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Advances in Applied Mathematics
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Contents
Preface
Chapter 1 General Formulas
1.1. Transforms Containing Arbitrary Functions
1.1.1. Basic formulas
1.1.2. f (axr) and the power function
1.1.3. f (axr) and elementary functions
1.1.4. Derivatives of f (x)
1.1.5. Integrals containing f (x)
Chapter 2 Elementary Functions
2.1. Algebraic Functions
2.1.1. (ar ? xr) + and (xr ? ar) +
2.1.2. (ax + b) and jx ? aj
2.1.3. (ax + b)ˆ (cx + d)˙
2.1.4. (a ? x)+ (bx + c) and (x ? a)+ (bx + c)
2.1.5. (ax + b)ˆ (cx + d)˙
2.1.6. (a ? x) ?1+ (xn + bn)r and (x ? a) ?1+ (xn + bn)r
2.1.7.?ax2 + bx + c (dx + e)
2.1.8. Algebraic functions of pax + b
2.1.9. Algebraic functions ofpax2 + bx + c
2.1.10. Various algebraic functions
2.2. The Exponential Function
2.2.1. e?axr?bxp
2.2.2. ebxm(a?x)n and algebraic functions
2.2.3. e'(x) and algebraic functions
2.3. Hyperbolic Functions
2.3.1. Rational functions of sinh x and cosh x
2.3.2. Hyperbolic and algebraic functions
2.3.3. Hyperbolic functions and eax
2.3.4. Hyperbolic functions and e'(x)
2.4. Trigonometric Functions
2.4.1. sin (ax + b) and cos (ax + b)
2.4.2. Trigonometric and algebraic functions
2.4.3. Trigonometric and the exponential functions
2.4.4. Trigonometric and hyperbolic functions
2.4.5. Products of trigonometric functions
2.4.6. sincn (bx) and elementary functions
2.5. The Logarithmic Function
2.5.1. ln (bx) and algebraic functions
2.5.2. ln (bx + c) and algebraic functions
2.5.3. lnax + bcx + d, ln ax + bcx + d and algebraic functions
2.5.4. ln?ax2 + bx + cand algebraic functions
2.5.5. lnax2 + bx + cdx2 + ex + fand algebraic functions
2.5.6. ln (' (x)) and algebraic functions
2.5.7. ln (' (x)) and the exponential function
2.5.8. The logarithmic and hyperbolic or trigonometric functionsNotation: =ˆ10˙.
2.5.9. Products of logarithms
2.6. Inverse Trigonometric Functions
2.6.1. arcsin (' (x)), arccos (' (x)), and algebraic functions
2.6.2. arcsin (' (x)), arccos (' (x)), and the exponential function
2.6.3. arccos (bx) and hyperbolic or trigonometric functionsNotation: =ˆ10˙.
2.6.4. Trigonometric functions of inverse trigonometric functions
2.6.5. arcsin (' (x)), arccos (' (x)), and the logarithmic function
2.6.6. arctan (' (x)) and arccot (bx)
2.6.7. arctan (' (x)) and the exponential function1 e?ax arctan (bx)2as ? (s) ?a1?sb? (s ? 1) 2F3
2.6.8. arctan (' (x)) and trigonometric functions
2.6.9. arctan (' (x)) and the logarithmic function
2.6.10. arccsc (' (x)) and algebraic functions
2.6.11. arcsec (bx) and algebraic functions
2.6.12. Products of inverse trigonometric functions
2.7. Inverse Hyperbolic Functions
2.7.1. arcsinhn (' (x)) and elementary functions
2.7.2. arccoshn (' (x)) and elementary functions
2.7.3. arctanh (ax) and elementary functions
2.7.4. arccoth (ax) and algebraic functions
2.7.5. arcsechn (' (x)) and elementary functions
2.7.6. arccschn (' (x)) and elementary functions
2.7.7. Hypebolic functions of inverse hyperbolic functions
Chapter 3 Special Functions
