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Special functions
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266
Table of contents
266
1. Elementary special functions
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Special functions
14
1.1. Exponentials
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Elementary special functions
1.2. Hyperbolic functions
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Elementary special functions
1.3. Inverse hyperbolic functions
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Elementary special functions
1.4. Logarithms
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Elementary special functions
1.5. Trigonometry
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Elementary special functions
1.6. Dirichlet function
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Elementary special functions
1.7. Double exponential function
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Elementary special functions
1.8. Exponential function
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Elementary special functions
1.9. Gudermannian function
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Elementary special functions
1.10. Kronecker delta
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Elementary special functions
1.11. Logarithm
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Elementary special functions
1.12. Ptolemy's table of chords
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Elementary special functions
1.13. Sigmoid function
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Elementary special functions
1.14. Soboleva modified hyperbolic tangent
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Elementary special functions
2. Elliptic functions
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Special functions
26
2.1. Elliptic curves
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Elliptic functions
2.2. Inverse Jacobi elliptic functions
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Elliptic functions
2.3. Inverse lemniscate functions
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Elliptic functions
2.4. Abel elliptic functions
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Elliptic functions
2.5. Carlson symmetric form
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Elliptic functions
2.6. Complex multiplication
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Elliptic functions
2.7. Dedekind eta function
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Elliptic functions
2.8. Dixon elliptic functions
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Elliptic functions
2.9. Elliptic function
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Elliptic functions
2.10. Elliptic integral
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Elliptic functions
2.11. Equianharmonic
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Elliptic functions
2.12. Fundamenta nova theoriae functionum ellipticarum
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Elliptic functions
2.13. Fundamental pair of periods
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Elliptic functions
2.14. Half-period ratio
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Elliptic functions
2.15. J-invariant
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Elliptic functions
2.16. Jacobi theta functions (notational variations)
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Elliptic functions
2.17. Landen's transformation
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Elliptic functions
2.18. Legendre's relation
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Elliptic functions
2.19. Lemniscate elliptic functions
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Elliptic functions
2.20. Modular lambda function
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Elliptic functions
2.21. Nome (mathematics)
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Elliptic functions
2.22. Picard–Fuchs equation
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Elliptic functions
2.23. Quarter period
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Elliptic functions
2.24. Theta function
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Elliptic functions
2.25. Weierstrass elliptic function
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Elliptic functions
2.26. Weierstrass functions
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Elliptic functions
3. Hypergeometric functions
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Special functions
17
3.1. Appell series
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Hypergeometric functions
3.2. Bilateral hypergeometric series
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Hypergeometric functions
3.3. Dougall's formula
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Hypergeometric functions
3.4. Elliptic hypergeometric series
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Hypergeometric functions
3.5. Frobenius solution to the hypergeometric equation
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Hypergeometric functions
3.6. General hypergeometric function
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Hypergeometric functions
3.7. Horn function
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Hypergeometric functions
3.8. Humbert series
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Hypergeometric functions
3.9. Hypergeometric function of a matrix argument
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Hypergeometric functions
3.10. Kampé de Fériet function
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Hypergeometric functions
3.11. Lauricella hypergeometric series
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Hypergeometric functions
3.12. Legendre function
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Hypergeometric functions
3.13. List of hypergeometric identities
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Hypergeometric functions
3.14. MacRobert E function
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Hypergeometric functions
3.15. Meijer G-function
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Hypergeometric functions
3.16. Riemann's differential equation
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Hypergeometric functions
3.17. Schwarz's list
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Hypergeometric functions
4. Special hypergeometric functions
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Special functions
15
4.1. Bessel function
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Special hypergeometric functions
4.2. Bessel–Clifford function
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Special hypergeometric functions
4.3. Coulomb wave function
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Special hypergeometric functions
4.4. Cunningham function
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Special hypergeometric functions
4.5. Error function
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Special hypergeometric functions
4.6. Incomplete Bessel functions
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Special hypergeometric functions
4.7. Kelvin functions
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Special hypergeometric functions
4.8. Lentz's algorithm
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Special hypergeometric functions
4.9. Logarithmic integral function
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Special hypergeometric functions
4.10. Solid harmonics
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Special hypergeometric functions
4.11. Sonine formula
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Special hypergeometric functions
4.12. Spherical harmonics
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Special hypergeometric functions
4.13. Table of spherical harmonics
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Special hypergeometric functions
4.14. Toronto function
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Special hypergeometric functions
4.15. Zonal spherical harmonics
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Special hypergeometric functions
5. Theta functions
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Special functions
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5.1. Jacobi form
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Theta functions
5.2. Metaplectic group
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Theta functions
5.3. Schottky problem
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Theta functions
5.4. Theta characteristic
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Theta functions
5.5. Theta divisor
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Theta functions
5.6. Theta function of a lattice
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Theta functions
6. Zeta and L-functions
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Special functions
87
6.1. Airy zeta function
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Zeta and L-functions
6.2. Apéry's theorem
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Zeta and L-functions
