Evaluate this integral - HW1

Max Woesner

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Homework #1 - Evaluate this integral

Problem Statement

Evaluate the integral $\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{j2\pi (n-m)t/T}dt$

Solution
For $n=m,\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{j2\pi (n-m)t/T}dt=\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}1dt=T{|}_{\frac{-T}{2}}^{\frac{T}{2}}=\frac{T}{2}-\frac{-T}{2}=T$
For $n\ne m,\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{j2\pi (n-m)t/T}dt=\frac{e^{j2\pi (n-m)t/T}}{\frac{j2\pi (n-m)}{T}}{|}_{\frac{-T}{2}}^{\frac{T}{2}}=\frac{e^{j\pi (n-m)}-e^{-j\pi (n-m)}}{\frac{j2\pi (n-m)}{T}}=0$
So, $\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{j2\pi (n-m)t/T}dt=\{\begin{array}{ll}T,& \text{for }n=m\\ 0,& \text{for }n\ne m\end{array}$

Alternate method:
$\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{j2\pi (n-m)t/T}dt=\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}[cos(2\pi (n-m)t/T)+jsin(2\pi (n-m)t/T)]dt$
For $n=m,\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}[cos(2\pi (n-m)t/T)+jsin(2\pi (n-m)t/T)]dt=\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}1dt=T{|}_{\frac{-T}{2}}^{\frac{T}{2}}=\frac{T}{2}-\frac{-T}{2}=T$
For $n\ne m,\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}[cos(2\pi (n-m)t/T)+jsin(2\pi (n-m)t/T)]dt=0$
So, $\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{j2\pi (n-m)t/T}dt=\{\begin{array}{ll}T,& \text{for }n=m\\ 0,& \text{for }n\ne m\end{array}$