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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 ( 0 ) t d t = ∫ − ∞ ∞ s ( t ) d t {\displaystyle s(0)=S(f)|_{f=0}=\int _{-\infty }^{\infty }s(t)e^{-j2\pi (0)t}dt=\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 
in time transforms in frequency to
transforms to