Evaluate this integral - HW1

Max Woesner

Back to my Home Page

Homework #1 - Evaluate this integral

Evaluate the integral $\int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt\!$
For $n=m,\int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt=\int _{-{\frac {T}{2}}}^{\frac {T}{2}}1dt=T{\Bigg |}_{\frac {-T}{2}}^{\frac {T}{2}}={\frac {T}{2}}-{\frac {-T}{2}}=T\!$
For $n\neq m,\int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt={\frac {e^{j2\pi (n-m)t/T}}{\frac {j2\pi (n-m)}{T}}}{\Bigg |}_{\frac {-T}{2}}^{\frac {T}{2}}={\frac {e^{j\pi (n-m)}-e^{-j\pi (n-m)}}{\frac {j2\pi (n-m)}{T}}}=0\!$
So, $\int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi (n-m)t/T}dt={\begin{cases}T,&{\mbox{for }}n=m\\0,&{\mbox{for }}n\neq m\end{cases}}\!$