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

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

Published $1968$.

Definition of the Laplace Transform of $F \left({t}\right)$

$32.1$: Definition of Laplace Transform: $\displaystyle \mathcal L \left\{ {F \left({t}\right)} \right\} = \int_0^\infty e^{-st} F\left({t}\right) \mathrm d t = f \left({s}\right)$

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 $f \left({s}\right)$

If $\mathcal L \left\{{ F \left({t}\right) }\right\} = f \left({s}\right)$, then we say that $F \left({t}\right) = \mathcal L^{-1} \left\{ {f \left({s}\right)} \right\}$ is the inverse Laplace transform of $f \left({s}\right)$.

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

Complex Inversion Formula

The inverse Laplace transform of $f \left({s}\right)$ can be found directly by methods of complex variable theory. The result is:

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

where $c$ is chosen so that all the singular points of $f \left({s}\right)$ lie to the left of the line $\operatorname{Re} \left({s}\right) = c$ 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 $-t \, \map F t$