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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 ( f ) | f = 0 = [ ∫ − ∞ ∞ s ( t ) e − j 2 π f t d t ] | f = 0 = ∫ − ∞ ∞ s ( t ) d t {\displaystyle s(0)=S(f)|_{f=0}=[\int _{-\infty }^{\infty }s(t)e^{-j2\pi ft}dt]|_{f=0}=\int _{-\infty }^{\infty }s(t)dt}
And ∫ − ∞ ∞ s ( t ) d t {\displaystyle \int _{-\infty }^{\infty }s(t)dt} in time transforms in frequency to ∫ − ∞ ∞ S ( f ) d f {\displaystyle \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(f)|_{f=0}=[\int _{-\infty }^{\infty }s(t)e^{-j2\pi ft}dt]|_{f=0}=\int _{-\infty }^{\infty }s(t)dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de5601dcf4b1f73e9f1dc392dae638418563bdf8)
in time transforms in frequency to
transforms to