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# Formelsammlung Mathematik: Integraltransformationen

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##### 1.1
${\displaystyle F(s)={\mathcal {F}}[f(t)](s)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)e^{-ist}\,dt}$

##### 1.2
${\displaystyle f(t)={\mathcal {F}}^{-1}[F(s)](t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }F(s)e^{ist}\,ds}$

##### 1.3
${\displaystyle {\mathcal {F}}[(f*g)(t)](s)={\sqrt {2\pi }}\cdot {\mathcal {F}}[f(t)](s)\cdot {\mathcal {F}}[g(t)](s)}$

##### 2.1
${\displaystyle F(s)={\mathcal {L}}[f(t)](s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}$

##### 2.2
${\displaystyle f(t)={\mathcal {L}}^{-1}[F(s)](t)={\frac {1}{2\pi i}}\int _{a-i\infty }^{a+i\infty }F(s)e^{st}\,ds}$

##### 2.3
${\displaystyle {\mathcal {L}}[(f*g)(t)](s)=F(s)\cdot G(s)}$

##### 3.1
${\displaystyle F(s)={\mathcal {M}}[f(t)](s)=\int _{0}^{\infty }f(t)\,t^{s-1}\,dt}$

##### 3.2
${\displaystyle f(t)={\mathcal {M}}^{-1}[F(s)](t)={\frac {1}{2\pi i}}\int _{a-i\infty }^{a+i\infty }F(s)\,t^{-s}\,ds}$

##### 3.3
${\displaystyle {\mathcal {M}}\left[(f*g)(t)\right](s)=F(s)\cdot G(s)}$