sqrt(pi)2^(nu)Gamma(nu+3/2))-z^(-nu)H (nu+1) ‎[525 bytes]
  186. (hist) ‎Paper:H.J. Haubold/Mittag-Leffler Functions and Their Applications ‎[525 bytes]
  187. (hist) ‎U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n ‎[525 bytes]
  188. (hist) ‎Arcsec ‎[526 bytes]
  189. (hist) ‎Faber F1 ‎[526 bytes]
  190. (hist) ‎Quotient rule for derivatives ‎[526 bytes]
  191. (hist) ‎Modular form ‎[528 bytes]
  192. (hist) ‎Recurrence relation for partition function with sum of divisors ‎[530 bytes]
  193. (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]
  194. (hist) ‎Chebyshev theta function ‎[531 bytes]
  195. (hist) ‎Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4 ‎[531 bytes]
  196. (hist) ‎Series for log(riemann zeta) over primes ‎[532 bytes]
  197. (hist) ‎Asymptotic formula for partition function ‎[533 bytes]
  198. (hist) ‎Value of polygamma at positive integer ‎[534 bytes]
  199. (hist) ‎Product representation of q-exponential E sub 1/q ‎[535 bytes]
  200. (hist) ‎Functional equation for Riemann zeta ‎[536 bytes]
  201. (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]
  202. (hist) ‎Dickson polynomial ‎[538 bytes]
  203. (hist) ‎Z/(1-sqrt(q))2Phi1(q,sqrt(q);sqrt(q^3);z)=Sum z^k/(1-q^(k-1/2)) ‎[538 bytes]
  204. (hist) ‎Li2(z)=zPhi(z,2,1) ‎[538 bytes]
  205. (hist) ‎2Phi1(q,-1;-q;z)=1+2Sum z^k/(1+q^k) ‎[539 bytes]
  206. (hist) ‎Series for log(z) for Re(z) greater than 0 ‎[540 bytes]
  207. (hist) ‎Q-exponential e sub q ‎[541 bytes]
  208. (hist) ‎Generating function for partition function ‎[541 bytes]
  209. (hist) ‎Relationship between logarithmic integral and exponential integral ‎[542 bytes]
  210. (hist) ‎Erfc ‎[542 bytes]
  211. (hist) ‎Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3 ‎[542 bytes]
  212. (hist) ‎H (1/2)(z)=sqrt(2/(pi z))(1-cos(z)) ‎[543 bytes]
  213. (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]
  214. (hist) ‎Digamma functional equation ‎[546 bytes]
  215. (hist) ‎Logarithmic derivative of Riemann zeta in terms of Mangoldt function ‎[546 bytes]
  216. (hist) ‎Riemann Siegel theta function ‎[548 bytes]
  217. (hist) ‎E is irrational ‎[548 bytes]
  218. (hist) ‎Arccsc ‎[549 bytes]
  219. (hist) ‎Recurrence relation for Struve function (2) ‎[549 bytes]
  220. (hist) ‎Exsecant ‎[550 bytes]
  221. (hist) ‎Binet's formula ‎[551 bytes]
  222. (hist) ‎Mertens ‎[555 bytes]
  223. (hist) ‎Genocchi numbers ‎[555 bytes]
  224. (hist) ‎Taylor series of log(1-z) ‎[555 bytes]
  225. (hist) ‎Elliptic E ‎[556 bytes]
  226. (hist) ‎Log((1+z)/(1-z)) as continued fraction ‎[556 bytes]
  227. (hist) ‎Q-sin sub q ‎[557 bytes]
  228. (hist) ‎Bernstein B ‎[558 bytes]
  229. (hist) ‎Sum of divisors ‎[559 bytes]
  230. (hist) ‎Knopp function ‎[559 bytes]
  231. (hist) ‎Bessel-Clifford ‎[559 bytes]
  232. (hist) ‎Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0 ‎[559 bytes]
  233. (hist) ‎1/B(n,m)=m((n+m-1) choose (n-1)) ‎[560 bytes]
  234. (hist) ‎1/B(n,m)=n((n+m-1) choose (m-1)) ‎[561 bytes]
  235. (hist) ‎Relationship between sinh and sin ‎[563 bytes]
  236. (hist) ‎H (3/2)(z)=sqrt(z/(2pi))(1+2/z^2)-sqrt(2/(pi z))(sin(z)+cos(z)/z) ‎[563 bytes]
  237. (hist) ‎Dedekind eta ‎[565 bytes]
  238. (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]
  239. (hist) ‎T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k) ‎[566 bytes]
  240. (hist) ‎Logarithm of product is a sum of logarithms ‎[567 bytes]
  241. (hist) ‎Hypergeometric 1F1 ‎[568 bytes]
  242. (hist) ‎Closed formula for physicist's Hermite polynomials ‎[568 bytes]
  243. (hist) ‎Asymptotic behavior of Sievert integral ‎[568 bytes]
  244. (hist) ‎(z/(1-q))2Phi1(q,q;q^2;z)=Sum z^k/(1-q^k) ‎[568 bytes]
  245. (hist) ‎Value of polygamma at 1/2 ‎[574 bytes]
  246. (hist) ‎Derivative of the logarithm ‎[574 bytes]
  247. (hist) ‎Riemann zeta at even integers ‎[574 bytes]
  248. (hist) ‎Hurwitz zeta ‎[575 bytes]
  249. (hist) ‎Alternating sum over bottom of binomial coefficient with top fixed equals 0 ‎[578 bytes]
  250. (hist) ‎Digamma at 1 ‎[578 bytes]

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