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Problem Statement
Evaluate the integral ∫ − T 2 T 2 e j 2 π ( n − m ) t / T d t {\displaystyle \int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt\!}
Solution For n = m , ∫ − T 2 T 2 e j 2 π ( n − m ) t / T d t = ∫ − T 2 T 2 1 d t = T | − T 2 T 2 = T 2 − − T 2 = T {\displaystyle n=m,\int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt=\int _{-{\frac {T}{2}}}^{\frac {T}{2}}1dt=T{\Bigg |}_{\frac {-T}{2}}^{\frac {T}{2}}={\frac {T}{2}}-{\frac {-T}{2}}=T\!} For n ≠ m , ∫ − T 2 T 2 e j 2 π ( n − m ) t / T d t = e j 2 π ( n − m ) t / T j 2 π ( n − m ) T | − T 2 T 2 = e j π ( n − m ) − e − j π ( n − m ) j 2 π ( n − m ) T = 0 {\displaystyle n\neq m,\int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt={\frac {e^{j2\pi (n-m)t/T}}{\frac {j2\pi (n-m)}{T}}}{\Bigg |}_{\frac {-T}{2}}^{\frac {T}{2}}={\frac {e^{j\pi (n-m)}-e^{-j\pi (n-m)}}{\frac {j2\pi (n-m)}{T}}}=0\!} So, ∫ − T 2 T 2 e j 2 π ( n − m ) t / T d t = { T , for n = m 0 , for n ≠ m {\displaystyle \int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt={\begin{cases}T,&{\mbox{for }}n=m\\0,&{\mbox{for }}n\neq m\end{cases}}\!} Alternate method: ∫ − T 2 T 2 e j 2 π ( n − m ) t / T d t = ∫ − T 2 T 2 [ c o s ( 2 π ( n − m ) t / T ) + j s i n ( 2 π ( n − m ) t / T ) ] d t {\displaystyle \int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt=\int _{-{\frac {T}{2}}}^{\frac {T}{2}}[cos(2\pi (n-m)t/T)+jsin(2\pi (n-m)t/T)]dt\!} For n = m , ∫ − T 2 T 2 [ c o s ( 2 π ( n − m ) t / T ) + j s i n ( 2 π ( n − m ) t / T ) ] d t = ∫ − T 2 T 2 1 d t = T | − T 2 T 2 = T 2 − − T 2 = T {\displaystyle n=m,\int _{-{\frac {T}{2}}}^{\frac {T}{2}}[cos(2\pi (n-m)t/T)+jsin(2\pi (n-m)t/T)]dt=\int _{-{\frac {T}{2}}}^{\frac {T}{2}}1dt=T{\Bigg |}_{\frac {-T}{2}}^{\frac {T}{2}}={\frac {T}{2}}-{\frac {-T}{2}}=T\!} For n ≠ m , ∫ − T 2 T 2 [ c o s ( 2 π ( n − m ) t / T ) + j s i n ( 2 π ( n − m ) t / T ) ] d t = 0 {\displaystyle n\neq m,\int _{-{\frac {T}{2}}}^{\frac {T}{2}}[cos(2\pi (n-m)t/T)+jsin(2\pi (n-m)t/T)]dt=0\!} So, ∫ − T 2 T 2 e j 2 π ( n − m ) t / T d t = { T , for n = m 0 , for n ≠ m {\displaystyle \int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt={\begin{cases}T,&{\mbox{for }}n=m\\0,&{\mbox{for }}n\neq m\end{cases}}\!}
![{\displaystyle \int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt=\int _{-{\frac {T}{2}}}^{\frac {T}{2}}[cos(2\pi (n-m)t/T)+jsin(2\pi (n-m)t/T)]dt\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab48349f12ac1961cfe09cb4762a9f3191adcf33)
![{\displaystyle n=m,\int _{-{\frac {T}{2}}}^{\frac {T}{2}}[cos(2\pi (n-m)t/T)+jsin(2\pi (n-m)t/T)]dt=\int _{-{\frac {T}{2}}}^{\frac {T}{2}}1dt=T{\Bigg |}_{\frac {-T}{2}}^{\frac {T}{2}}={\frac {T}{2}}-{\frac {-T}{2}}=T\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d93da93452a60462aa0c83502042ae62ffe3e016)
![{\displaystyle n\neq m,\int _{-{\frac {T}{2}}}^{\frac {T}{2}}[cos(2\pi (n-m)t/T)+jsin(2\pi (n-m)t/T)]dt=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ba342f9ebf1b555c1f0bbeb0e9951bb6cb1be12)