List of Figures 1 Algebraic and Analytic Methods 1.4.1 Piecewise continuous function on [a,b). 1.4.2 Convex function fβ‘(x). 1.6.1 Right-hand rule for cross products. 1.10.1 Domain D. 2 Asymptotic Approximations 2.10.1 Contour π. 2.11.1 |erfcβ‘(50β’cβ‘(ΞΈ))|. 3 Numerical Methods 3.1.1 Representation of data in the binary interchange formats for binary32, binary64 and binary128. 3.11.1 Error of the minimax rational approximation to J0β‘(x) 4 Elementary Functions 4.2.1 Branch cut for lnβ‘z and zΞ±. 4.3.1 lnβ‘x and ex. (i) z-plane (ii) w-plane A B C CΒ― D DΒ― E EΒ― F z 0 r r+iβ’Ο rβiβ’Ο iβ’Ο βiβ’Ο βr+iβ’Ο βrβiβ’Ο βr w 1 er βer+iβ’0 βerβiβ’0 β1+iβ’0 β1βiβ’0 βeβr+iβ’0 βeβrβiβ’0 eβr 4.3.2 Conformal mapping of exponential and logarithm. 4.3.3 lnβ‘(x+iβ’y). 4.3.4 ex+iβ’y. 4.13.1 Branches W0β‘(x), WΒ±1β‘(xβ0β’i) of the Lambert W-function. 4.13.2 The Wβ‘(z) function on the first 5 Riemann sheets. 4.15.1 sinβ‘x and cosβ‘x. 4.15.2 Arcsinβ‘x and Arccosβ‘x. 4.15.3 tanβ‘x and cotβ‘x. 4.15.4 arctanβ‘x and arccotβ‘x. 4.15.5 cscβ‘x and secβ‘x. 4.15.6 arccscβ‘x and arcsecβ‘x. βββ (i) z-plane βββββββββββββββββ (ii) w-plane A B C CΒ― D DΒ― E EΒ― F z 0 12β’Ο 12β’Ο+iβ’r 12β’Οβiβ’r iβ’r βiβ’r β12β’Ο+iβ’r β12β’Οβiβ’r β12β’Ο w 0 1 coshβ‘r+iβ’0 coshβ‘rβiβ’0 iβ’sinhβ‘r βiβ’sinhβ‘r βcoshβ‘r+iβ’0 βcoshβ‘rβiβ’0 β1 4.15.7 Conformal mapping of sine and inverse sine. 4.15.8 sinβ‘(x+iβ’y). 4.15.9 arcsinβ‘(x+iβ’y). 4.15.10 tanβ‘(x+iβ’y). 4.15.11 arctanβ‘(x+iβ’y). 4.15.12 cscβ‘(x+iβ’y). 4.15.13 arccscβ‘(x+iβ’y). 4.16.1 Quadrants for the angle ΞΈ. (i) arcsinβ‘z and arccosβ‘z (ii) arctanβ‘z (iii) arccscβ‘z and arcsecβ‘z (iv) arccotβ‘z 4.23.1 Branch cuts for the inverse trigonometric functions. 4.29.1 sinhβ‘x and coshβ‘x. 4.29.2 arcsinhβ‘x and arccoshβ‘x. 4.29.3 tanhβ‘x and cothβ‘x. 4.29.4 arctanhβ‘x and arccothβ‘x. 4.29.5 cschβ‘x and sechβ‘x. 4.29.6 arccschβ‘x and arcsechβ‘x. (i) arcsinhβ‘z (ii) arccoshβ‘z (iii) arctanhβ‘z (iv) arccschβ‘z (v) arcsechβ‘z (vi) arccothβ‘z 4.37.1 Branch cuts for the inverse hyperbolic functions. 4.42.1 Planar right triangle. 4.42.2 Planar triangle. 4.42.3 Spherical triangle. 5 Gamma Function 5.3.1 Ξβ‘(x), 1/Ξβ‘(x). 5.3.2 lnβ‘Ξβ‘(x). 5.3.3 Οβ‘(x). 5.3.4 |Ξβ‘(x+iβ’y)|. 5.3.5 1/|Ξβ‘(x+iβ’y)|. 5.3.6 |Οβ‘(x+iβ’y)|. 5.9.1 Contour for Hankelβs loop integral. 5.12.1 Contour for first loop integral for the beta function. 5.12.2 Contour for second loop integral for the beta function. 5.12.3 Contour for Pochhammerβs integral. 6 Exponential, Logarithmic, Sine, and Cosine Integrals 6.3.1 E1β‘(x), Eiβ‘(x), 0<xβ€2. 6.3.2 Siβ‘(x),Ciβ‘(x), 0β€xβ€15. 6.3.3 |E1β‘(x+iβ’y)|, β4β€xβ€4, β4β€yβ€4. 6.16.1 Gibbs phenomenon. 6.16.2 liβ‘(x), Οβ‘(x), x=10,20,β¦,1000. 7 Error Functions, Dawsonβs and Fresnel Integrals 7.3.1 erfcβ‘x, erfcβ‘(10β’x), β3β€xβ€3. 7.3.2 Fβ‘(x), β3.5β€xβ€3.5. 7.3.3 Cβ‘(x), Sβ‘(x), 0β€xβ€4. 7.3.4 |β±β‘(x)|2, β8β€xβ€8. 7.3.5 |erfβ‘(x+iβ’y)|, β3β€xβ€3, β3β€yβ€3. 7.3.6 |erfcβ‘(x+iβ’y)|, β3β€xβ€3, β3β€yβ€3. 7.18.1 Repeated integrals of the scaled complementary error function. 7.19.1 Voigt function π΄β‘(x,t), t=0.1, 2.5, 5, 10. 7.19.2 Voigt function π΅β‘(x,t), t=0.1, 2.5, 5, 10. 7.20.1 Cornuβs spiral. 8 Incomplete Gamma and Related Functions 8.3.1 Ξβ‘(a,x), a = 0.25, 1, 2, 2.5, 3. 8.3.2 Ξ³β‘(a,x), a = 0.25, 0.5, 0.75, 1. 8.3.3 Ξ³β‘(a,x), a = 1, 2, 2.5, 3. 8.3.4 Ξ³ββ‘(a,x) (= xβaβ’Pβ‘(a,x)), a = 0.25, 0.5, 0.75, 1, 2. 8.3.5 xβaβΞ³ββ‘(a,x) (= xβaβ’Qβ‘(a,x)), a = 0.25, 0.5, 1, 2. 8.3.6 Ξ³ββ‘(a,x) (= xβaβ’Pβ‘(a,x)), β4β€xβ€4, β5β€aβ€4. 8.3.7 xβaβΞ³ββ‘(a,x) (= xβaβ’Qβ‘(a,x)), 0β€xβ€4, β5β€aβ€5. 8.3.8 Ξβ‘(0.25,x+iβ’y), β3β€xβ€3, β3β€yβ€3. 8.3.9 Ξ³β‘(0.25,x+iβ’y), β3β€xβ€3, β3β€yβ€3. 8.3.10 Ξ³ββ‘(0.25,x+iβ’y), β3β€xβ€3, β3β€yβ€3. 8.3.11 Ξβ‘(1,x+iβ’y), β3β€xβ€3, β3β€yβ€3. 8.3.12 Ξ³β‘(1,x+iβ’y), β3β€xβ€3, β3β€yβ€3. 