3.1. The Gamma ? (z), Psi (z), and Zeta (z) Functions
3.1.1. ? (' (x))
3.1.2. (ax + b)
3.1.3. (n) (ax + b)
3.1.4. (; ax + b)
3.2. The Polylogarithm Lin (z)
3.2.1. Lin (bx) and algebraic functions
3.2.2. Lin (bx) and the logarithmic or inverse trigonometric functions
3.3. The Exponential Integral Ei (z)
3.3.1. Ei (' (x)) and algebraic functions
3.3.2. Ei (' (x)) and the exponential function
3.3.3. Ei (bx) and hyperbolic or trigonometric functions
3.3.4. eax lnn x Ei (bx)
3.3.5. Products of Ei (ax)
3.4. The Sine si (z), Si (z), and Cosine ci (z) Integrals
3.4.1. si (ax), Si (ax), and ci (ax)
3.4.2. si (bx), ci (bx), and algebraic functions
3.4.3. si (bx), ci (bx), and the exponential function
3.4.4. si (bx), ci (bx), and trigonometric functions
3.4.5. Si (bx) and the logarithmic or inverse trigonometric functions
3.4.6. Si (bx), si (bx), ci (bx), and Ei (?axr)
3.4.7. si2 (bx) + ci2 (bx) and trigonometric functions
3.4.8. Products of si (bx) and ci (bx)
3.5. Hyperbolic Sine shi (z) and Cosine chi (z) Integrals
3.5.1. shi (bx), chi (bx), and algebraic functions
3.5.2. shi (bx), chi (bx), and the exponential function
3.5.3. shi (bx) and the logarithmic or inverse trigonometric functions
3.6. erf (z), erfc (z), and er (z)
3.6.1. erf (ax + b), erfc?ax + bx?1
3.6.2. erf (bx), erfc (bx), and algebraic functions
3.6.3. erf (bx), erfc (bx), and the exponential function
3.6.4. erf (bx), erfc (bx), er (bx), and algebraic or the exponential functions
3.6.5. erf (' (x)), erfc (' (x)), and algebraic functions
3.6.7. erf (bx), erfc (bx), and trigonometric functionsNotation: =ˆ10˙.
3.6.8. erfc (bx), er (bx), and the exponential or trigonometric functionsNotation: =ˆ10˙.
3.6.9. erf (bx), erfc (bx), and the logarithmic function
3.6.10. erf (ax) and inverse trigonometric functions
3.6.11. erf (bx) and Ei??ax2
3.6.12. erf (bx), erfc (bx), and si (ax), ci (ax), Si (ax)
3.6.13. Products of erf (ax), erfc (bx), er(cx)
3.6.14. Products of erf (ax), erfc (bx), er (cx), and algebraic functions
3.6.15. Products of erf (ax), erfc (bx), er (cx), and the exponential function
3.6.16. Products of erf (ax), erfc (bx), er (cx), and the logarithmic function
3.6.17. Products of erf (ax), erfc (bx), er (cx), and inverse trigonometric func-tions
3.7. The Fresnel Integrals S (z) and C (z)
3.7.1. S (' (x)), C (' (x)), and algebraic functionsNotation: =ˆ10˙.
3.7.2. S (bx), C (bx), and the exponential function
3.7.3. S (' (x)), C (' (x)), and trigonometric functions
3.7.4. S (bx), C (bx), and the logarithmic function
3.7.5. S (bx), C (bx), and si (ax), ci (ax)
3.7.6. S (bx), C (bx), and erf (apx), erfc (apx)
3.7.7. Products of S (bx) and C (bx)
3.8. The Incomplete Gamma Function ? (; z) and (; z)
3.8.1. ? (; ax), (; ax), and algebraic functions
3.8.2. ? (; ax), (; ax), and the exponential function
3.8.3. ? (; ax), (; ax), and trigonometric functions
3.8.4. ? (; ax), (; ax), and the logarithmic function
3.8.5. (; ax) and inverse trigonometric functions
3.8.6. ? (; ax), (; ax), and Ei (bx)
3.8.7. ? (; ax), (; ax), and erf (bxr), erfc (bxr), er (bxr)
