Applied Mathematics/Laplace Transforms

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The Laplace transform is an integral transform which is widely used in physics and engineering. Laplace transform is denoted as \displaystyle\mathcal{L} \left\{f(t)\right\}.

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace.

Definition[edit]

For a function f(t), using Napier's constant"e" and complex number "s", the Laplace transform F(s) is defined as follow:

F(s) = \mathcal{L} \left\{f(t)\right\}(s)=\int_0^{\infty} e^{-st} f(t) \,dt

The parameter s is a complex number:

s = \sigma + i \omega, \, with real numbers σ and ω.

This F(s) is the Laplace transform of f(t).

Examples of Laplace transform[edit]

Examples of Laplace transform
function result of Laplace transform
C (constant) \frac{C}{s}
t \frac{1}{s^2}
t^n (n is natural number) \frac{n!}{s^{n+1}}
\frac{t^{n-1}}{(n-1)!} \frac{1}{s^{n}}
e^{at} \frac{1}{s-a}
e^{-at} \frac{1}{s+a}
{\rm cos}\ \omega t \frac{s}{s^2 + {\omega}^2}
{\rm sin}\ \omega t \frac{\omega}{s^2 + {\omega}^2}
 \frac{t^{n-1}}{\Gamma(n)} \frac{1}{s^{n}} (n>0)
 \delta (t-a) (Delta function) e^{-as}
 H(t-a) (Heaviside function)  \frac {e^{-as}}{s}

Examples of calculation[edit]

(1)Suppose f(t)=C (C = constant)
\int_0^{\infty} e^{-st} C \,dt
=\frac{C}{s}
=F(s)


(2)Suppose f(t)=e^{-at}
\int_0^{\infty} e^{-st} \cdot e^{-at} \,dt
=\int_0^{\infty} e^{-(s+a)t} \,dt
=\left[\frac{-e^{-(s+a)t}}{s+a}\right]_0^\infty
=\frac{1}{s+a}