8.3.13 Ξ³ββ‘(1,x+iβ’y), β3β€xβ€3, β3β€yβ€3. 8.3.14 Ξβ‘(2.5,x+iβ’y), β2.2β€xβ€3, β3β€yβ€3. 8.3.15 Ξ³β‘(2.5,x+iβ’y), β2.2β€xβ€3, β3β€yβ€3. 8.3.16 Ξ³ββ‘(2.5,x+iβ’y), β3β€xβ€3, β3β€yβ€3. 8.19.1 Epβ‘(x), 0β€xβ€3, 0β€pβ€8. 8.19.2 E12β‘(x+iβ’y), β4β€xβ€4, β4β€yβ€4. 8.19.3 E1β‘(x+iβ’y), β4β€xβ€4, β4β€yβ€4. 8.19.4 E32β‘(x+iβ’y), β3β€xβ€3, β3β€yβ€3. 8.19.5 E2β‘(x+iβ’y), β3β€xβ€3, β3β€yβ€3. 9 Airy and Related Functions 9.3.1 Aiβ‘(x), Biβ‘(x), Mβ‘(x). 9.3.2 Aiβ²β‘(x), Biβ²β‘(x), Nβ‘(x). 9.3.3 Aiβ‘(x+iβ’y). 9.3.4 Biβ‘(x+iβ’y). 9.3.5 Aiβ²β‘(x+iβ’y). 9.3.6 Biβ²β‘(x+iβ’y). 9.12.1 Giβ‘(x), Giβ²β‘(x). 9.12.2 Hiβ‘(x), Hiβ²β‘(x). 9.13.1 Paths β0, β1, β2, β3. 9.13.2 Paths β1, β2, β3. 10 Bessel Functions 10.3.1 J0β‘(x), Y0β‘(x), J1β‘(x), Y1β‘(x), 0β€xβ€10. 10.3.2 J5β‘(x), Y5β‘(x), M5β‘(x), 0β€xβ€15. 10.3.3 J5β²β‘(x), Y5β²β‘(x), N5β‘(x), 0β€xβ€15. 10.3.4 ΞΈ5β‘(x), Ο5β‘(x), 0β€xβ€15. 10.3.5 JΞ½β‘(x), 0β€xβ€10, 0β€Ξ½β€5. 10.3.6 YΞ½β‘(x), 0<xβ€10, 0β€Ξ½β€5. 10.3.7 JΞ½β²β‘(x), 0β€xβ€10, 0β€Ξ½β€5. 10.3.8 YΞ½β²β‘(x), 0.2β€xβ€10, 0β€Ξ½β€5. 10.3.9 J0β‘(x+iβ’y), β10β€xβ€10, β4β€yβ€4. 10.3.10 H0(1)β‘(x+iβ’y), β10β€xβ€5, β2.8β€yβ€4. 10.3.11 J1β‘(x+iβ’y), β10β€xβ€10, β4β€yβ€4. 10.3.12 H1(1)β‘(x+iβ’y), β10β€xβ€5, β2.8β€yβ€4. 10.3.13 J5β‘(x+iβ’y), β10β€xβ€10, β4β€yβ€4. 10.3.14 H5(1)β‘(x+iβ’y), β20β€xβ€10, β4β€yβ€4. 10.3.15 J5.5β‘(x+iβ’y), β10β€xβ€10, β4β€yβ€4. 10.3.16 H5.5(1)β‘(x+iβ’y), β20β€xβ€10, β4β€yβ€4. 10.3.17 J~1/2β‘(x), Y~1/2β‘(x), 0.01β€xβ€10. 10.3.18 J~1β‘(x), Y~1β‘(x), 0.01β€xβ€10. 10.3.19 J~5β‘(x), Y~5β‘(x), 0.01β€xβ€10. 10.20.1 z-plane. 10.20.2 ΞΆ-plane. 10.20.3 Domain π. 10.21.1 Zeros of Ynβ‘(nβ’z) in |phβ‘z|β€Ο. 10.21.2 Zeros of Hn(1)β‘(nβ’z) in |phβ‘z|β€Ο. 10.21.3 Zeros of Ynβ‘(nβ’z) in |phβ‘z|β€Ο. 10.21.4 Zeros of Hn(1)β‘(nβ’z) in |phβ‘z|β€Ο. 10.21.5 Zeros of Ynβ‘(nβ’z) in |phβ‘z|β€Ο. 10.21.6 Zeros of Hn(1)β‘(nβ’z) in |phβ‘z|β€Ο. 10.23.1 Grafβs and Gegenbauerβs addition theorems. 10.26.1 I0β‘(x), I1β‘(x), K0β‘(x), K1β‘(x), 0β€xβ€3. 10.26.2 eβxβ’I0β‘(x), eβxβ’I1β‘(x), exβ’K0β‘(x), exβ’K1β‘(x), 0β€xβ€10. 10.26.3 IΞ½β‘(x), 0β€xβ€5, 0β€Ξ½β€4. 10.26.4 KΞ½β‘(x), 0.1β€xβ€5, 0β€Ξ½β€4. 10.26.5 IΞ½β²β‘(x), 0β€xβ€5, 0β€Ξ½β€4. 10.26.6 KΞ½β²β‘(x), 0.3β€xβ€5, 0β€Ξ½β€4. 10.26.7 I~1/2β‘(x), K~1/2β‘(x), 0.01β€xβ€3. 10.26.8 I~1β‘(x), K~1β‘(x), 0.01β€xβ€3. 10.26.9 I~5β‘(x), K~5β‘(x), 0.01β€xβ€3. 10.26.10 K~5β‘(x), 0.01β€xβ€3. 10.41.1 z-plane. 10.41.2 Ξ·-plane. 10.48.1 πnβ‘(x), n=0β’(1)β’4, 0β€xβ€12. 10.48.2 πnβ‘(x), n=0β’(1)β’4, 0<xβ€12. 10.48.3 π5β‘(x), π5β‘(x), π52β‘(x)+π52β‘(x), 0β€xβ€12. 10.48.4 π5β²β‘(x), π5β²β‘(x), π5β²2β‘(x)+π5β²2β‘(x), 0β€xβ€12. 10.48.5 π0(1)β‘(x), π0(2)β‘(x), π0β‘(x), 0β€xβ€4. 10.48.6 π1(1)β‘(x),π1(2)β‘(x),π1β‘(x), 0β€xβ€4. 10.48.7 π5(1)β‘(x), π5(2)β‘(x), π5β‘(x), 0β€xβ€8. 10.62.1 berβ‘x, beiβ‘x, berβ²β‘x, beiβ²β‘x, 0β€xβ€8. 10.62.2 kerβ‘x, keiβ‘x, kerβ²β‘x, keiβ²β‘x, 0β€xβ€8. 10.62.3 eβx/2β’berβ‘x, eβx/2β’beiβ‘x, eβx/2β’Mβ‘(x), 0β€xβ€8. 10.62.4 ex/2β’kerβ‘x, ex/2β’keiβ‘x, ex/2β’Nβ‘(x), 0β€xβ€8. 11 Struve and Related Functions 11.3.1 πΞ½β‘(x), 0β€xβ€12, Ξ½=0,12,1,32,2,3. 11.3.2 πΞ½β‘(x), 0<xβ€16, Ξ½=0,12,1,32,2,3. 11.3.3 πΞ½β‘(x), 0β€xβ€12, Ξ½=β3,β2,β32,β1,β12. 11.3.4 πΞ½β‘(x), 0<xβ€16, Ξ½=β4,β3,β2,β1,0. 11.3.5 πΞ½β‘(x), 0β€xβ€8, β4β€Ξ½β€4. 11.3.6 πΞ½β‘(x), 0β€xβ€8, β4β€Ξ½β€4. 11.3.7 |π0β‘(x+iβ’y)|, β8β€xβ€8, β3β€yβ€3. 11.3.8 |π0β‘(x+iβ’y)|, β8β€xβ€8, β3β€yβ€3. 11.3.9 |π12β‘(x+iβ’y)|, β8β€xβ€8, β3β€yβ€3. 11.3.10 |π12β‘(x+iβ’y)|, β8β€xβ€8, β3β€yβ€3. 11.3.11 |π1β‘(x+iβ’y)|, β8.5β€xβ€8.5, β3β€yβ€3. 11.3.12 |π1β‘(x+iβ’y)|, β8β€xβ€8, β3β€yβ€3. 11.3.13 πΞ½β‘(x), 0β€x<4.38, Ξ½=0,12,1,32,2,3. 11.3.14 πΞ½β‘(x), 0β€xβ€16, Ξ½=0,12,1,32,2,3. 11.3.15 πΞ½β‘(x), 0β€x<4.25, Ξ½=β3,β2,β32,β1,β12. 11.3.16 πΞ½β‘(x), 0<xβ€16, Ξ½=β3,β2,β32,β1,β12. 11.3.17 πΞ½β‘(x), 0β€xβ€5.6, β4β€Ξ½β€4. 11.3.18 πΞ½β‘(x), 0β€xβ€8, β4β€Ξ½β€4. 