Formelsammlung Mathematik: Integraltransformationen

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1.1

$F(s)=ℱ[f(t)](s)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-ist}dt$

1.2

$f(t)={ℱ}^{-1}[F(s)](t)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(s)e^{ist}ds$

1.3

$ℱ[(f*g)(t)](s)=\sqrt{2\pi }\cdot ℱ[f(t)](s)\cdot ℱ[g(t)](s)$

2.1

$F(s)=ℒ[f(t)](s)=\underset{0}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-st}dt$

2.2

$f(t)={ℒ}^{-1}[F(s)](t)=\frac{1}{2\pi i}\underset{a-i\mathrm{\infty }}{\overset{a+i\mathrm{\infty }}{\int }}F(s)e^{st}ds$

2.3

$ℒ[(f*g)(t)](s)=F(s)\cdot G(s)$

3.1

$F(s)=ℳ[f(t)](s)=\underset{0}{\overset{\mathrm{\infty }}{\int }}f(t)t^{s-1}dt$

3.2

$f(t)={ℳ}^{-1}[F(s)](t)=\frac{1}{2\pi i}\underset{a-i\mathrm{\infty }}{\overset{a+i\mathrm{\infty }}{\int }}F(s)t^{-s}ds$

3.3

$ℳ[(f*g)(t)](s)=F(s)\cdot G(s)$