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Problem Statement
6(a) Show F [ ∫ − ∞ t s ( λ ) d λ ] = S ( f ) j 2 π f if S ( 0 ) = 0 {\displaystyle {\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]={\frac {S(f)}{j2\pi f}}{\mbox{ if }}S(0)=0} . HINT: S ( 0 ) = S ( f ) | f = 0 = ∫ − ∞ ∞ s ( t ) e − j 2 π ( f → 0 ) t d t = ∫ − ∞ ∞ s ( t ) d t {\displaystyle S(0)=S(f)\vert _{_{f=0}}=\int _{-\infty }^{\infty }s(t)e^{-j2\pi (f\rightarrow 0)t}\,dt=\int _{-\infty }^{\infty }s(t)\,dt}
6(b) If S ( 0 ) ≠ 0 {\displaystyle S(0)\neq 0} can you find F [ ∫ − ∞ t s ( λ ) d λ ] {\displaystyle {\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]} in terms of S ( 0 ) {\displaystyle \displaystyle S(0)} ?
Answer a) S ( 0 ) = S ( f ) | f = 0 = ∫ − ∞ ∞ s ( t ) e − j 2 π f t d t = ∫ − ∞ ∞ s ( t ) d t {\displaystyle S(0)=S(f)|_{f=0}=\int _{-\infty }^{\infty }s(t)e^{-j2\pi ft}dt=\int _{-\infty }^{\infty }s(t)dt}
Remember dummy variable λ = t − t 0 {\displaystyle \lambda =t-t_{0}} Then s ( λ ) = s ( t − t 0 ) {\displaystyle s(\lambda )=s(t-t_{0})}
![{\displaystyle {\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]={\frac {S(f)}{j2\pi f}}{\mbox{ if }}S(0)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e30ea2d81a3c0586359e282685bc252899dbaab)
. HINT: 
can you find ![{\displaystyle {\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0af3b7173a594a013131920839a34842568e41f3)
in terms of 
?
Then 