11.3.19 |πβ12β‘(x+iβ’y)|, β3β€xβ€3, β3β€yβ€3. 11.3.20 |π12β‘(x+iβ’y)|, β3β€xβ€3, β3β€yβ€3. 11.10.1 πΞ½β‘(x), β8β€xβ€8, Ξ½=0,12,1,32. 11.10.2 πΞ½β‘(x), β8β€xβ€8, Ξ½=0,12,1,32. 11.10.3 πΞ½β‘(x), β10β€xβ€10, 0β€Ξ½β€5. 11.10.4 πΞ½β‘(x), β10β€xβ€10, 0β€Ξ½β€5. 12 Parabolic Cylinder Functions 12.3.1 Uβ‘(a,x), a = 0.5, 2, 3.5, 5, 8. 12.3.2 Vβ‘(a,x), a = 0.5, 2, 3.5, 5, 8. 12.3.3 Uβ‘(a,x), a = β0.5, β2, β3.5, β5. 12.3.4 Vβ‘(a,x), a = β0.5, β2, β3.5, β5. 12.3.5 Uβ‘(β8,x), UΒ―β‘(β8,x), Fβ‘(β8,x), β4β’2β€xβ€4β’2. 12.3.6 Uβ²β‘(β8,x), UΒ―β²β‘(β8,x), Gβ‘(β8,x), β4β’2β€xβ€4β’2. 12.3.7 Uβ‘(a,x), β2.5β€aβ€2.5, β2.5β€xβ€2.5. 12.3.8 Vβ‘(a,x), β2.5β€aβ€2.5, β2.5β€xβ€2.5. 12.3.9 Uβ‘(3.5,x+iβ’y), β3.6β€xβ€5, β5β€yβ€5. 12.3.10 Uβ‘(β3.5,x+iβ’y), β5β€xβ€5, β3.5β€yβ€3.5. 12.14.1 kβ1/2β’Wβ‘(3,x), k1/2β’Wβ‘(3,βx), F~β‘(3,x), 0β€xβ€8. 12.14.2 kβ1/2β’Wβ²β‘(3,x), k1/2β’Wβ²β‘(3,βx), G~β‘(3,x), 0β€xβ€8. 12.14.3 kβ1/2β’Wβ‘(β3,x), k1/2β’Wβ‘(β3,βx), F~β‘(β3,x), 0β€xβ€8. 12.14.4 kβ1/2β’Wβ²β‘(β3,x),k1/2β’Wβ²β‘(β3,βx), G~β‘(β3,x), 0β€xβ€8. 13 Confluent Hypergeometric Functions 13.4.1 Contour of integration in (13.4.11). 13.7.1 Regions for error bounds of U. 14 Legendre and Related Functions 14.4.1 π―Ξ½0β‘(x), Ξ½=0,12,1,2,4. 14.4.2 π°Ξ½0β‘(x), Ξ½=0,12,1,2,4. 14.4.3 π―Ξ½β1/2β‘(x), Ξ½=0,12,1,2,4. 14.4.4 π°Ξ½1/2β‘(x), Ξ½=0,12,1,2,4. 14.4.5 π―Ξ½β1β‘(x), Ξ½=0,12,1,2,4. 14.4.6 π°Ξ½1β‘(x), Ξ½=0,12,1,2,4. 14.4.7 π―0βΞΌβ‘(x), ΞΌ=0,12,1,2,4. 14.4.8 π°0ΞΌβ‘(x), ΞΌ=0,12,1,2,4. 14.4.9 π―1/2βΞΌβ‘(x), ΞΌ=0,12,1,2,4. 14.4.10 π°1/2ΞΌβ‘(x), ΞΌ=0,12,1,2,4. 14.4.11 π―1βΞΌβ‘(x), ΞΌ=0,12,1,2,4. 14.4.12 π°1ΞΌβ‘(x), ΞΌ=0,12,1,2,4. 14.4.13 π―Ξ½0β‘(x), 0β€Ξ½β€10,β1<x<1. 14.4.14 π°Ξ½0β‘(x), 0β€Ξ½β€10,β1<x<1. 14.4.15 π―0βΞΌβ‘(x), 0β€ΞΌβ€10,β1<x<1. 14.4.16 π°0ΞΌβ‘(x), 0β€ΞΌβ€6.2,β1<x<1. 14.4.17 PΞ½0β‘(x), Ξ½=0,12,1,2,4. 14.4.18 πΈΞ½0β‘(x), Ξ½=0,12,1,2,4. 14.4.19 PΞ½β1/2β‘(x), Ξ½=0,12,1,2,4. 14.4.20 πΈΞ½1/2β‘(x), Ξ½=0,12,1,2,4. 14.4.21 PΞ½β1β‘(x), Ξ½=0,12,1,2,4. 14.4.22 πΈΞ½1β‘(x), Ξ½=0,12,1,2,4. 14.4.23 P0βΞΌβ‘(x), ΞΌ=0,12,1,2,4. 14.4.24 πΈ0ΞΌβ‘(x), ΞΌ=0,2,4,8. 14.4.25 P1/2βΞΌβ‘(x), ΞΌ=0,12,1,2,4. 14.4.26 πΈ1/2ΞΌβ‘(x), ΞΌ=0,2,4,8. 14.4.27 P1βΞΌβ‘(x), ΞΌ=0,12,1,2,4. 14.4.28 πΈ1ΞΌβ‘(x), ΞΌ=0,2,4,8. 14.4.29 PΞ½0β‘(x), 0β€Ξ½β€10, 1<x<10. 14.4.30 πΈΞ½0β‘(x), 0β€Ξ½β€10, 1<x<10. 14.4.31 P0βΞΌβ‘(x), 0β€ΞΌβ€10, 1<x<10. 14.4.32 πΈ0ΞΌβ‘(x), 0β€ΞΌβ€10, 1<x<10. 14.20.1 π―β12+iβ’Ο0β‘(x), Ο=0,1,2,4,8. 14.20.2 π°^β12+iβ’Ο0β‘(x), Ο=0,12,1,2,4. 14.20.3 π―β12+iβ’Οβ1/2β‘(x), Ο=0,1,2,4,8. 14.20.4 π°^β12+iβ’Οβ1/2β‘(x), Ο=12,1,2,4. 14.20.5 π―β12+iβ’Οβ1β‘(x), Ο=0,1,2,4,8. 14.20.6 π°^β12+iβ’Οβ1β‘(x), Ο=0,12,1,2,4. 14.20.7 π―β12+iβ’Οβ2β‘(x),Ο=0,1,2,4,8. 14.20.8 π°^β12+iβ’Οβ2β‘(x), Ο=0,12,1,2,4. 14.22.1 P1/20β‘(x+iβ’y), β5β€xβ€5, β5β€yβ€5. 14.22.2 P1/2β1/2β‘(x+iβ’y), β5β€xβ€5, β5β€yβ€5. 14.22.3 P1/2β1β‘(x+iβ’y), β5β€xβ€5, β5β€yβ€5. 14.22.4 πΈ00β‘(x+iβ’y), β5β€xβ€5, β5β€yβ€5. 15 Hypergeometric Function 15.3.1 Fβ‘(43,916;145;x),β100β€xβ€1. 15.3.2 Fβ‘(5,β10;1;x),β0.023β€xβ€1. 15.3.3 Fβ‘(1,β10;10;x),β3β€xβ€1. 15.3.4 Fβ‘(5,10;1;x),β1β€xβ€0.022. 15.3.5 Fβ‘(43,916;145;x+iβ’y),0β€xβ€2,β0.5β€yβ€0.5. 15.3.6 Fβ‘(β3,35;u+iβ’v;12),β6β€uβ€2,β2β€vβ€2. 15.3.7 |π β‘(β3,35;u+iβ’v;12)|,β6β€uβ€2,β2β€vβ€2. 15.6.1 Contour of integration in (15.6.5). 16 Generalized Hypergeometric Functions & Meijer G-Function Case (i) Case (ii) Case (iii) 16.17.1 Integration path L for the Meijer G-function. 18 Orthogonal Polynomials 18.4.1 Jacobi polynomials Pn(1.5,β0.5)β‘(x), n=1,2,3,4,5. 18.4.2 Jacobi polynomials Pn(1.25,0.75)β‘(x), n=7,8. 18.4.3 Chebyshev polynomials Tnβ‘(x), n=1,2,3,4,5. 18.4.4 Legendre polynomials Pnβ‘(x), n=1,2,3,4,5. 18.4.5 Laguerre polynomials Lnβ‘(x), n=1,2,3,4,5. 18.4.6 Laguerre polynomials L3(Ξ±)β‘(x), Ξ±=0,1,2,3,4. 18.4.7 Monic Hermite polynomials hnβ‘(x)=2βnβ’Hnβ‘(x), n=1,2,3,4,5. 