3.8.8. Products of ? (; ax) and (; ax)
3.9. The Parabolic Cylinder Function D
3.9.1. D (bx) and elementary functionsNotation: =ˆ10˙.
3.9.2. D (bx) and erf (ax), erfc (ax)
3.9.3. Products of D (bxr)
3.10. The Bessel Function J (z)
3.10.1. J (bx) and algebraic functions
3.10.2. J (' (x)) and algebraic functions
3.10.3. J (' (x)) and the exponential function
3.10.4. J (bx) and trigonometric functionsNotation: =ˆ10˙.
3.10.5. J (bx) and the logarithmic function
3.10.6. J (bx) and inverse trigonometric functionsNotation: =ˆ10˙.
3.10.7. J (bx) and Ei (axr)
3.10.8. J (bx) and si (axr), Si (ax), or ci (axr)Notation: =ˆ10˙.
3.10.9. J (bx) and erf (axr), erfc (axr), or er (axr)Notation: =ˆ10˙.
3.10.10. J (bx) and S (axr), C (axr)Notation: =ˆ10˙.
3.10.11. J (bx) and ? (; axr), (; axr)Notation: =10.
3.10.12. J (bx) and D (axr)Notation: =ˆ10˙.
3.10.13. Products of J (ax)
3.10.14. J (bx) J (cx) and the exponential or trigonometric functionsNotation: =ˆ10˙.
3.10.15. J (bx) J (bx) and the logarithmic function
3.10.16. J (bx) J (bx) and inverse trigonometric functions
3.10.17. J (bx) J (bx) and Ei (?axr)
3.10.18. J (bx) J (bx) and erfc (ax), erf (a=x), ? (; ax)
3.10.19. J (' (x)) J ( (x))
3.10.20. J (' (x)) J ( (x)) and algebraic functions
3.10.21. J (axr) J (bxr) J (cx)
3.11. The Bessel Function Y (z)
3.11.1. Y (bx) and algebraic functions
3.11.2. Y ('(x)) and algebraic functions
3.11.3. Y (bx) and the exponential function
3.11.4. Y (bx) and trigonometric functionsNotation: =ˆ10˙.
3.11.5. Y (bx) and the logarithmic function
3.11.6. Y (bx) and Ei (axr)
3.11.7. Y (bx) and si (ax), ci (ax)
3.11.8. Y (bx) and erf (ax), erfc (ax), er (ax)
3.11.9. Y (bx) and S (ax), C (ax)Notation: =ˆ10˙.
3.11.10. Y (bx) and (; ax), ? (; ax)
3.11.11. Y (bx) and D (axr)
3.11.12. Y (' (x)) and J ( (x))
3.11.13. Y (bx), J (bx), and trigonometric functionsNotation: =ˆ10˙.
3.11.14. Y (bx), J (bx), and S (ax), C (ax)Notation: =ˆ10˙.
3.11.15. Y (ax) and J (bx) J (cx)Notation: =ˆ10˙.
3.11.16. Products of Y (' (x))
3.12. The Hankel Functions H(1) (z) and H(2) (z)
3.12.1. H(1) (ax), H(2) (ax)
3.12.2. H(1) (bx), H(2) (bx), and the exponential function
3.12.3. H(1) (ax), H(2) (ax), and trigonometric functions
3.12.4. H(1) (bx), H(2) (bx), and J (ax)
3.12.5. Products of H(1) (ax) and H(2) (ax)
3.13. The Modi ed Bessel Function I (z)
3.13.1. I (' (x)) and algebraic functions
3.13.2. I (' (x)) and the exponential function
3.13.3. I (ax) and trigonometric functionsNotation: =ˆ10˙.
3.13.4. I (ax) and the logarithmic function
3.13.5. I (ax) and inverse trigonometric functions
3.13.6. I (ax) and Ei (bxr)
3.13.7. I (ax) and si (bx), ci (bx)
3.13.8. I (ax) and erf (bxr), erfc (bxr)
3.13.9. I (ax) and S (bx), C (bx)
3.13.10. I (ax) and (; bx), ? (; bxr)
3.13.11. I (ax) and D (bxr)
3.13.12. I (ax) and J (bxr), Y (bxr)
3.13.13. Products of I (' (x))
3.14. The Macdonald Function K (z)
3.14.1. K (axr) and algebraic functions
3.14.2. K (' (x)) and algebraic functions
3.14.3. K (' (x)) and the exponential function