6.3. Arakawa–Kaneko zeta function
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Zeta and L-functions
6.4. Arithmetic zeta function
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Zeta and L-functions
6.5. Artin L-function
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Zeta and L-functions
6.6. Artin conductor
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Zeta and L-functions
6.7. Artin–Mazur zeta function
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Zeta and L-functions
6.8. Automorphic L-function
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Zeta and L-functions
6.9. Barnes zeta function
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Zeta and L-functions
6.10. Basel problem
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Zeta and L-functions
6.11. Beurling zeta function
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Zeta and L-functions
6.12. Birch and Swinnerton-Dyer conjecture
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Zeta and L-functions
6.13. Brumer–Stark conjecture
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Zeta and L-functions
6.14. Chowla–Mordell theorem
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Zeta and L-functions
6.15. Clausen function
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Zeta and L-functions
6.16. Dedekind zeta function
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Zeta and L-functions
6.17. Dirichlet's theorem on arithmetic progressions
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Zeta and L-functions
6.18. Dirichlet L-function
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Zeta and L-functions
6.19. Dirichlet beta function
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Zeta and L-functions
6.20. Dirichlet character
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Zeta and L-functions
6.21. Dirichlet eta function
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Zeta and L-functions
6.22. Divisor function
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Zeta and L-functions
6.23. Dwork conjecture
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Zeta and L-functions
6.24. Eichler–Shimura congruence relation
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Zeta and L-functions
6.25. Equivariant L-function
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Zeta and L-functions
6.26. Euler product
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Zeta and L-functions
6.27. Explicit formulae for L-functions
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Zeta and L-functions
6.28. Feller–Tornier constant
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Zeta and L-functions
6.29. Functional equation (L-function)
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Zeta and L-functions
6.30. Gan–Gross–Prasad conjecture
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Zeta and L-functions
6.31. Generalized Riemann hypothesis
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Zeta and L-functions
6.32. Goss zeta function
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Zeta and L-functions
6.33. Grand Riemann hypothesis
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Zeta and L-functions
6.34. Hadjicostas's formula
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Zeta and L-functions
6.35. Hardy–Littlewood zeta-function conjectures
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Zeta and L-functions
6.36. Hasse–Weil zeta function
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Zeta and L-functions
6.37. Hecke character
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Zeta and L-functions
6.38. Hideo Shimizu
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Zeta and L-functions
6.39. Hilbert–Pólya conjecture
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Zeta and L-functions
6.40. Hurwitz zeta function
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Zeta and L-functions
6.41. Igusa zeta function
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Zeta and L-functions
6.42. L-function
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Zeta and L-functions
6.43. Langlands program
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Zeta and L-functions
6.44. Langlands–Deligne local constant
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Zeta and L-functions
6.45. Lefschetz zeta function
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Zeta and L-functions
6.46. Lerch zeta function
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Zeta and L-functions
6.47. Li's criterion
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Zeta and L-functions
6.48. Lindelöf hypothesis
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Zeta and L-functions
6.49. List of zeta functions
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Zeta and L-functions
6.50. Local Langlands conjectures
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Zeta and L-functions
6.51. Local zeta function
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Zeta and L-functions
6.52. Matsumoto zeta function
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Zeta and L-functions
6.53. Montgomery's pair correlation conjecture
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Zeta and L-functions
6.54. Motivic L-function
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Zeta and L-functions
6.55. Multiple zeta function
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Zeta and L-functions
6.56. P-adic L-function
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Zeta and L-functions
6.57. Prime zeta function
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Zeta and L-functions
6.58. Proof of the Euler product formula for the Riemann zeta function
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Zeta and L-functions
6.59. Ramanujan tau function
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Zeta and L-functions
6.60. Ramanujan–Petersson conjecture
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Zeta and L-functions
6.61. Rankin–Selberg method
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Zeta and L-functions
6.62. Riemann Xi function
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Zeta and L-functions
6.63. Riemann hypothesis
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Zeta and L-functions
6.64. Riemann zeta function
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Zeta and L-functions
6.65. Riemann–Siegel formula
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Zeta and L-functions
6.66. Riemann–Siegel theta function
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Zeta and L-functions
6.67. Riesz function
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Zeta and L-functions
6.68. Ruelle zeta function
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Zeta and L-functions
6.69. Selberg's zeta function conjecture
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Zeta and L-functions
6.70. Selberg class
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Zeta and L-functions
6.71. Shimizu L-function
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Zeta and L-functions
6.72. Shimura variety
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Zeta and L-functions
6.73. Shintani zeta function
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Zeta and L-functions
6.74. Siegel zero
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Zeta and L-functions
6.75. Special values of L-functions
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Zeta and L-functions
6.76. Standard L-function
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Zeta and L-functions
6.77. Stieltjes constants
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Zeta and L-functions
6.78. Subgroup growth
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Zeta and L-functions
6.79. Tate's thesis
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Zeta and L-functions
6.80. Turing's method
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Zeta and L-functions
6.81. Waldspurger's theorem
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Zeta and L-functions
6.82. Weil's criterion
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Zeta and L-functions
6.83. Weil conjectures
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Zeta and L-functions
6.84. Z function
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Zeta and L-functions
6.85. ZetaGrid
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Zeta and L-functions
6.86. Zeta function (operator)
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Zeta and L-functions
6.87. Zeta function universality
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Zeta and L-functions
7. Airy function
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Special functions
8. Anger function
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Special functions
9. Baer function
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Special functions
10. Barnes integral
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Special functions
11. Bateman Manuscript Project
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Special functions
12. Bateman function
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Special functions