18.4.8 Laguerre polynomials L3(Ξ±)β‘(x), 0β€Ξ±β€3, 0β€xβ€10. 18.4.9 Laguerre polynomials L4(Ξ±)β‘(x), 0β€Ξ±β€3, 0β€xβ€10. 18.21.1 Askey scheme. 18.39.1 Graphs of the first and fourth excited state eigenfunctions of the harmonic oscillator, for β=k=m=1, of (18.39.13), in Ο1β’(x), Ο4β’(x) and those of the rational potential of (18.39.19), in Ο^3β’(x), Ο^6β’(x). Both sets satisfy the Sturm oscillation theorem. 18.39.2 CoulombβPollaczek weight functions, xβ[β1,1], (18.39.50) for s=10, l=0, and Z=Β±1. For Z=+1 the weight function, red curve, has an essential singularity at x=β1, as all derivatives vanish as xββ1+; the green curve is β«β1xwCPβ‘(y)β’dy, to be compared with its histogram approximation in Β§18.40(ii). For Z=β1 the weight function, blue curve, is non-zero at x=β1, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as kββ, xkββ1β. 18.40.1 Histogram approximations to the Repulsive CoulombβPollaczek, RCP, weight function integrated over [β1,x), see Figure 18.39.2 for an exact result, for Z=+1, shown for N=12 and N=120. 18.40.2 Derivative Rule inversions for wRCPβ‘(x) carried out via Lagrange and PWCF interpolations. Shown are the absolute errors of approximation (18.40.8) at the points xi,N, i=1,2,β¦,N for N=40. For the derivative rule Lagrange interpolation (red points) gives βΌ15 digits in the central region, while PWCF interpolation (blue points) gives βΌ25. 19 Elliptic Integrals 19.3.1 Kβ‘(k), Eβ‘(k), β2β€k2β€1. 19.3.2 RCβ‘(x,1), RCβ‘(x,β1), 0β€xβ€5. 19.3.3 Fβ‘(Ο,k), β1β€k2β€2, 0β€sin2β‘Οβ€1. 19.3.4 Eβ‘(Ο,k), β1β€k2β€2, 0β€sin2β‘Οβ€1. 19.3.5 Ξ β‘(Ξ±2,k), β2β€k2<1, β2β€Ξ±2β€2. 19.3.6 Ξ β‘(Ο,2,k), β1β€k2β€3, 0β€sin2β‘Ο<1. 19.3.7 Kβ‘(k), β2β€ββ‘(k2)β€2, β2β€ββ‘(k2)β€2. 19.3.8 Eβ‘(k), β2β€ββ‘(k2)β€2, β2β€ββ‘(k2)β€2. 19.3.9 ββ‘(Kβ‘(k)), β2β€ββ‘(k2)β€2, β2β€ββ‘(k2)β€2. 19.3.10 ββ‘(Kβ‘(k)), β2β€ββ‘(k2)β€2, β2β€ββ‘(k2)β€2. 19.3.11 ββ‘(Eβ‘(k)), β2β€ββ‘(k2)β€2, β2β€ββ‘(k2)β€2. 19.3.12 ββ‘(Eβ‘(k)), β2β€ββ‘(k2)β€2, β2β€ββ‘(k2)β€2. 19.17.1 RFβ‘(x,y,1), 0β€xβ€1, y=0,β0.1,β0.5,β1. 19.17.2 RGβ‘(x,y,1), 0β€xβ€1, y=0,β0.1,β0.5,β1. 19.17.3 RDβ‘(x,y,1), 0β€xβ€2, y=0,β0.1,β1,β5,β25. 19.17.4 RJβ‘(x,y,1,2), 0β€xβ€1, y=0,β0.1,β0.5,β1. 19.17.5 RJβ‘(x,y,1,0.5), 0β€xβ€1, y=0,β0.1,β0.5,β1. 19.17.6 RJβ‘(x,y,1,β0.5), 0β€xβ€1, y=0,β0.1,β0.5,β1. 19.17.7 RJβ‘(0.5,y,1,p), y=0,β0.01,β0.05,β0.2,β1, β1β€p<0. 19.17.8 RJβ‘(0,y,1,p), 0β€yβ€1, β1β€pβ€2. 20 Theta Functions 20.2.1 Fundamental parallelogram. 20.3.1 ΞΈjβ‘(Οβ’x,0.15), 0β€xβ€2, j=1,2,3,4. 20.3.2 ΞΈ1β‘(Οβ’x,q), 0β€xβ€2, q = 0.05, 0.5, 0.7, 0.9. 20.3.3 ΞΈ2β‘(Οβ’x,q), 0β€xβ€2, q = 0.05, 0.5, 0.7, 0.9. 20.3.4 ΞΈ3β‘(Οβ’x,q), 0β€xβ€2, q = 0.05, 0.5, 0.7, 0.9. 20.3.5 ΞΈ4β‘(Οβ’x,q), 0β€xβ€2, q = 0.05, 0.5, 0.7, 0.9. 20.3.6 ΞΈ1β‘(x,q), 0β€qβ€1, x = 0, 0.4, 5, 10, 40. 20.3.7 ΞΈ2β‘(x,q), 0β€qβ€1, x = 0, 0.4, 5, 10, 40. 20.3.8 ΞΈ3β‘(x,q), 0β€qβ€1, x = 0, 0.4, 5, 10, 40. 20.3.9 ΞΈ4β‘(x,q), 0β€qβ€1, x = 0, 0.4, 5, 10, 40. 20.3.10 ΞΈ1β‘(Οβ’x,q), 0β€xβ€2, 0β€qβ€0.99. 20.3.11 ΞΈ2β‘(Οβ’x,q), 0β€xβ€2, 0β€qβ€0.99. 20.3.12 ΞΈ3β‘(Οβ’x,q), 0β€xβ€2, 0β€qβ€0.99. 20.3.13 ΞΈ4β‘(Οβ’x,q), 0β€xβ€2, 0β€qβ€0.99. 20.3.14 ΞΈ1β‘(Οβ’x+iβ’y,0.12), β1β€xβ€1, β1β€yβ€2.3. 20.3.15 ΞΈ2β‘(Οβ’x+iβ’y,0.12), β1β€xβ€1, β1β€yβ€2.3. 20.3.16 ΞΈ3β‘(Οβ’x+iβ’y,0.12), β1β€xβ€1, β1β€yβ€1.5. 20.3.17 ΞΈ4β‘(Οβ’x+iβ’y,0.12), β1β€xβ€1, β1β€yβ€1.5. 20.3.18 ΞΈ1β‘(0.1|u+iβ’v), β1β€uβ€1, 0.005β€vβ€0.5. 20.3.19 ΞΈ2β‘(0|u+iβ’v), β1β€uβ€1, 0.005β€vβ€0.1. 20.3.20 ΞΈ3β‘(0|u+iβ’v), β1β€uβ€1, 0.005β€vβ€0.1. 20.3.21 ΞΈ4β‘(0|u+iβ’v), β1β€uβ€1, 0.005β€vβ€0.1. 21 Multidimensional Theta Functions (a1) (b1) (c1) (a2) (b2) (c2) (a3) (b3) (c3) 21.4.1 ΞΈ^β‘(π³|π) parametrized by (21.4.1). 21.4.2 ββ‘ΞΈ^β‘(x+iβ’y,0|π1), 0β€xβ€1, 0β€yβ€5. 21.4.3 |ΞΈ^β‘(x+iβ’y,0|π1)|, 0β€xβ€1, 0β€yβ€2. 21.4.4 ΞΈ^β‘(iβ’x,iβ’y|π1), 0β€xβ€4, 0β€yβ€4. 21.4.5 ββ‘ΞΈ^β‘(x+iβ’y,0,0|π2), 0β€xβ€1, 0β€yβ€3. 