3.14.4. K (ax) and hyperbolic or trigonometric functionsNotation: =ˆ10˙.
3.14.5. K (ax) and the logarithmic function
3.14.6. K (ax) and Ei (bxr)
3.14.7. K (ax) and Si (bx), si (bx), ci (bx)
3.14.8. K (ax) and erf (bxr), er (bxr), erfc (bxr)
3.14.9. K (ax) and S (bx), C (bx)Notation: =ˆ10˙.
3.14.10. K (ax) and ? (; bx), (; bx)
3.14.11. K (ax) and D (bpx)Notation: =ˆ10˙.
3.14.12. K (' (x)) and J ( (x))Notation: =ˆ10˙.
3.14.13. K (' (x)) and Y ( (x))
3.14.14. K (ax) and J (ax), Y (ax)
3.14.15. K (' (x)) and I ( (x))
3.14.16. K (ax), I (' (x)), and the exponential functionNotation: =ˆ10˙.
3.14.17. K (ax) and I (ax), J (bx)
3.14.18. Products of K (' (x))
3.14.19. Products of K (axr) and the exponential function
3.14.20. Products of K (axr) and trigonometric or hyperbolic functionsNotation: =ˆ10˙.
3.14.21. Products of K (ax) and erf (bpx), er (bpx)Notation: =ˆ10˙.
3.14.22. Products of K (ax) and S (cx), C (cx)Notation: =ˆ10˙.
3.14.23. Products of K (ax) and J (bxr), I (cxr)
3.15. The Struve Functions H (z) and L (z)
3.15.1. H (bx), L (bx), and algebraic functions
3.15.2. H (bx), L (bx), and the exponential function
3.15.3. H (bx), L (bx), and trigonometric functionsNotation: =ˆ10˙.
3.15.4. H (bx), L (bx), and the logarithmic or inverse trigonometric functions
3.15.5. H (bx), L (bx), and ? (; ax)
3.15.6. H (bx), L (bx), and Ei??ax2, erfc (axr), D (ax)
3.15.7. H (bx) and J (ax)
3.15.8. H(bx), L (bx), and K (axr)
3.15.9. H (' (x)) ? Y (' (x)), I (' (x)) ? L (' (x))
3.16. The Anger J (z) and Weber E (z) Functions
3.16.1. J (' (x)), E (' (x)), and algebraic functions
3.16.2. J (bx), E (bx), and the exponential or trigonometric functions
3.16.3. J (bx), E (bx), and Ei??ax2or erfc (ax)
3.16.4. J (bx), E (bx), and J (ax)
3.17. The Kelvin Functions ber (z), bei (z), and ker (z), kei (z)
3.17.1. ber (bx), bei (bx), ker (bx), kei (bx), and algebraic functions
3.17.2. ber (bx), bei (bx), ker (bx), kei (bx), and the exponential function
3.17.3. ker (bx), kei (bx), and trigonometric functionsNotation: =ˆ10˙.
3.17.4. ber (bx), bei (bx), ker (bx), kei (bx), and Ei (?axr)
3.17.5. ber (bx), bei (bx), ker (bx), kei (bx), and the Bessel functions
3.17.6. ' (x) (ber2 (bx) + bei2 (bx)) and ker2 (bx) + kei2 (bx)
3.17.7. Products of ber (bx), bei (bx), ker (bx), kei (bx)
3.18. The Airy Functions Ai (z) and Bi (z)
3.18.1. Ai (bx), Ai0 (bx), Bi (bx), and algebraic functions
3.18.2. Ai (bx), Ai0 (bx), Bi (bx), and the exponential function
3.18.3. Ai (bx) and trigonometric functionsNotation: =ˆ10˙.
3.18.4. Ai (bx), Ai0 (bx), Bi (bx), and special functions
3.18.5. Products of Airy functions
3.19. The Legendre Polynomials Pn (z)
3.19.1. Pn (' (x)) and algebraic functions
3.19.2. Pn (bx) and the exponential function
3.19.3. Pn (ax + b) and Ei (cxr)
3.19.4. Pn (ax + b) and si (cxr), ci (cxr)
3.19.5. Pn (ax + b) and erf (cxr), erfc (cxr)
3.19.6. Products of Pn (axr + b)
3.20. The Chebyshev Polynomials Tn (z)
3.20.1. Tn (' (x)) and algebraic functions
