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This Fourier Transform property says s ( 0 ) {\displaystyle s(0)\!} becomes ∫ − ∞ ∞ S ( f ) d f {\displaystyle \int _{-\infty }^{\infty }S(f)df}
s ( 0 ) = s ( t ) | t = 0 = F − 1 [ S ( f ) ] ) | t = 0 {\displaystyle s(0)=s(t)|_{t=0}={\mathcal {F}}^{-1}\left[S(f)\right])|_{t=0}}
Evaluating the inverse Fourier transform at time zero
s ( 0 ) = ∫ − ∞ ∞ S ( f ) e j 2 π f ( 0 ) d f = ∫ − ∞ ∞ S ( f ) d f {\displaystyle s(0)=\int _{-\infty }^{\infty }S(f)e^{j2\pi f(0)}df=\int _{-\infty }^{\infty }S(f)df}
Therefore this process shows that s ( 0 ) {\displaystyle s(0)\!} is ∫ − ∞ ∞ S ( f ) d f {\displaystyle \int _{-\infty }^{\infty }S(f)df}
becomes 
![{\displaystyle s(0)=s(t)|_{t=0}={\mathcal {F}}^{-1}\left[S(f)\right])|_{t=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f19a8736f9b71134613aad997d291ba1b946b46c)
is