13. Bickley–Naylor functions
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Special functions
14. Bounded type (mathematics)
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Special functions
15. Boxcar function
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Special functions
16. Buchstab function
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Special functions
17. Böhmer integral
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Special functions
18. Cantor function
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Special functions
19. Carotid–Kundalini function
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Special functions
20. Chandrasekhar's H-function
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Special functions
21. Chandrasekhar's X- and Y-function
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Special functions
22. Chapman function
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Special functions
23. Clausen's formula
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Special functions
24. Complete Fermi–Dirac integral
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Special functions
25. Confluent hypergeometric function
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Special functions
26. Conical function
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Special functions
27. Coshc function
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Special functions
28. Crenel function
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Special functions
29. Dawson function
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Special functions
30. Debye function
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Special functions
31. Dickman function
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Special functions
32. Einstein function
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Special functions
33. Entire function
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Special functions
34. Exponential integral
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Special functions
35. Ferrers function
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Special functions
36. Fox H-function
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Special functions
37. Fresnel integral
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Special functions
38. Goodwin–Staton integral
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Special functions
39. Griewank function
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Special functions
40. Hankel contour
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Special functions
41. Harish-Chandra's Ξ function
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Special functions
42. Heaviside step function
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Special functions
43. Herglotz–Zagier function
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Special functions
44. Heun function
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Special functions
45. Holonomic function
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Special functions
46. Hough function
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Special functions
47. Hyperbolastic functions
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Special functions
48. Incomplete Bessel K function/generalized incomplete gamma function
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Special functions
49. Incomplete Fermi–Dirac integral
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Special functions
50. Incomplete polylogarithm
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Special functions
51. Inverse tangent integral
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Special functions
52. Jacobi elliptic functions
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Special functions
53. Jacobi zeta function
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Special functions
54. Kontorovich–Lebedev transform
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Special functions
55. Kummer's function
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Special functions
56. Lambert W function
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Special functions
57. Lamé function
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Special functions
58. Legendre chi function
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Special functions
59. Legendre form
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Special functions
60. List of eponyms of special functions
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Special functions
61. Logit
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Special functions
62. Lommel function
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Special functions
63. Mathieu function
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Special functions
64. Mayer f-function
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Special functions
65. Minkowski's question-mark function
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Special functions
66. Mittag-Leffler function
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Special functions
67. Modular form
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Special functions
68. Neville theta functions
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Special functions
69. Oblate spheroidal wave function
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Special functions
70. Painlevé transcendents
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Special functions
71. Parabolic cylinder function
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Special functions
72. Pochhammer contour
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Special functions
73. Prolate spheroidal wave function
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Special functions
74. Q-function
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Special functions
75. Ramp function
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Special functions
76. Real analytic Eisenstein series
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Special functions
77. Rectangular function
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Special functions
78. Ruler function
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Special functions
79. Scorer's function
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Special functions
80. Selberg integral
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Special functions
81. Sinhc function
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Special functions
82. Spence's function
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Special functions
83. Spin-weighted spherical harmonics
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Special functions
84. Step function
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Special functions
85. Struve function
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Special functions
86. Strömgren integral
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Special functions
87. Student's t-distribution
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Special functions
88. Synchrotron function
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Special functions
89. Tak (function)
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Special functions
90. Tanc function
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Special functions
91. Tanhc function
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Special functions
92. Thomae's function
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Special functions
93. Tracy–Widom distribution
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Special functions
94. Transcendental function
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Special functions
95. Transport function
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Special functions
96. Triangular function
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Special functions
97. Trigonometric integral
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Special functions
98. Voigt profile
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Special functions
99. Walsh function
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Special functions
100. Whipple formulae
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Special functions
101. Wright omega function
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Special functions
1. Elementary special functions
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Special functions
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1.1. Exponentials
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Elementary special functions
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1.2. Hyperbolic functions
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Elementary special functions
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1.3. Inverse hyperbolic functions
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Elementary special functions
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1.4. Logarithms
