Tables of Laplace Transforms

This material represents a collection of integrals of the Laplace- and inverse Laplace Transform type. The usef- ness of this kind of information as a tool in various branches of Mathematics is firmly established. Previous publications include the contributions by A. Erdelyi and Roberts and Kaufmann (see References). Special consideration is given to results involving higher functions as integrand and it is believed that a substantial amount of them is presented here for the first time. Greek letters denote complex parameters within the given range of validity. Latin letters denote (unless otherwise stated) real positive parameters and a possible extension to complex values by analytic continuation will often pose no serious problem. The authors are indebted to Mrs. Jolan Eross for her tireless effort and patience while typing this manu­ script. Oregon State University Corvallis, Oregon Eastern Michigan University Ypsilanti, Michigan The Authors Contents Part I. Laplace Transforms In troduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1 General Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. 2 Algebraic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1. 3 Powers of Arbitrary Order. . . . . . . . . . . . . . . . . . . . . . . . 21 1. 4 Sectionally Rational- and Rows of Delta Functions 28 1. 5 Exponential Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1. 6 Logarithmic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1. 7 Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 54 1. 8 Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . 81 1. 9 Hyperbolic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 1. 10 Inverse Hyperbolic Functions. . . . . . . . . . . . . . . . . . . . . 99 1. 11 Orthogonal Polynomials . . . . . . . ?. . . . . . . . . . . . . . . . . . . 103 1. 12 Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1. 13 Bessel Functions of Order Zero and Unity . . . . . . . . . 119 1. 14 Bessel Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 1. 15 Modified Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . .

I. Laplace Transforms.- 1.1 General Formulas.- 1.2 Algebraic Functions.- 1.3 Powers of Arbitrary Order.- 1.4 Sectionally Rational- and Rows of Delta Functions.- 1.5 Exponential Functions.- 1.6 Logarithmic Functions.- 1.7 Trigonometric Functions.- 1.8 Inverse Trigonometric Functions.- 1.9 Hyperbolic Functions.- 1.10 Inverse Hyperbolic Functions.- 1.11 Orthogonal Polynomials.- 1.12 Legendre Functions.- 1.13Bessel Functions of Order Zero and Unity.- 1.14 Bessel Functions.- 1.15 Modified Bessel Functions.- 1.16 Functions Related to Bessel Functions and Kelvin Functions.- 1.17 Whittaker Functions and Special Cases.- 1.18 Elliptic Functions.- 1.19 Gauss’ Hypergeometric Function.- 1.20 Miscellaneous Functions.- 1.21 Generalized Hypergeometric Functions.- II. Inverse Laplace Transforms.- 2.1 General Formulas.- 2.2 Rational Functions.- 2.3 Irrational Algebraic Functions.- 2.4 Powers of Arbitrary Order.- 2.5 Exponential Functions.- 2.6 Logarithmic Functions.- 2.7 Trigonometric- and Inverse Functions.- 2.8 Hyperbolic- and Inverse Functions.- 2.9 Orthogonal Polynomials.- 2.10 Gamma Function and Related Functions.- 2.11 Legendre Functions.- 2.12 Bessel Functions.- 2.13 Modified Bessel Functions.- 2.14 Functions Related to Bessel Functions and Kelvin Functions.- 2.15 Special Cases of Whittaker Functions.- 2.16 Parabolic Cylinder Functions and Whittaker Functions.- 2.17 Elliptic Integrals and Elliptic Functions.- 2.18 Gauss’ Hypergeometric Functions.- 2.19 Generalized Hypergeometric Functions.- 2.20 Miscellaneous Functions.