Differential Equations - Laplace Transforms: Table

	$f\left(t\right)=L^{-1}\left\{F\left(s\right)\right\}$	$F\left(s\right)=L\left\{f\left(t\right)\right\}$
	1	$\frac{1}{s}$
	$e^{at}$	$\frac{1}{s-a}$
	$t^n,n=1,2,3,\dots$	$\frac{n!}{s^{n+1}}$
	$t^p$, $p>-1$	$\frac{\Gamma \left(p+1\right)}{s^{p+1}}$
	$\sqrt{t}$	$\frac{\sqrt{\pi }}{2s^{\frac{3}{2}}}$
	$t^{n-\frac{1}{2}},n=1,2,3,\dots$	$\frac{1\cdot 3\cdot 5\cdots \left(2n-1\right)\sqrt{\pi }}{2^n{s}^{n+\frac{1}{2}}}$
	$\sin \left(at\right)$	$\frac{a}{s^2+a^2}$
	$\cos \left(at\right)$	$\frac{s}{s^2+a^2}$
	$t\sin \left(at\right)$	$\frac{2as}{{\left(s^2+a^2\right)}^2}$
	$t\cos \left(at\right)$	$\frac{s^2-a^2}{{\left(s^2+a^2\right)}^2}$
	$\sin \left(at\right)-at\cos \left(at\right)$	$\frac{2a^3}{{\left(s^2+a^2\right)}^2}$
	$\sin \left(at\right)+at\cos \left(at\right)$	$\frac{2a{s}^2}{{\left(s^2+a^2\right)}^2}$
	$\cos \left(at\right)-at\sin \left(at\right)$	$\frac{s\left(s^2-a^2\right)}{{\left(s^2+a^2\right)}^2}$
	$\cos \left(at\right)+at\sin \left(at\right)$	$\frac{s\left(s^2+3a^2\right)}{{\left(s^2+a^2\right)}^2}$
	$\sin \left(at+b\right)$	$\frac{s\sin \left(b\right)+a\cos \left(b\right)}{s^2+a^2}$
	$\cos \left(at+b\right)$	$\frac{s\cos \left(b\right)-a\sin \left(b\right)}{s^2+a^2}$
	$\sinh \left(at\right)$	$\frac{a}{s^2-a^2}$
	$\cosh \left(at\right)$	$\frac{s}{s^2-a^2}$
	$e^{at}\sin \left(bt\right)$	$\frac{b}{{\left(s-a\right)}^2+b^2}$
	$e^{at}\cos \left(bt\right)$	$\frac{s-a}{{\left(s-a\right)}^2+b^2}$
	$e^{at}\sinh \left(bt\right)$	$\frac{b}{{\left(s-a\right)}^2-b^2}$
	$e^{at}\cosh \left(bt\right)$	$\frac{s-a}{{\left(s-a\right)}^2-b^2}$
	$t^n{e}^{at},n=1,2,3,\dots$	$\frac{n!}{{\left(s-a\right)}^{n+1}}$
	$f\left(ct\right)$	$\frac{1}{c}F\left(\frac{s}{c}\right)$
	$u_c\left(t\right)=u\left(t-c\right)$ Heaviside Function	$\frac{e^{-cs}}{s}$
	$\delta \left(t-c\right)$ Dirac Delta Function	$e^{-cs}$
	$u_c\left(t\right)f\left(t-c\right)$	$e^{-cs}F\left(s\right)$
	$u_c\left(t\right)g\left(t\right)$	$e^{-cs}L\left\{g\left(t+c\right)\right\}$
	$e^{ct}f\left(t\right)$	$F\left(s-c\right)$
	$t^{n}f\left(t\right),n=1,2,3,\dots$	${\left(-1\right)}^n{F}^{\left(n\right)}\left(s\right)$
	$\frac{1}{t}f\left(t\right)$	${\int }_s^{\infty }F\left(u\right)du$
	${\int }_0^{t}f\left(v\right)dv$	$\frac{F\left(s\right)}{s}$
	${\int }_0^{t}f\left(t-\tau \right)g\left(\tau \right)d\tau$	$F\left(s\right)G\left(s\right)$
	$f\left(t+T\right)=f\left(t\right)$	$\frac{{\int }_0^T{e}^{-st}f\left(t\right)dt}{1-e^{-sT}}$
	$f^{\prime }\left(t\right)$	$sF\left(s\right)-f\left(0\right)$
	$f^{\prime \prime }\left(t\right)$	$s^{2}F\left(s\right)-sf\left(0\right)-f^{\prime }\left(0\right)$
	$f^{\left(n\right)}\left(t\right)$	$s^{n}F\left(s\right)-s^{n-1}f\left(0\right)-s^{n-2}{f}^{\prime }\left(0\right)\cdots -s{f}^{\left(n-2\right)}\left(0\right)-f^{\left(n-1\right)}\left(0\right)$

Table Notes

1. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas.
2. Recall the definition of hyperbolic functions.$$\cosh \left(t\right)=\frac{e^t+e^{-t}}{2}\sinh \left(t\right)=\frac{e^t-e^{-t}}{2}$$
3. Be careful when using “normal” trig function vs. hyperbolic functions. The only difference in the formulas is the “$+a^2$” for the “normal” trig functions becomes a “$-a^2$” for the hyperbolic functions!
4. Formula #4 uses the Gamma function which is defined as$$\Gamma \left(t\right)={\int }_0^{\infty }{e}^{-x}{x}^{t-1}dx$$If $n$ is a positive integer then,
   $$\Gamma \left(n+1\right)=n!$$The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function
   $$\begin{array}{c}\Gamma \left(p+1\right)=p\Gamma \left(p\right)\\ p\left(p+1\right)\left(p+2\right)\cdots \left(p+n-1\right)=\frac{\Gamma \left(p+n\right)}{\Gamma \left(p\right)}\\ \Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }\end{array}$$