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Elementary special functions
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1.5. Trigonometry
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Elementary special functions
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1.6. Dirichlet function
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Elementary special functions
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1.7. Double exponential function
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Elementary special functions
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1.8. Exponential function
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Elementary special functions
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1.9. Gudermannian function
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Elementary special functions
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1.10. Kronecker delta
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Elementary special functions
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1.11. Logarithm
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Elementary special functions
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1.12. Ptolemy's table of chords
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Elementary special functions
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1.13. Sigmoid function
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Elementary special functions
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1.14. Soboleva modified hyperbolic tangent
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Elementary special functions
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2. Elliptic functions
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Special functions
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2.1. Elliptic curves
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Elliptic functions
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2.2. Inverse Jacobi elliptic functions
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Elliptic functions
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2.3. Inverse lemniscate functions
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Elliptic functions
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2.4. Abel elliptic functions
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Elliptic functions
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2.5. Carlson symmetric form
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Elliptic functions
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2.6. Complex multiplication
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Elliptic functions
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2.7. Dedekind eta function
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Elliptic functions
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2.8. Dixon elliptic functions
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Elliptic functions
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2.9. Elliptic function
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Elliptic functions
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2.10. Elliptic integral
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Elliptic functions
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2.11. Equianharmonic
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Elliptic functions
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2.12. Fundamenta nova theoriae functionum ellipticarum
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Elliptic functions
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2.13. Fundamental pair of periods
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Elliptic functions
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2.14. Half-period ratio
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Elliptic functions
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2.15. J-invariant
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Elliptic functions
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2.16. Jacobi theta functions (notational variations)
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Elliptic functions
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2.17. Landen's transformation
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Elliptic functions
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2.18. Legendre's relation
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Elliptic functions
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2.19. Lemniscate elliptic functions
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Elliptic functions
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2.20. Modular lambda function
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Elliptic functions
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2.21. Nome (mathematics)
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Elliptic functions
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2.22. Picard–Fuchs equation
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Elliptic functions
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2.23. Quarter period
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Elliptic functions
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2.24. Theta function
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Elliptic functions
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2.25. Weierstrass elliptic function
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Elliptic functions
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2.26. Weierstrass functions
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Elliptic functions
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3. Hypergeometric functions
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Special functions
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3.1. Appell series
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Hypergeometric functions
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3.2. Bilateral hypergeometric series
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Hypergeometric functions
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3.3. Dougall's formula
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Hypergeometric functions
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3.4. Elliptic hypergeometric series
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Hypergeometric functions
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3.5. Frobenius solution to the hypergeometric equation
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Hypergeometric functions
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3.6. General hypergeometric function
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Hypergeometric functions
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3.7. Horn function
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Hypergeometric functions
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3.8. Humbert series
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Hypergeometric functions
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3.9. Hypergeometric function of a matrix argument
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Hypergeometric functions
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3.10. Kampé de Fériet function
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Hypergeometric functions
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3.11. Lauricella hypergeometric series
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Hypergeometric functions
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3.12. Legendre function
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Hypergeometric functions
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3.13. List of hypergeometric identities
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Hypergeometric functions
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3.14. MacRobert E function
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Hypergeometric functions
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3.15. Meijer G-function
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Hypergeometric functions
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3.16. Riemann's differential equation
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Hypergeometric functions
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3.17. Schwarz's list
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Hypergeometric functions
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4. Special hypergeometric functions
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Special functions
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4.1. Bessel function
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Special hypergeometric functions
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4.2. Bessel–Clifford function
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Special hypergeometric functions
