Fourier transformation of $e^-at$ from $0$ to $T$

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I have to find the fourier transform of $g(t)$. Picture is added at the link below.



https://www.mathcha.io/editor/KkvMFWBUm3TXrhr0



I have tried this way
beginalign
G(f)&=int ^infty _-infty g( t) e^-jomega t, dt\
&=int ^infty _-infty e^-at rectleft(fractT -frac12right) e^-jomega t, dt\
&=int ^T_0 e^-at e^-jomega t, dt\
&=frac1-e^-( a+jomega ) Ta+jomega
endalign
where $$
rect(t) ~is ~unit ~rectangular ~function\
rect(fractT-frac12) = begingather*
begincases
1 & textfor 0le tle T\
0 & textfor elsewhere
endcases
endgather*
$$



But i am not sure whether it's right or not.






fourier-transform







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  • What is "rect"?
    – Lord Shark the Unknown
    Sep 2 at 6:41










  • Then I'd say that both "rect" and the second line of your calculation are unnecessary.
    – Lord Shark the Unknown
    Sep 2 at 6:55














up vote
0
down vote

favorite












I have to find the fourier transform of $g(t)$. Picture is added at the link below.



https://www.mathcha.io/editor/KkvMFWBUm3TXrhr0



I have tried this way
beginalign
G(f)&=int ^infty _-infty g( t) e^-jomega t, dt\
&=int ^infty _-infty e^-at rectleft(fractT -frac12right) e^-jomega t, dt\
&=int ^T_0 e^-at e^-jomega t, dt\
&=frac1-e^-( a+jomega ) Ta+jomega
endalign
where $$
rect(t) ~is ~unit ~rectangular ~function\
rect(fractT-frac12) = begingather*
begincases
1 & textfor 0le tle T\
0 & textfor elsewhere
endcases
endgather*
$$



But i am not sure whether it's right or not.






fourier-transform







share|cite|improve this question























  • What is "rect"?
    – Lord Shark the Unknown
    Sep 2 at 6:41










  • Then I'd say that both "rect" and the second line of your calculation are unnecessary.
    – Lord Shark the Unknown
    Sep 2 at 6:55












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have to find the fourier transform of $g(t)$. Picture is added at the link below.



https://www.mathcha.io/editor/KkvMFWBUm3TXrhr0



I have tried this way
beginalign
G(f)&=int ^infty _-infty g( t) e^-jomega t, dt\
&=int ^infty _-infty e^-at rectleft(fractT -frac12right) e^-jomega t, dt\
&=int ^T_0 e^-at e^-jomega t, dt\
&=frac1-e^-( a+jomega ) Ta+jomega
endalign
where $$
rect(t) ~is ~unit ~rectangular ~function\
rect(fractT-frac12) = begingather*
begincases
1 & textfor 0le tle T\
0 & textfor elsewhere
endcases
endgather*
$$



But i am not sure whether it's right or not.






fourier-transform







share|cite|improve this question















I have to find the fourier transform of $g(t)$. Picture is added at the link below.



https://www.mathcha.io/editor/KkvMFWBUm3TXrhr0



I have tried this way
beginalign
G(f)&=int ^infty _-infty g( t) e^-jomega t, dt\
&=int ^infty _-infty e^-at rectleft(fractT -frac12right) e^-jomega t, dt\
&=int ^T_0 e^-at e^-jomega t, dt\
&=frac1-e^-( a+jomega ) Ta+jomega
endalign
where $$
rect(t) ~is ~unit ~rectangular ~function\
rect(fractT-frac12) = begingather*
begincases
1 & textfor 0le tle T\
0 & textfor elsewhere
endcases
endgather*
$$



But i am not sure whether it's right or not.






fourier-transform




fourier-transform






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Sep 2 at 7:12

























asked Sep 2 at 6:34









Leolime

11




11











  • What is "rect"?
    – Lord Shark the Unknown
    Sep 2 at 6:41










  • Then I'd say that both "rect" and the second line of your calculation are unnecessary.
    – Lord Shark the Unknown
    Sep 2 at 6:55
















  • What is "rect"?
    – Lord Shark the Unknown
    Sep 2 at 6:41










  • Then I'd say that both "rect" and the second line of your calculation are unnecessary.
    – Lord Shark the Unknown
    Sep 2 at 6:55















What is "rect"?
– Lord Shark the Unknown
Sep 2 at 6:41




What is "rect"?
– Lord Shark the Unknown
Sep 2 at 6:41












Then I'd say that both "rect" and the second line of your calculation are unnecessary.
– Lord Shark the Unknown
Sep 2 at 6:55




Then I'd say that both "rect" and the second line of your calculation are unnecessary.
– Lord Shark the Unknown
Sep 2 at 6:55















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