21.7.1 A basis of cycles for a genus 2 surface. 21.9.1 Two-dimensional periodic waves in a shallow water wave tank. 21.9.2 Contour plot of a two-phase solution of Equation (21.9.3). 22 Jacobian Elliptic Functions 22.3.1 snβ‘(x,k), cnβ‘(x,k), dnβ‘(x,k), k=0.4, β3β’Kβ‘β€xβ€3β’Kβ‘, Kβ‘=1.6399β’β¦. 22.3.2 snβ‘(x,k), cnβ‘(x,k), dnβ‘(x,k), k=0.7, β3β’Kβ‘β€xβ€3β’Kβ‘, Kβ‘=1.8456β’β¦. 22.3.3 snβ‘(x,k), cnβ‘(x,k), dnβ‘(x,k), k=0.99, β3β’Kβ‘β€xβ€3β’Kβ‘, Kβ‘=3.3566β’β¦. 22.3.4 snβ‘(x,k), cnβ‘(x,k), dnβ‘(x,k), k=0.999999, β3β’Kβ‘β€xβ€3β’Kβ‘, Kβ‘=7.9474β’β¦. 22.3.5 dsβ‘(x,k), sdβ‘(x,k), dcβ‘(x,k), k=0.4, β2β’Kβ‘β€xβ€2β’Kβ‘, Kβ‘=1.6399β’β¦. 22.3.6 dsβ‘(x,k), sdβ‘(x,k), dcβ‘(x,k), k=0.7, β2β’Kβ‘β€xβ€2β’Kβ‘, Kβ‘=1.8456β’β¦. 22.3.7 dsβ‘(x,k), sdβ‘(x,k), dcβ‘(x,k), k=0.99, β2β’Kβ‘β€xβ€2β’Kβ‘, Kβ‘=3.3566β’β¦. 22.3.8 dsβ‘(x,k), sdβ‘(x,k), dcβ‘(x,k), k=0.999999, β2β’Kβ‘β€xβ€2β’Kβ‘, Kβ‘=7.9474β’β¦. 22.3.9 csβ‘(x,k), nsβ‘(x,k), scβ‘(x,k), k=0.4, β2β’Kβ‘β€xβ€2β’Kβ‘, Kβ‘=1.6399β’β¦. 22.3.10 csβ‘(x,k), nsβ‘(x,k), scβ‘(x,k), k=0.7, β2β’Kβ‘β€xβ€2β’Kβ‘, Kβ‘=1.8456β’β¦. 22.3.11 csβ‘(x,k), nsβ‘(x,k), scβ‘(x,k), k=0.99, β2β’Kβ‘β€xβ€2β’Kβ‘, Kβ‘=3.3566β’β¦. 22.3.12 csβ‘(x,k), nsβ‘(x,k), scβ‘(x,k), k=0.999999, β2β’Kβ‘β€xβ€2β’Kβ‘, Kβ‘=7.9474β’β¦. 22.3.13 snβ‘(x,k) for k=1βeβn, n=0 to 20, β5β’Οβ€xβ€5β’Ο. 22.3.14 cnβ‘(x,k) for k=1βeβn, n=0 to 20, β5β’Οβ€xβ€5β’Ο. 22.3.15 dnβ‘(x,k) for k=1βeβn, n=0 to 20, β5β’Οβ€xβ€5β’Ο. 22.3.16 snβ‘(x+iβ’y,k), k=0.99, β3β’Kβ‘β€xβ€3β’Kβ‘, 0β€yβ€4β’Kβ²β‘. 22.3.17 cnβ‘(x+iβ’y,k), k=0.99, β3β’Kβ‘β€xβ€3β’Kβ‘, 0β€yβ€4β’Kβ²β‘. 22.3.18 dnβ‘(x+iβ’y,k), k=0.99, β3β’Kβ‘β€xβ€3β’Kβ‘, 0β€yβ€4β’Kβ²β‘. 22.3.19 cdβ‘(x+iβ’y,k), k=0.99, β3β’Kβ‘β€xβ€3β’Kβ‘, 0β€yβ€4β’Kβ²β‘. 22.3.20 dcβ‘(x+iβ’y,k), k=0.99, β3β’Kβ‘β€xβ€3β’Kβ‘, 0β€yβ€4β’Kβ²β‘. 22.3.21 nsβ‘(x+iβ’y,k), k=0.99, β3β’Kβ‘β€xβ€3β’Kβ‘. 22.3.22 ββ‘snβ‘(x,k). 22.3.23 ββ‘snβ‘(x,k). 22.3.24 snβ‘(x+iβ’y,k). 22.3.25 snβ‘(5,k). 22.3.26 Density plot of |snβ‘(5,k)|. 22.3.27 Density plot of |snβ‘(10,k)|. 22.3.28 Density plot of |snβ‘(20,k)|. 22.3.29 Density plot of |snβ‘(30,k)|. (a) snβ‘(z,k) (b) cnβ‘(z,k) (c) dnβ‘(z,k) 22.4.1 Poles, zeros of the principal Jacobian elliptic functions. 22.4.2 Fundamental unit cell. 22.16.1 amβ‘(x,k), 0β€xβ€10β’Ο, k=0.4,0.7,0.99,0.999999. 22.16.2 β°β‘(x,k), 0β€xβ€10β’Ο, k=0.4,0.7,0.99,0.999999. 22.16.3 Zβ‘(x|k), 0β€xβ€10β’Ο, k=0.4,0.7,0.99,0.999999. 22.19.1 amβ‘(x,k), 0β€xβ€10β’Ο, k=0.5,0.9999,1.0001,2. 23 Weierstrass Elliptic and Modular Functions 23.4.1 ββ‘(x;g2β‘,0), 0β€xβ€9, g2β‘ = 0.1, 0.2, 0.5, 0.8. 23.4.2 ββ‘(x;0,g3β‘), 0β€xβ€9, g3β‘ = 0.1, 0.2, 0.5, 0.8. 23.4.3 ΞΆβ‘(x;g2β‘,0), 0β€xβ€8, g2β‘ = 0.1, 0.2, 0.5, 0.8. 23.4.4 ΞΆβ‘(x;0,g3β‘), 0β€xβ€8, g3β‘ = 0.1, 0.2, 0.5, 0.8. 23.4.5 Οβ‘(x;g2β‘,0), β5β€xβ€5, g2β‘ = 0.1, 0.2, 0.5, 0.8. 23.4.6 Οβ‘(x;0,g3β‘), β5β€xβ€5, g3β‘ = 0.1, 0.2, 0.5, 0.8. 23.4.7 ββ‘(x), 0β€xβ€9, k2 = 0.2, 0.8, 0.95, 0.99. 23.4.8 ββ‘(x+iβ’y), β2β’Kβ‘(k)β€xβ€2β’Kβ‘(k), 0β€yβ€6β’Kβ²β‘(k), k2=0.9. 23.4.9 ββ‘(x+iβ’y;1,4β’i), β3.8β€xβ€3.8, β3.8β€yβ€3.8. 23.4.10 ΞΆβ‘(x+iβ’y;1,0), β5β€xβ€5, β5β€yβ€5. 23.4.11 Οβ‘(x+iβ’y;1,i), β2.5β€xβ€2.5, β2.5β€yβ€2.5. 23.4.12 ββ‘(3.7;a+iβ’b,0), β5β€aβ€3, β4β€bβ€4. 23.5.1 Rhombic lattice. ββ‘(2β’Ο3)=Ο1. 23.5.2 Equianharmonic lattice. 2β’Ο3=eΟβ’i/3β’2β’Ο1, 2β’Ο1β2β’Ο3=eβΟβ’i/3β’2β’Ο1. 23.16.1 Ξ»β‘(iβ’y), Jβ‘(iβ’y), Ξ·β‘(iβ’y), 0β€yβ€3. 23.16.2 Ξ»β‘(x+iβ’y), β0.25β€xβ€0.25, 0.005β€yβ€0.1. 23.16.3 Ξ·β‘(x+iβ’y), β0.0625β€xβ€0.0625, 0.0001β€yβ€0.07. 24 Bernoulli and Euler Polynomials 24.3.1 Bernoulli polynomials Bnβ‘(x), n=2,3,β¦,6. 24.3.2 Euler polynomials Enβ‘(x), n=2,3,β¦,6. 25 Zeta and Related Functions 25.3.1 ΞΆβ‘(x), ΞΆβ²β‘(x), β20β€xβ€10. 25.3.2 ΞΆβ‘(x), ΞΆβ²β‘(x), β12β€xβ€β2. 25.3.3 |ΞΆβ‘(x+iβ’y)|, β4β€xβ€4, β10β€yβ€40. 25.3.4 Zβ‘(t), 0β€tβ€50. 25.3.5 Zβ‘(t), 1000β€tβ€1050. 