Laplace Transform

The Laplace transform maps a time-domain function f(t) to a complex-frequency-domain function F(s) by an exponential-weighted integral, converting differential equations into algebraic ones. It turns differentiation into multiplication by s (with initial-value terms), so linear ODEs with constant coefficients become rational equations in s. The solution is recovered by inverse-transforming back to the t domain via the table or partial fractions. The transform is linear. Entries 24-26 give the scaling and shifting rules: frequency-shift by an exponential factor in t, time-shift by multiplication by e^-cs in s. The table pairs each f(t) with its F(s). Reading right-to-left gives the inverse transform.

#	$f\left(t\right)=L^{-1}\left\{F\left(s\right)\right\}$	$F\left(s\right)=L\left\{f\left(t\right)\right\}$
1	1	$\frac{1}{s}$
2	$e^{at}$	$\frac{1}{s-a}$
3	$t^{n}\text{ }\text{ }\text{ }\text{ }\left(n=1,2,3...\right)$	$\frac{n!}{s^{\left(n+1\right)}}$
4	$t^{p}\left(p>-1\right)$	$\frac{\ulcorner\left(p+1\right)}{s^{\left(p+1\right)}}$
5	$\sqrt{t}$	$\frac{\sqrt{\pi}}{2\text{ }s^{\frac{3}{2}}}$
6	$t^{\frac{n-1}{2}}\text{ }\text{ }\text{ }\left(n=1,2,3...\right)$	$\frac{1\times3\times5...\left(2n-1\right)\sqrt{\pi}}{2^{n}s^{\frac{n+1}{2}}}$
7	$\sin\left(at\right)$	$\frac{a}{s^{2}+a^{2}}$
8	$\cos\left(at\right)$	$\frac{s}{s^{2}+a^{2}}$
9	$t\text{ }\sin\left(at\right)$	$\frac{2as}{\left(s^{2}+a^{2}\right)^{2}}$
10	$t\text{ }\cos\left(at\right)$	$\frac{s^{2}-a^{2}}{\left(s^{2}+a^{2}\right)^{2}}$
11	$\sin\left(at\right)-at\text{ }\cos\left(at\right)$	$\frac{2a^{3}}{\left(s^{2}+a^{2}\right)^{2}}$
12	$\sin\left(at\right)+at\text{ }\cos\left(at\right)$	$\frac{2as^{2}}{\left(s^{2}+a^{2}\right)^{2}}$
13	$\cos\left(at\right)-at\text{ }sin\left(at\right)$	$\frac{s\left(s^{2}-a^{2}\right)}{\left(s^{2}+a^{2}\right)^{2}}$
14	$\cos\left(at\right)+at\text{ }sin\left(at\right)$	$\frac{s\left(s^{2}+3a^{2}\right)}{\left(s^{2}+a^{2}\right)^{2}}$
15	$\sin\left(at+b\right)$	$\frac{s\text{ }\sin\left(b\right)+a\text{ }\cos\left(b\right)}{s^{2}+a^{2}}$
16	$\cos\left(at+b\right)$	$\frac{s\text{ }\cos\left(b\right)-a\text{ }\sin\left(b\right)}{s^{2}+a^{2}}$
17	$\sinh\left(at\right)$	$\frac{a}{s^{2}-a^{2}}$
18	$\cosh\left(at\right)$	$\frac{s}{s^{2}-a^{2}}$
19	$e^{at}\sin\left(bt\right)$	$\frac{b}{\left(s-a\right)^{2}+b}$
20	$e^{at}\cos\left(bt\right)$	$\frac{s-a}{\left(s-a\right)^{2}+b}$
21	$e^{at}\sinh\left(bt\right)$	$\frac{b}{\left(s-a\right)^{2}-b}$
22	$e^{at}\cosh\left(bt\right)$	$\frac{s-a}{\left(s-a\right)^{2}-b}$
23	$t^{n}e^{at}\text{ }\text{ }\text{ }\text{ }\left(n=1,2,3...\right)$	$\frac{n!}{\left(s-a\right)^{\left(n+1\right)}}$
24	$f\left(ct\right)$	$\left(\frac{1}{c}\right)F\left(\frac{s}{c}\right)$
25	$u_{c}\left(t\right)=u\left(t-c\right)$	$\frac{e^{-cs}}{s}$
26	$\wp\left(t-c\right)$	$e^{-cs}$

Notes:

1. The list is not inclusive and only contauin some of the most commonly used Laplace transforms and Formulase.

2. deffinition of hyperbolic functions

The hyperbolic functions in entries 17-22:

3. Be coreful while using “normal” trigonometric function vs hyperbolic trigonometric function. The only difference in the formaulas is “+a²” forthe “normal” trigonometric functions becomes “+a²” for the hyperbolic trigonometric functions.

4. Formula #4 uses Gamma Function which is defined as

The Gamma function extends the factorial to non-integer arguments:

if n is a positive intrger then,

At positive integers it reduces to the factorial:

The Gamma function extends the normal factorial. Two quick facts.

Recurrence relation:

Half-integer value: