Laplace Transforms Involving Fractional Operations

Like integer-order systems, the Laplace transform and its inversion are very important. In this appendix, the Laplace transform definition is given first. Then some essential special functions are described. Finally, tables of inversions of Laplace transforms involving fractional and irrational-order operators are given [243].

C.1 Laplace Transforms

For a time-domain function , its Laplace transform in the -domain is defined as

C.1

where is the notation of Laplace transform.

If the Laplace transform of a signal is , the inverse Laplace transform of is defined as

C.2

where is greater than the real part of all the poles of function .

C.2 Special Functions for Laplace Transforms

Some special functions for Laplace transforms are listed in Table C.1.

Special functions	Definition
Mittag–Leffler function
Dawson function
erf function
erfc function
Hermite polynomial
Bessel function	is the solution to
Extended Bessel function

C.3 Laplace Transform Tables

Inversions of Laplace transforms involving fractional and irrational operators are collected in Table C.2.