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This Fourier Transform property says s ( 0 ) {\displaystyle s(0)} transforms to ∫ − ∞ ∞ S ( f ) d f {\displaystyle \int _{-\infty }^{\infty }S(f)df}
s ( 0 ) = s ( t ) | t = 0 = F − 1 [ S ( f ) ] {\displaystyle s(0)=s(t)|_{t=0}={\mathcal {F}}^{-1}\left[S(f)\right]}
s ( 0 ) = s ( t ) | t = 0 = ∫ − ∞ ∞ S ( f ) e j 2 π f ( 0 ) d f = ∫ − ∞ ∞ S ( f ) d f {\displaystyle s(0)=s(t)|_{t=0}=\int _{-\infty }^{\infty }S(f)e^{j2\pi f(0)}df=\int _{-\infty }^{\infty }S(f)df}
The result is s ( 0 ) {\displaystyle s(0)} transforms to ∫ − ∞ ∞ S ( f ) d f {\displaystyle \int _{-\infty }^{\infty }S(f)df}
transforms to 
![{\displaystyle s(0)=s(t)|_{t=0}={\mathcal {F}}^{-1}\left[S(f)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c683328c1f15e58de287c10176416518dc140f)
transforms to