25.3.6 Zβ‘(t), 10000β€tβ€10050. 25.11.1 ΞΆβ‘(x,a), a = 0.3, 0.5, 0.8, 1, β20β€xβ€10. 25.11.2 ΞΆβ‘(x,a), β19.5β€xβ€10, 0.02β€aβ€1. 25.12.1 Li2β‘(x), β20β€x<1. 25.12.2 |Li2β‘(x+iβ’y)|, β20β€xβ€20, β20β€yβ€20. 26 Combinatorial Analysis 26.9.1 Ferrers graph of the partition 7+4+3+3+2+1. 26.9.2 The partition 5+5+3+2 represented as a lattice path. 26.12.1 A plane partition of 75. 28 Mathieu Functions and Hillβs Equation 28.2.1 Eigenvalues anβ‘(q), bnβ‘(q) of Mathieuβs equation. 28.3.1 ce2β’nβ‘(x,1), 0β€xβ€Ο/2, n=0,1,2,3. 28.3.2 ce2β’nβ‘(x,10), 0β€xβ€Ο/2, n=0,1,2,3. 28.3.3 ce2β’n+1β‘(x,1), 0β€xβ€Ο/2, n=0,1,2,3. 28.3.4 ce2β’n+1β‘(x,10), 0β€xβ€Ο/2, n=0,1,2,3. 28.3.5 se2β’n+1β‘(x,1), 0β€xβ€Ο/2, n=0,1,2,3. 28.3.6 se2β’n+1β‘(x,10), 0β€xβ€Ο/2, n=0,1,2,3. 28.3.7 se2β’nβ‘(x,1), 0β€xβ€Ο/2, n=1,2,3,4. 28.3.8 se2β’nβ‘(x,10), 0β€xβ€Ο/2, n=1,2,3,4. 28.3.9 ce0β‘(x,q), 0β€xβ€2β’Ο, 0β€qβ€10. 28.3.10 se1β‘(x,q), 0β€xβ€2β’Ο, 0β€qβ€10. 28.3.11 ce1β‘(x,q), 0β€xβ€2β’Ο, 0β€qβ€10. 28.3.12 se2β‘(x,q), 0β€xβ€2β’Ο, 0β€qβ€10. 28.3.13 ce2β‘(x,q), 0β€xβ€2β’Ο, 0β€qβ€10. 28.5.1 fe0β‘(x,0.5), ce0β‘(x,0.5), 0β€xβ€2β’Ο. 28.5.2 fe0β‘(x,1), ce0β‘(x,1), 0β€xβ€2β’Ο. 28.5.3 fe1β‘(x,0.5), ce1β‘(x,0.5), 0β€xβ€2β’Ο. 28.5.4 fe1β‘(x,1), ce1β‘(x,1), 0β€xβ€2β’Ο. 28.5.5 ge1β‘(x,0.5), se1β‘(x,0.5), 0β€xβ€2β’Ο. 28.5.6 ge1β‘(x,1), se1β‘(x,1), 0β€xβ€2β’Ο. 28.7.1 Branch point of the eigenvalues a0β‘(iβ’q^) and a2β‘(iβ’q^): 0β€q^β€2.5. 28.13.1 λνβ‘(q), Ξ½=0.5β’(1)β’3.5; anβ‘(q),bnβ‘(q), n=0,1,2,3,4 (aβs), n=1,2,3,4 (bβs). 28.13.2 λνβ‘(q), β2<Ξ½<2, 0β€qβ€10. 28.13.3 ceΞ½β‘(x,1), β1<Ξ½<1, 0β€xβ€2β’Ο. 28.13.4 seΞ½β‘(x,1), 0<Ξ½<1, 0β€xβ€2β’Ο. 28.13.5 meiβ’ΞΌβ‘(x,1), 0.1β€ΞΌβ€0.4, βΟβ€xβ€Ο. 28.17.1 Stability chart for eigenvalues of Mathieuβs equation (28.2.1). 28.21.1 Mc0(1)β‘(x,h), 0β€hβ€3, 0β€xβ€2. 28.21.2 Mc1(1)β‘(x,h), 0β€hβ€3, 0β€xβ€2. 28.21.3 Mc0(2)β‘(x,h) 0.1β€hβ€2, 0β€xβ€2. 28.21.4 Mc1(2)β‘(x,h), 0.2β€hβ€2, 0β€xβ€2. 28.21.5 Ms1(1)β‘(x,h), 0β€hβ€3, 0β€xβ€2. 28.21.6 Ms1(2)β‘(x,h), 0.2β€hβ€2, 0β€xβ€2. 29 LamΓ© Functions 29.2.1 Singularities of LamΓ©βs equation. 29.4.1 aΞ½mβ‘(0.5), bΞ½m+1β‘(0.5), m=0,1,2,3. 29.4.2 aΞ½3β‘(0.5)βbΞ½3β‘(0.5). 29.4.3 a1.5mβ‘(k2), b1.5m+1β‘(k2). 29.4.4 aΞ½mβ‘(0.1), bΞ½m+1β‘(0.1), m=0,1,2,3. 29.4.5 aΞ½mβ‘(0.9), bΞ½m+1β‘(0.9), m=0,1,2,3. 29.4.6 aΞ½2β‘(0.5)βbΞ½2β‘(0.5). 29.4.7 aΞ½4β‘(0.5)βbΞ½4β‘(0.5) 29.4.8 a2.5mβ‘(k2), b2.5m+1β‘(k2). 29.4.9 aΞ½0β‘(k2). 29.4.10 bΞ½1β‘(k2). 29.4.11 aΞ½1β‘(k2). 29.4.12 bΞ½2β‘(k2). 29.4.13 πΈπ1.5mβ‘(x,0.5), β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. 29.4.14 πΈπ 1.5mβ‘(x,0.5), β2β’Kβ‘β€xβ€2β’Kβ‘, m=1,2,3. 29.4.15 πΈπ1.5mβ‘(x,0.1), β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. 29.4.16 πΈπ 1.5mβ‘(x,0.1), β2β’Kβ‘β€xβ€2β’Kβ‘, m=1,2,3. 29.4.17 πΈπ1.5mβ‘(x,0.9), β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. 29.4.18 πΈπ 1.5mβ‘(x,0.9), β2β’Kβ‘β€xβ€2β’Kβ‘, m=1,2,3. 29.4.19 πΈπ2.5mβ‘(x,0.1), β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. 29.4.20 πΈπ 2.5mβ‘(x,0.1), β2β’Kβ‘β€xβ€2β’Kβ‘, m=1,2,3. 29.4.21 πΈπ2.5mβ‘(x,0.5), β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. 29.4.22 πΈπ 2.5mβ‘(x,0.5), β2β’Kβ‘β€xβ€2β’Kβ‘, m=1,2,3. 29.4.23 πΈπ2.5mβ‘(x,0.9), β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. 29.4.24 πΈπ 2.5mβ‘(x,0.9), β2β’Kβ‘β€xβ€2β’Kβ‘, m=1,2,3. 29.4.25 πΈπ1.50β‘(x,k2). 29.4.26 πΈπ 1.51β‘(x,k2). 29.4.27 πΈπ1.51β‘(x,k2). 29.4.28 πΈπ 1.52β‘(x,k2). 29.4.29 πΈπ2.50β‘(x,k2). 29.4.30 πΈπ 2.51β‘(x,k2). 29.4.31 πΈπ2.51β‘(x,k2). 29.4.32 πΈπ 2.52β‘(x,k2). 29.13.1 a2mβ‘(k2), b2mβ‘(k2). 29.13.2 a1mβ‘(k2), b1mβ‘(k2). 29.13.3 a3mβ‘(k2), b3mβ‘(k2). 29.13.4 a4mβ‘(k2), b4mβ‘(k2). 29.13.5 π’πΈ4mβ‘(x,0.1) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. Kβ‘=1.61244β’β¦. 29.13.6 π’πΈ4mβ‘(x,0.9) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. Kβ‘=2.57809β’β¦. 