# Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 32

## Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 32

Published $1968$.

### Definition of the Laplace Transform of $\map F t$

$32.1$: Definition of Laplace Transform: $\displaystyle \laptrans {\map F t} = \int_0^\infty e^{-s t} \map F t \rd t = \map f s$

In general $f \left({s}\right)$ will exist for $s > a$ where $a$ is some constant $\mathcal L$ is called the Laplace transform operator.

### Definition of the Inverse Laplace Transform of $\map f s$

If $\laptrans {\map F t} = \map f s$, then we say that $\map F t = \invlaptrans {\map f s}$ is the inverse Laplace transform of $\map f s$.

$\mathcal L^{-1}$ is called the inverse Laplace transform operator.

### Complex Inversion Formula

The inverse Laplace transform of $\map f s$ can be found directly by methods of complex variable theory. The result is:

$32.2$: Definition of Inverse Laplace Transform: $\displaystyle \map F t = \frac 1 {2 \pi i} \int_{c \mathop - i \, \infty}^{c \mathop + i \, \infty} e^{s t} \map f s \rd s = \frac 1 {2 \pi i} \lim_{T \mathop \to \infty} \int_{c \mathop - i \, T}^{c \mathop + i \, T} e^{s t} \map f s \rd s$

where $c$ is chosen so that all the singular points of $\map f s$ lie to the left of the line $\map \Re s = c$ in the complex $s$ plane.

### Table of General Properties of Laplace Transforms

In the following, $s$ and $t$ are the independent variables of the real functions $f$ and $F$ respectively.

$f$ denotes results of the Laplace transform on functions denoted with $F$.

$a$ and $b$ are constants.

$32.3$: Laplace Transform of $a \map {F_1} t + b \map {F_2} t$
$32.4$: Laplace Transform of $a \, \map F {a t}$
$32.5$: Laplace Transform of $e^{a t} \, \map F t$
$32.6$: Laplace Transform of $\map F {t - a}$
$32.7$: Laplace Transform of $\map {F'} t$
$32.8$: Laplace Transform of $\map {F''} t$
$32.9$: Laplace Transform of $\map {F^{\paren n} } t$
$32.10$: Laplace Transform of $-t \, \map F t$