Laplace Transform Table

Laplace transform is a mathematical tool that converts a function of a real variable to a function of a complex variable s (complex frequency). It is used for solving differential equations. Here is the Laplace transform table.

Contents show

S.No.	Function	Laplace transform
1.	1	$\frac{1}{s}$
2.	e^at	$\frac{1}{s-a}$
3.	e⁻^at	$\frac{1}{s+a}$
4.	tⁿ , where n= 1,2,3,……	$\frac{n!}{s^{n+1}}$
5.	t^p , p > -1	$\frac{\Gamma (p+1)}{s^{p+1}}$
6.	tⁿ e^at, where n= 1,2,3,……	$\frac{n!}{(s-a)^{n+1}}$
7.	$\sqrt{t}$	$\frac{\sqrt{\pi}}{{2s^{\frac{3}{2}}}}$
8.	t^n-½ , where n= 1,2,3,……	$\frac{1 \times 3\times 5......\times (2n+1)\sqrt{\pi}}{{2^ns^{n+\frac{1}{2}}}}$
9.	sin(at)	$\frac{a}{s^2+a^2}$
10.	cos(at)	$\frac{s}{s^2+a^2}$
11.	t sin(at)	$\frac{2as}{(s^2+a^2)^2}$
12.	t cos(at)	$\frac{s^2-a^2}{(s^2+a^2)^2}$
13.	sin(at) − at cos(at)	$\frac{2a^3}{(s^2+a^2)^2}$
14.	sin(at) + at cos(at)	$\frac{2as^2}{(s^2+a^2)^2}$
15.	cos(at) − at sin(at)	$\frac{s(s^2-a^2)}{(s^2+a^2)^2}$
16.	cos(at) + at sin(at)	$\frac{s(s^2+3a^2)}{(s^2+a^2)^2}$
17.	sin(at + b)	$\frac{s(sin(b))+acos(b)}{s^2+a^2}$
18.	cos(at + b)	$\frac{s(cos(b))-asin(b)}{s^2+a^2}$
19.	sinh(at)	$\frac{a}{s^2-a^2}$
20.	cosh(at)	$\frac{s}{s^2-a^2}$
21.	e^at sin(bt)	$\frac{b}{(s-a)^2+b^2}$
22.	e^at cos(bt)	$\frac{s-a}{(s-a)^2+b^2}$
23.	e^at sinh(bt)	$\frac{b}{(s-a)^2-b^2}$
24.	e^at cosh(bt)	$\frac{s-a}{(s-a)^2-b^2}$
25.	t sinh(bt)	$\frac{2bs}{(s^2-b^2)^2}$
26.	t cosh(bt)	$\frac{s^2+b^2}{(s^2-b^2)^2}$

Laplace Transform Properties Table

Let F(s) be the laplace transform of f(t) i.e.

$f(t)\xrightarrow[]{laplace\; transform}F(s)$

S.No.	Function	Laplace transform
1.	f(at)	$\frac{1}{a}F(\frac{s}{a})$
2.	f(t-a)	$e^{-as}F(s)$
3.	e^at f(t)	$F(s-a)$
4.	tⁿ f(t)	$(-1)^n\frac{\mathrm{d^2} F(s)}{\mathrm{d} t^2}$
5.	$\frac{1}{t}f(t)$	$\int_{s}^{\infty}F(u)du$
6.	$\int_{0}^{t}f(v)dv$	$\frac{F(s)}{s}$
7.	$\int_{0}^{t}f(t-\tau)g(\tau)d\tau$	$F(s)G(s)$
8.	$\frac{\mathrm{d} f(t)}{\mathrm{d} t}=f_{t}^{'}$	$sF(s)-f(0)$
9.	$\frac{\mathrm{d^2} f(t)}{\mathrm{d} t^2}=f_{t}^{''}$	$s^2F(s)-sf(0)-f^{'}(0)$
10.	$\frac{\mathrm{d^n} f(t)}{\mathrm{d} t^n}=f_{t}^{(n)}$	$s^nF(s)-s^{n-1}f(0)-s^{n-2}f^{'}(0).....-f^{(n-1)}(0)$