3.20.2. Tn (bx) and the exponential function
3.20.3. Tn (bx) and hyperbolic functions
3.20.4. Tn (ax + b) and trigonometric functions
3.20.5. Tn (ax + b) and the logarithmic function
3.20.6. Tn (bx) and inverse trigonometric functions
3.20.7. Tn (ax + b) and Ei (cxr)
3.20.8. Tn (ax + b) and si (cxr), ci (cxr)
3.20.9. Tn (ax + b) and erf (cxr), erfc (cxr)
3.20.10. Tn (bx) and ? (; ax), (; ax)
3.20.11. Tn (' (x)) and J (cxr), I (cx)
3.20.12. Tn (' (x)) and K (cxr)
3.20.13. Tn (bx) and H (ax), L (ax)
3.20.14. Tn (ax + b) and Pm (' (x))
3.20.15. Products of Tn (' (x))
3.21. The Chebyshev Polynomials Un (z)
3.21.1. Un (' (x)) and algebraic functions
3.21.2. Products of Un (' (x))
3.22. The Hermite Polynomials Hn (z)
3.22.1. Hn (bx) and algebraic functions
3.22.2. Hn (bx) and the exponential function
3.22.3. Hn (bx) and trigonometric functions
3.22.4. Hn (bx) and the logarithmic function
3.22.5. Hn (bx) and inverse trigonometric functions
3.22.6. Hn (bx) and Ei (axr)
3.22.7. Hn (bx) and si (axr), ci (axr)
3.22.8. Hn (bx) and erf (axr), erfc (axr)
3.22.9. Hn (bx) and S (axr), C (axr)
3.22.10. Hn (bx) and (; axr), ? (; axr)
3.22.11. Hn (bx) and J (axr), I (axr)
3.22.12. Hn (bx) and Y (axr), K (axr)
3.22.13. Hn (bx) and Pm (' (x))
3.22.14. Hn (bx) and Tm (' (x)), Um (' (x))
3.22.15. Products of Hn (bx)
3.23. The Laguerre Polynomials Ln (z)
3.23.1. Ln (bx) and algebraic functions
3.23.2. Ln (bx) and the exponential function
3.23.3. Ln (bx) and trigonometric functions
3.23.4. Ln (bx) and the logarithmic function
3.23.5. Lm (bxr) and Ei (axr)
3.23.6. Ln (bx) and si (axr), ci (axr)
3.23.7. Ln (bx) and erf (axr), erfc (axr)
3.23.8. Ln (bx) and S (axr), C (axr)
3.23.9. Ln (bx) and (; axr), ? (; axr)
3.23.10. Ln (bx) and J (axr), I (axr)
3.23.11. Ln (bx) and Y (axr), K (axr)
3.23.12. Ln (bxr) and Pn (axp + c)
3.23.13. Ln (bx) and Tn (ax + c), Un (ax + c)
3.23.14. Ln (bxr) and Hn (ax)
3.23.15. Products of Ln (bx)
3.24. The Gegenbauer Polynomials Cn (z)
3.24.1. Cn (' (x)) and algebraic functions
3.24.2. Cn (bx) and the exponential function
3.24.3. Cn (bx) and hyperbolic functions
3.24.4. Cn (ax + b) and trigonometric functions
3.24.5. Cn (bx) and the logarithmic function
3.24.6. Cn (bx) and inverse trigonometric functions
3.24.7. Cn (ax + b) and Ei (axr)
3.24.8. Cn (ax + b) and si (ax), ci (ax)
3.24.9. Cn (ax + b) and erf (ax), erfc (ax)
3.24.10. Cn (bx) and ? (; ax), (; ax)
3.24.11. Cn (bx) and Bessel functions
3.24.12. Cn (bx) and H (ax), L (ax)
3.24.13. Cn (ax + b) and Pm (cxr + d)
3.24.14. Cn (bx) and Hm (ax)
3.24.15. Cn (bx) and Lm (axr)
3.24.16. Products of C-n (bx)
3.25. The Jacobi Polynomials P(ˆ; ˙)n (z)
3.25.1. P(ˆ; ˙)n (' (x)) and algebraic functions
3.25.2. P(ˆ; ˙)n (' (x)) and the exponential function
3.25.3. P(ˆ; ˙)n (' (x)) and trigonometric functions
3.25.4. P(ˆ; ˙)n (' (x)) and the logarithmic function
3.25.5. P(ˆ; ˙)n (' (x)) and Ei (bx)
3.25.6. P(ˆ; ˙)n (' (x)) and si (bpx), ci (bpx)
3.25.7. P(ˆ; ˙)n (' (x)) and erf (bxr), erfc (bxr)
3.25.8. P(ˆ; ˙)n (' (x)) and (; bx)