29.13.7 π πΈ5mβ‘(x,0.1) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. Kβ‘=1.61244β’β¦. 29.13.8 π πΈ5mβ‘(x,0.9) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. Kβ‘=2.57809β’β¦. 29.13.9 ππΈ5mβ‘(x,0.1) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. Kβ‘=1.61244β’β¦. 29.13.10 ππΈ5mβ‘(x,0.9) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. Kβ‘=2.57809β’β¦. 29.13.11 ππΈ5mβ‘(x,0.1) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. Kβ‘=1.61244β’β¦. 29.13.12 ππΈ5mβ‘(x,0.9) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1,2. Kβ‘=2.57809β’β¦. 29.13.13 π ππΈ4mβ‘(x,0.1) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1. Kβ‘=1.61244β’β¦. 29.13.14 π ππΈ4mβ‘(x,0.9) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1. Kβ‘=2.57809β’β¦. 29.13.15 π ππΈ4mβ‘(x,0.1) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1. Kβ‘=1.61244β’β¦. 29.13.16 π ππΈ4mβ‘(x,0.9) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1. Kβ‘=2.57809β’β¦. 29.13.17 πππΈ4mβ‘(x,0.1) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1. Kβ‘=1.61244β’β¦. 29.13.18 πππΈ4mβ‘(x,0.9) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1. Kβ‘=2.57809β’β¦. 29.13.19 π πππΈ5mβ‘(x,0.1) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1. Kβ‘=1.61244β’β¦. 29.13.20 π πππΈ5mβ‘(x,0.9) for β2β’Kβ‘β€xβ€2β’Kβ‘, m=0,1. Kβ‘=2.57809β’β¦. 29.13.21 |π’πΈ41β‘(x+iβ’y,0.1)| for β3β’Kβ‘β€xβ€3β’Kβ‘, 0β€yβ€2β’Kβ²β‘. Kβ‘=1.61244β’β¦, Kβ²β‘=2.57809β’β¦. 29.13.22 |π’πΈ41β‘(x+iβ’y,0.5)| for β3β’Kβ‘β€xβ€3β’Kβ‘, 0β€yβ€2β’Kβ²β‘. Kβ‘=Kβ²β‘=1.85407β’β¦. 29.13.23 |π’πΈ41β‘(x+iβ’y,0.9)| for β3β’Kβ‘β€xβ€3β’Kβ‘, 0β€yβ€2β’Kβ²β‘. Kβ‘=2.57809β’β¦, Kβ²β‘=1.61244β’β¦. 30 Spheroidal Wave Functions 30.7.1 Ξ»n0β‘(Ξ³2), n=0,1,2,3, β10β€Ξ³2β€10. 30.7.2 Ξ»n1β‘(Ξ³2), n=1,2,3,4, β10β€Ξ³2β€10. 30.7.3 Ξ»n5β‘(Ξ³2), n=5,6,7,8, β40β€Ξ³2β€40. 30.7.4 Ξ»n10β‘(Ξ³2), n=10,11,12,13, β50β€Ξ³2β€150. 30.7.5 π―πn0β‘(x,4), n=0,1,2,3, β1β€xβ€1. 30.7.6 π―πn0β‘(x,β4), n=0,1,2,3, β1β€xβ€1. 30.7.7 π―πn1β‘(x,30), n=1,2,3,4, β1β€xβ€1. 30.7.8 π―πn1β‘(x,β30), n=1,2,3,4, β1β€xβ€1. 30.7.9 π―π20β‘(x,Ξ³2), β1β€xβ€1, β50β€Ξ³2β€50. 30.7.10 π―π31β‘(x,Ξ³2), β1β€xβ€1, β50β€Ξ³2β€50. 30.7.11 π°πn0β‘(x,4), n=0,1,2,3, β1<x<1. 30.7.12 π°πn0β‘(x,β4), n=0,1,2,3, β1<x<1. 30.7.13 π°πn1β‘(x,4), for n=1,2,3,4, β1<x<1. 30.7.14 π°πn1β‘(x,β4), n=1,2,3,4, β1<x<1. 30.7.15 π°π10β‘(x,Ξ³2),β1<x<1,β10β€Ξ³2β€10. 30.7.16 |ππ 00β‘(x+iβ’y,4)|, β2β€xβ€2, β2β€yβ€2. 30.7.17 |ππ 00β‘(x+iβ’y,β4)|, β2β€xβ€2, β2β€yβ€2. 30.7.18 |ππ 11β‘(x+iβ’y,4)|, β2β€xβ€2, β2β€yβ€2. 30.7.19 |ππ 11β‘(x+iβ’y,β4)|, β2β€xβ€2, β2β€yβ€2. 30.7.20 |ππ 00β‘(x+iβ’y,4)|, β2β€xβ€2, β2β€yβ€2. 30.7.21 |ππ 00β‘(x+iβ’y,β4)|, β1.8β€xβ€1.8, β2β€yβ€2. 30.11.1 Sn0β’(1)β‘(x,2), n=0,1, 1β€xβ€10. 30.11.2 Sn0β’(1)β‘(iβ’y,2β’i), n=0,1, 0β€yβ€10. 30.11.3 Sn1β’(1)β‘(x,2), n=1,2, 1β€xβ€10. 30.11.4 Sn1β’(1)β‘(iβ’y,2β’i), n=1,2, 0β€yβ€10. 32 PainlevΓ© Transcendents 32.3.1 wkβ‘(x), β12β€xβ€1.33, k=0.5, 0.75, 1, 1.25. 32.3.2 wkβ‘(x), β12β€xβ€2.43, k=β0.5, β0.25, 0, 1, 2. 32.3.3 wkβ‘(x), β12β€xβ€0.73, k=1.85185β3, 1.85185β5. 32.3.4 wkβ‘(x), β12β€xβ€2.3, k=β0.45142β7, β0.45142β8. 32.3.5 wkβ‘(x), kβ’Aiβ‘(x), β10β€xβ€4, k=0.5. 32.3.6 wkβ‘(x), β10β€xβ€4, k=0.999, 1.001. 32.3.7 ukβ‘(x;β12), β12β€xβ€4, k=0.33554β691, 0.33554β692. 32.3.8 ukβ‘(x;12), β12β€xβ€4, k=0.47443. 32.3.9 ukβ‘(x;32), β12β€xβ€4, k=0.38736, 0.38737. 32.3.10 ukβ‘(x;52), β12β€xβ€4, k=0.24499β2, 0.24499β3. 33 Coulomb Functions 33.3.1 Fββ‘(Ξ·,Ο), Gββ‘(Ξ·,Ο), β=0, Ξ·=β2. 33.3.2 Fββ‘(Ξ·,Ο), Gββ‘(Ξ·,Ο), β=0, Ξ·=0. 33.3.3 Fββ‘(Ξ·,Ο), Gββ‘(Ξ·,Ο), β=0, Ξ·=2. 33.3.4 Fββ‘(Ξ·,Ο), Gββ‘(Ξ·,Ο), β=0, Ξ·=2. 33.3.5 Fββ‘(Ξ·,Ο), Gββ‘(Ξ·,Ο), Mββ‘(Ξ·,Ο), β=0, Ξ·=15/2. 33.3.6 Fββ‘(Ξ·,Ο), Gββ‘(Ξ·,Ο), Mββ‘(Ξ·,Ο), β=5, Ξ·=0. 33.3.7 F0β‘(Ξ·,Ο), β2β€Ξ·β€2, 0β€Οβ€5. 33.3.8 G0β‘(Ξ·,Ο), β2β€Ξ·β€2, 0<Οβ€5. 33.15.1 fβ‘(Ο΅,β;r),hβ‘(Ο΅,β;r), β=0,Ο΅=4. 