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4.3. Coulomb wave function
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Special hypergeometric functions
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4.4. Cunningham function
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Special hypergeometric functions
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4.5. Error function
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Special hypergeometric functions
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4.6. Incomplete Bessel functions
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Special hypergeometric functions
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4.7. Kelvin functions
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Special hypergeometric functions
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4.8. Lentz's algorithm
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Special hypergeometric functions
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4.9. Logarithmic integral function
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Special hypergeometric functions
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4.10. Solid harmonics
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Special hypergeometric functions
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4.11. Sonine formula
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Special hypergeometric functions
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4.12. Spherical harmonics
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Special hypergeometric functions
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4.13. Table of spherical harmonics
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Special hypergeometric functions
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4.14. Toronto function
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Special hypergeometric functions
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4.15. Zonal spherical harmonics
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Special hypergeometric functions
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5. Theta functions
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Special functions
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5.1. Jacobi form
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Theta functions
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5.2. Metaplectic group
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Theta functions
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5.3. Schottky problem
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Theta functions
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5.4. Theta characteristic
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Theta functions
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5.5. Theta divisor
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Theta functions
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5.6. Theta function of a lattice
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Theta functions
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6. Zeta and L-functions
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87
Special functions
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wiki
6.1. Airy zeta function
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Zeta and L-functions
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6.2. Apéry's theorem
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Zeta and L-functions
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6.3. Arakawa–Kaneko zeta function
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Zeta and L-functions
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6.4. Arithmetic zeta function
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Zeta and L-functions
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6.5. Artin L-function
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Zeta and L-functions
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6.6. Artin conductor
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Zeta and L-functions
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6.7. Artin–Mazur zeta function
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Zeta and L-functions
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6.8. Automorphic L-function
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6.9. Barnes zeta function
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6.10. Basel problem
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6.11. Beurling zeta function
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6.12. Birch and Swinnerton-Dyer conjecture
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6.13. Brumer–Stark conjecture
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6.14. Chowla–Mordell theorem
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6.15. Clausen function
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6.16. Dedekind zeta function
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6.17. Dirichlet's theorem on arithmetic progressions
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6.18. Dirichlet L-function
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6.19. Dirichlet beta function
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6.20. Dirichlet character
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6.21. Dirichlet eta function
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6.22. Divisor function
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6.23. Dwork conjecture
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6.24. Eichler–Shimura congruence relation
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6.25. Equivariant L-function
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6.26. Euler product
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6.27. Explicit formulae for L-functions
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6.28. Feller–Tornier constant
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6.29. Functional equation (L-function)
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6.30. Gan–Gross–Prasad conjecture
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6.31. Generalized Riemann hypothesis
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6.32. Goss zeta function
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6.33. Grand Riemann hypothesis
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6.34. Hadjicostas's formula
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6.35. Hardy–Littlewood zeta-function conjectures
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6.36. Hasse–Weil zeta function
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6.37. Hecke character
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6.38. Hideo Shimizu
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6.39. Hilbert–Pólya conjecture
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6.40. Hurwitz zeta function
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6.41. Igusa zeta function
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6.42. L-function
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6.43. Langlands program
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6.44. Langlands–Deligne local constant
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6.45. Lefschetz zeta function
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6.46. Lerch zeta function
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6.47. Li's criterion
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6.48. Lindelöf hypothesis
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6.49. List of zeta functions
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6.50. Local Langlands conjectures
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6.51. Local zeta function
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6.52. Matsumoto zeta function
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6.53. Montgomery's pair correlation conjecture
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6.54. Motivic L-function
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6.55. Multiple zeta function
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6.56. P-adic L-function
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6.57. Prime zeta function
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6.58. Proof of the Euler product formula for the Riemann zeta function
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6.59. Ramanujan tau function
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6.60. Ramanujan–Petersson conjecture
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6.61. Rankin–Selberg method
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6.62. Riemann Xi function
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6.63. Riemann hypothesis
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6.64. Riemann zeta function
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6.65. Riemann–Siegel formula