3.25.9. P(ˆ; ˙)n (' (x)) and I (bxr), J (bxr)
3.25.10. P(ˆ; ˙)n (' (x)) and K (bxr)
3.25.11. P(ˆ; ˙)n (' (x)) and Pm ( (x))
3.25.12. P(ˆ; ˙)n (' (x)) and Tm ( (x))
3.25.13. P(ˆ; ˙)n (' (x)) and Um ( (x))
3.25.14. P(ˆ; ˙)n (' (x)) and Hm (bpx)
3.25.15. P(ˆ; ˙)n (' (x)) and Lm (bx)
3.25.16. P(ˆ; ˙)n (' (x)) and Cm ( (x))
3.25.17. Products of P(ˆ; ˙)n (ax + b)
3.26. The Complete Elliptic Integrals K(z), E(z), and D(z)
3.26.1. K(' (x))
3.26.2. K(' (x)) and algebraic functions
3.26.3. (a ? x)K(' (x)) and algebraic functions
3.26.4. (x ? a)K(' (x)) and algebraic functions
3.26.5. E(' (x)) and algebraic functions
3.26.6. (a ? x)E(' (x)) and algebraic functions
3.26.7. (x ? a)E(' (x)) and algebraic functions
3.26.8. K(' (x)), E(' (x)), and the exponential function
3.26.9. K(' (x)), E(' (x)), and hyperbolic or trigonometric functions
3.26.10. K(' (x)), E(' (x)), and the logarithmic function
3.26.11. K(' (x)), E(' (x)), and inverse trigonometric functions
3.26.12. K(' (x)), E(' (x)), and Li2 (ax)
3.26.13. K(' (x)), E(' (x)), and Si (axr), shi (axr)
3.26.14. K(' (x)), E(' (x)), and ci (ax), chi (ax)
3.26.15. K(' (x)), E(' (x)), and erf (axr)
3.26.16. K(' (x)), E(' (x)), and S (apx), C (apx)
3.26.17. K(' (x)), E(' (x)), and (; ax)
3.26.18. K(' (x)), E(' (x)), and J (bxr), I (bxr)
3.26.19. K(' (x)), E(' (x)), and H (bxr), L (bxr)
3.26.20. K(bx), E(bx), and Tn (ax)
3.26.21. K(' (x)), E(' (x)), and Ln (ax), Hn (axr)
3.26.22. K(bx), E(bx), and Cn (ax)
3.26.23. D(' (x)) and various functions
3.26.24. Products of K(' (x))
3.26.25. Products of K(' (x)) and E(' (x))
3.26.25. Products of K(' (x)) and E(' (x))
3.26.26. Products of E(' (x))
3.26.27. Products containing D(' (x))
3.27. The Hypergeometric Function 0F1 (b; z)
3.27.1. 0F1 (b; !x) and the exponential function
3.27.2. 0F1 (b; !x) and trigonometric functions
3.27.3. 0F1 (b; !x) and sinc (pax)
3.27.4. 0F1 (b; !x) and the Bessel functions
3.27.5. 0F1 (b; !x) and ker (pax), kei (pax)
3.27.6. 0F1 (b; !x) and Ai ( 3pax), Ai0 ( 3pax)
3.28. The Kummer Conuent Hypergeometric Function 1F1 (a; b; z)
3.28.1. 1F1 (a; b; !x) and algebraic functions
3.28.2. 1F1 (a; b; !x) and the exponential function
3.28.3. 1F1 (a; b; !x) and trigonometric functions
3.28.4. 1F1 (a; b; !x) and the logarithmic function
3.28.5. 1F1 (a; b; !x) and erf (˙px), erfc (˙px)
3.28.6. 1F1 (a; b; !x) and the Bessel functions
3.28.7. 1F1 (a; b; !x) and the Struve functions
3.28.8. 1F1 (a; b; !x) and Pn (' (x))
3.28.9. 1F1 (a; b; !x) and Tn (' (x))
3.28.10. 1F1 (a; b; !x) and Un (' (x))
3.28.11. 1F1 (a; b; !x) and Hn (˙px)
3.28.12. 1F1 (a; b; !x) and Ln (˙x)
3.28.13. 1F1 (a; b; !x) and Cn (' (x))
3.28.14. 1F1 (a; b; !x) and P(ˆ; ˙)n (' (x))
3.28.15. Products of 1F1 (a; b; !xr)
3.29. The Tricomi Conuent Hypergeometric Function (a; b; z)
3.29.1. (a; b; !x) and algebraic functions
3.29.2. (a; b; !x) and the exponential function
3.29.3. (a; b; !x) and trigonometric functions
3.29.4. (a; b; !x) and the logarithmic function
3.29.5. (a; b; !x) and Ei (˙x)