33.15.2 fβ‘(Ο΅,β;r),hβ‘(Ο΅,β;r), β=1,Ο΅=4. 33.15.3 fβ‘(Ο΅,β;r),hβ‘(Ο΅,β;r), β=0,Ο΅=β1/Ξ½2,Ξ½=1.5. 33.15.4 fβ‘(Ο΅,β;r),hβ‘(Ο΅,β;r), β=0,Ο΅=β1/Ξ½2,Ξ½=2. 33.15.5 fβ‘(Ο΅,β;r),hβ‘(Ο΅,β;r), β=0,Ο΅=β1/Ξ½2,Ξ½=2.5. 33.15.6 fβ‘(Ο΅,β;r), β=0,β2<Ο΅<2,β15<r<15. 33.15.7 hβ‘(Ο΅,β;r), β=0,β2<Ο΅<2,β15<r<15. 33.15.8 fβ‘(Ο΅,β;r), β=1,β2<Ο΅<2,β15<r<15. 33.15.9 hβ‘(Ο΅,β;r), β=1,β2<Ο΅<2,β15<r<15. 33.15.10 sβ‘(Ο΅,β;r), β=0,β0.15<Ο΅<0.10,0<r<65. 33.15.11 cβ‘(Ο΅,β;r), β=0,β0.15<Ο΅<0.10,0<r<65. 34 3j, 6j, 9j Symbols 34.2.1 Angular momenta jr and projective quantum numbers. 34.4.1 Tetrahedron corresponding to 6β’j symbol. 36 Integrals with Coalescing Saddles (a) Density plot. (b) 3D plot. 36.3.1 Modulus of Pearcey integral |Ξ¨2β‘(x,y)|. (a) Density plot. (b) 3D plot. 36.3.2 Modulus of swallowtail canonical integral function |Ξ¨3β‘(x,y,3)|. (a) Density plot. (b) 3D plot. 36.3.3 Modulus of swallowtail canonical integral function |Ξ¨3β‘(x,y,0)|. (a) Density plot. (b) 3D plot. 36.3.4 Modulus of swallowtail canonical integral function |Ξ¨3β‘(x,y,β3)|. (a) Density plot. (b) 3D plot. 36.3.5 Modulus of swallowtail canonical integral function |Ξ¨3β‘(x,y,β7.5)|. (a) Density plot. (b) 3D plot. 36.3.6 Modulus of elliptic umbilic canonical integral function |Ξ¨(E)β‘(x,y,0)|. (a) Density plot. (b) 3D plot. 36.3.7 Modulus of elliptic umbilic canonical integral function |Ξ¨(E)β‘(x,y,2)|. (a) Density plot. (b) 3D plot. 36.3.8 Modulus of elliptic umbilic canonical integral function |Ξ¨(E)β‘(x,y,4)|. (a) Density plot. (b) 3D plot. 36.3.9 Modulus of hyperbolic umbilic canonical integral function |Ξ¨(H)β‘(x,y,0)|. (a) Density plot. (b) 3D plot. 36.3.10 Modulus of hyperbolic umbilic canonical integral function |Ξ¨(H)β‘(x,y,1)|. (a) Density plot. (b) 3D plot. 36.3.11 Modulus of hyperbolic umbilic canonical integral function |Ξ¨(H)β‘(x,y,2)|. (a) Density plot. (b) 3D plot. 36.3.12 Modulus of hyperbolic umbilic canonical integral function |Ξ¨(H)β‘(x,y,3)|. (a) Contour plot, at intervals of Ο/4. (b) Density plot. 36.3.13 Phase of Pearcey integral phβ‘Ξ¨2β‘(x,y). (a) phβ‘Ξ¨3β‘(x,y,3). (b) phβ‘Ξ¨3β‘(x,y,0). (c) phβ‘Ξ¨3β‘(x,y,β3). (d) phβ‘Ξ¨3β‘(x,y,β7.5). 36.3.14 Density plots of phase of swallowtail canonical integrals. (a) Contour plot. (b) Density plot. 36.3.15 Phase of elliptic umbilic canonical integral phβ‘Ξ¨(E)β‘(x,y,0). (a) Contour plot. (b) Density plot. 36.3.16 Phase of elliptic umbilic canonical integral phβ‘Ξ¨(E)β‘(x,y,2). (a) Contour plot. (b) Density plot. 36.3.17 Phase of elliptic umbilic canonical integral phβ‘Ξ¨(E)β‘(x,y,4). (a) Contour plot. (b) Density plot. 36.3.18 Phase of hyperbolic umbilic canonical integral phβ‘Ξ¨(H)β‘(x,y,0). (a) Contour plot. (b) Density plot. 36.3.19 Phase of hyperbolic umbilic canonical integral phβ‘Ξ¨(H)β‘(x,y,1). (a) Contour plot. (b) Density plot. 36.3.20 Phase of hyperbolic umbilic canonical integral phβ‘Ξ¨(H)β‘(x,y,2). (a) Contour plot. (b) Density plot. 36.3.21 Phase of hyperbolic umbilic canonical integral phβ‘Ξ¨(H)β‘(x,y,3). 36.4.1 Bifurcation set of cusp catastrophe. 36.4.2 Bifurcation set of swallowtail catastrophe. 36.4.3 Bifurcation set of elliptic umbilic catastrophe. 36.4.4 Bifurcation set of hyperbolic umbilic catastrophe. 36.5.1 Cusp catastrophe. 36.5.2 Swallowtail catastrophe with z<0. 36.5.3 Swallowtail catastrophe with z=0. 36.5.4 Swallowtail catastrophe with z>0. 36.5.5 Elliptic umbilic catastrophe with z=constant. 36.5.6 Hyperbolic umbilic catastrophe with z=constant. 36.5.7 Sheets of the Stokes surface for the swallowtail catastrophe (colored and with mesh) and the bifurcation set (gray). 36.5.8 Sheets of the Stokes surface for the elliptic umbilic catastrophe. 36.5.9 Sheets of the Stokes surface for the hyperbolic umbilic catastrophe 36.13.1 Kelvinβs ship wave pattern.