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6.66. Riemann–Siegel theta function
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6.67. Riesz function
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6.68. Ruelle zeta function
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6.69. Selberg's zeta function conjecture
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6.70. Selberg class
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6.71. Shimizu L-function
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6.72. Shimura variety
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6.73. Shintani zeta function
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6.74. Siegel zero
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6.75. Special values of L-functions
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6.76. Standard L-function
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6.77. Stieltjes constants
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6.78. Subgroup growth
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6.79. Tate's thesis
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6.80. Turing's method
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6.81. Waldspurger's theorem
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6.82. Weil's criterion
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6.83. Weil conjectures
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6.84. Z function
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6.85. ZetaGrid
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6.86. Zeta function (operator)
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6.87. Zeta function universality
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7. Airy function
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8. Anger function
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9. Baer function
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10. Barnes integral
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11. Bateman Manuscript Project
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12. Bateman function
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13. Bickley–Naylor functions
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14. Bounded type (mathematics)
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15. Boxcar function
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16. Buchstab function
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17. Böhmer integral
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18. Cantor function
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19. Carotid–Kundalini function
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20. Chandrasekhar's H-function
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21. Chandrasekhar's X- and Y-function
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22. Chapman function
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23. Clausen's formula
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24. Complete Fermi–Dirac integral
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25. Confluent hypergeometric function
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26. Conical function
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27. Coshc function
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28. Crenel function
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29. Dawson function
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30. Debye function
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31. Dickman function
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32. Einstein function
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33. Entire function
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34. Exponential integral
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35. Ferrers function
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36. Fox H-function
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37. Fresnel integral
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38. Goodwin–Staton integral
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39. Griewank function
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40. Hankel contour
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41. Harish-Chandra's Ξ function
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42. Heaviside step function
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43. Herglotz–Zagier function
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44. Heun function
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45. Holonomic function
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46. Hough function
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47. Hyperbolastic functions
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48. Incomplete Bessel K function/generalized incomplete gamma function
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49. Incomplete Fermi–Dirac integral
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50. Incomplete polylogarithm
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51. Inverse tangent integral
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52. Jacobi elliptic functions
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53. Jacobi zeta function
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54. Kontorovich–Lebedev transform
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55. Kummer's function
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56. Lambert W function
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57. Lamé function
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58. Legendre chi function
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59. Legendre form
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60. List of eponyms of special functions
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61. Logit
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62. Lommel function
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63. Mathieu function
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64. Mayer f-function
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65. Minkowski's question-mark function
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66. Mittag-Leffler function
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67. Modular form
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68. Neville theta functions
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69. Oblate spheroidal wave function
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70. Painlevé transcendents
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71. Parabolic cylinder function
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72. Pochhammer contour
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73. Prolate spheroidal wave function
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74. Q-function
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75. Ramp function
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76. Real analytic Eisenstein series
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77. Rectangular function
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78. Ruler function
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79. Scorer's function
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80. Selberg integral
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81. Sinhc function
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82. Spence's function
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83. Spin-weighted spherical harmonics
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84. Step function
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85. Struve function
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86. Strömgren integral
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87. Student's t-distribution
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88. Synchrotron function
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89. Tak (function)
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90. Tanc function
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91. Tanhc function
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92. Thomae's function
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93. Tracy–Widom distribution
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94. Transcendental function
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95. Transport function
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96. Triangular function
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97. Trigonometric integral
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98. Voigt profile
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99. Walsh function
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100. Whipple formulae
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101. Wright omega function
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Ancestors
Combinatorics
Fields of mathematics
Mathematics
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