3.29.6. (a; b; !x) and erf (˙px), erfc (˙px)
3.29.7. (a; b; !x) and the Bessel functions
3.29.8. (a;b;!x)andPn('(x))
3.29.9. (a; b; !x) and Tn (' (x))
3.29.10. (a; b; !x) and Un (' (x))
3.29.11. (a; b; !x) and Hn (˙px)
3.29.12. (a; b; !x) and Ln (˙x)
3.29.13. (a; b; !x) and Cn (' (x))
3.29.14. (a; b; !x) and P(; )n (' (x))
3.29.15. (a; b; !x) and K(' (x)), E(' (x))
3.29.16. (a; b; !x) and 1F1 (a; b; ˙x)
3.29.17. Products of (a; b; !x)
3.30. The Whittaker Functions Mˆ; ˙ (z) and Wˆ; ˙ (z)
3.30.1. Wˆ; ˙ (ax)
3.30.2. Mˆ; ˙ (ax), Wˆ; ˙ (bx), and the exponential function
3.30.3. Wˆ; ˙ (ax) and hyperbolic functions
3.30.4. Wˆ; ˙ (ax) and L˙ˆ (bx)
3.30.5. Wˆ; ˙ (ax) and 1F1 (b; c; dx), (b; c; dx)
3.30.6. Products of M; (ax) and W; (bx)
3.31. The Gauss Hypergeometric Function 2F1 (a; b; c; z)
3.31.1. 2F1 (a; b; c; !x) and algebraic functions
3.31.2. 2F1a; b; c;!xand algebraic functions
3.31.3. 2F1 (a; b; c; !xr) and various functions
3.31.4. 2F1a; b; c;! ? x!and algebraic functions
3.31.5. 2F1a; b; c;!x + !and algebraic functions
3.31.6. 2F1a; b; c;x ? !xand algebraic functions
3.31.7. 2F1a; b; c;xx + !and algebraic functions
3.31.8. 2F1a; b; c;4!x?x + !2and algebraic functions
3.31.9. 2F1a; b; c; ?4!x(x ? !)2and algebraic functions
3.31.10. 2F1a; b; c; 1x3 + 1x2 + 1x + 1 2x3 + 2x2 + 2x + 2and algebraic functions
3.31.11. 2F1a; b; c;!1x + ˙1!2x + ˙2and algebraic functions
3.31.12. 2F1a; b; c;px ?px + !2pxand algebraic functions
3.31.13. 2F1a; b; c;p! ?px + !2p!and algebraic functions
3.31.14. 2F1a; b; c;px + ! ?pxpx + ! +pxand algebraic functions
3.31.15. 2F1a; b; c;px + ! ?p!px + ! +p!and algebraic functions
3.31.16. 2F1a; b; c;x ? 2p!px + ! + 2!xand algebraic functions
3.31.17. 2F1a; b; c;2x ? 2pxpx + ! + !!and algebraic functions
3.31.18. 2F1a; b; c;2x ? 2pxpx + ! + !2px?px ?px + !and algebraic functions
3.31.19. 2F1a; b; c;x ? 2p!px + ! + 2!2p!?p! ?px + !and algebraic functions
3.31.20. 2F1a; b; c;x ?px2 + !22xand algebraic functions
3.31.21. 2F1a; b; c;! ?px2 + !22!and algebraic functions
3.31.22. 2F1a; b; c;px2 + !2 ? xpx2 + !2 + xand algebraic functions
3.31.23. 2F1a; b; c;px2 + !2 ? !px2 + !2 + !and algebraic functions
3.31.24. 2F1a; b; c;x2 ? 2!px2 + !2 + 2!2x2and algebraic functions
3.31.25. 2F1a; b; c;2x2 ? 2xpx2 + !2 + !2!2and algebraic functions
3.31.26. 2F1a; b; c;2x2 ? 2xpx2 + !2 + !22x?x ?px2 + !2and algebraic functions
3.31.27. 2F1a; b; c;x2 ? 2!px2 + !2 + 2!22!?! ?px2 + !2and algebraic functions
3.31.28. 2F1 (a; b; c; ' (x)) and algebraic functions
3.31.29. 2F1 (a; b; c; ' (x)) and the exponential function
3.31.30. 2F1 (a; b; c; !x + ˙) and trigonometric functions
3.31.31. 2F1 (a; b; c; ' (x)) and the Bessel functions
3.31.32. 2F21 (a; b; c; ' (x))
3.31.33. 2F1a1; b1; c1; ?x!2F1a2; b2; c2; ?x!and algebraic functions
3.31.34. 2F1a1; b1c1; 1 ? !1x2F1a2; b2c2; 1 ? !2xand algebraic functions
3.31.35. 2F1a1; b1c1;p!?px+!2p!2F1a2; b2c2;p!?px+!2p!and algebraic functions
3.31.36. 2F1a1; b1c1;px?px+!2px2F1a2; b2c2;px?px+!2pxand algebraic functions
3.31.37. 2F1a1; b1c1; ?2px(pxpx+!)!2F1a2; b2c2; ?2px(px+px+!)!and algebraicfunctions
3.31.38. 2F1a1; b1c1;2p!(px+!?p!)x2F1a2; b2c2; ?2p!(px+!+p!)xand algebraic functions
3.31.39. 2F1a1; b1c1;2p!(p!+p!?x)x2F1a2; b2c2; ?2p!(p!+p!?x)xand algebraic functions
3.32. The Generalized Hypergeometric Function 3F2a1; a2 ; a3b1; b2; z
3.32.1. 3F2 a1; a2; a3b1; b2; ' (x)and algebraic functions
3.33. The Generalized Hypergeometric Functions pFq ((ap) ; (bq) ; z)
3.33.1. pFq ((ap) ; (bq) ; ' (x)) and algebraic functions
3.33.2. pFq ((ap) ; (bq) ; !xr) and the exponential function
3.33.3. pFq ((ap) ; (bq) ; !xr) and the logarithmic function
3.33.4. pFq ((ap) ; (bq) ; !x) and inverse trigonometric functions
3.33.5. pFq ((ap) ; (bq) ; !x) and Ei (˙xr)
3.33.6. pFq ((ap) ; (bq) ; !x) and erfc (˙xr)
3.33.7. pFq ((ap) ; (bq) ; !x) and ? (; xr)
3.33.8. pFq ((ap) ; (bq) ; !xr) and J (˙x), Y (˙x)
3.33.9. pFq ((ap) ; (bq) ; !x) and K (˙xr)
3.33.10. pFq ((ap) ; (bq) ; !x) and Ai (˙xr)
3.33.11. pFq ((ap) ; (bq) ; !xr) and Pn (' (x))
3.33.12. pFq ((ap) ; (bq) ; !xr) and Tn (' (x))
3.33.13. pFq ((ap) ; (bq) ; !xr) and Un (' (x))
3.33.14. pFq ((ap) ; (bq) ; !x) and Hn (˙xr)
3.33.15. pFq ((ap) ; (bq) ; !x) and Ln (˙xr)
3.33.16. pFq ((ap) ; (bq) ; !x) and Cn (' (x))
3.33.17. pFq ((ap) ; (bq) ; !xr) and P( ; )n (' (x))
3.33.18. pFq ((ap) ; (bq) ; !xr) and K(' (x)), E(' (x))
3.33.19. pFq ((ap) ; (bq) ; !xr) and P (' (x)), P (' (x))
3.33.20. pFq ((ap) ; (bq) ; !xr) and Q (' (x))
3.33.21. pFq ((ap) ; (bq) ; !xr) and (a; b; ˙x)
3.33.22. pFq ((ap) ; (bq) ; !xr) and 2F1 (a; b; ' (x))
3.33.23. Products of pFq ((ap) ; (bq) ; !xr)
3.34. The Appell Functions
3.34.1. The Appell and algebraic functions
3.35. The Humbert Functions
3.35.1. The Humbert and algebraic functions
3.35.2. The Humbert and the exponential functions
3.36. The Meijer G-Function
3.36.1. Gmnpq !x (ap)(bq)
3.36.2. Gmnpq !x (ap)(bq) and algebraic functions
3.36.3. Gmnpq !x (ap)(bq) and the exponential function
3.36.4. Gmnpq !x (ap)(bq) and trigonometric functions
3.36.5. Gmnpq !x (ap)(bq) and the Bessel functions
3.36.5. Gmnpq !x (ap)(bq) and the Bessel functions
3.36.6. Gmnpq !x (ap)(bq) and orthogonal polynomials
3.36.7. Gmnpq !x (ap)(bq) and the Legendre function
3.36.8. Gmnpq !x (ap)(bq) and the Struve function
3.36.9. Gmnpq !x (ap)(bq) and the Whittaker functions
3.36.10. Gmnpq !x (ap)(bq) and hypergeometric functions
3.36.11. Products of two Meijer's G-functions
3.37. Various Special Functions
3.37.1. The exponential integral E (z)
3.37.2. The theta functions j (b; ax)
3.37.3. The generalized Fresnel integrals S (z; ) and C (z; )
3.37.4. The integral Bessel functions
3.37.5. The Lommel functions
3.37.6. The Owen and H-functions
3.37.7. The Bessel{Maitland and generalized Bessel{Maitland functions
3.37.8. Other functions
Appendix I Some Properties of the Mellin Transforms
Appendix II Conditions of Convergence
Bibliography
Index of Notations for Functions and Constants
Index of Notations for Symbols
