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## Homework Statement

Sawtooth signal with To = 1, at T=0, x = 0, at T=1, x =1

verify:

[itex]

a_{k} = \left\{\begin{matrix}

\frac{1}{2}, for k=0; & \\\frac{j}{2\pi k}, for k \neq 0;

&

\end{matrix}\right.

[/itex]

## Homework Equations

[itex]\frac{1}{T_{0}} \int_{0}^{T_{0}} te^{-j(2\pi/T_{0}))kt}dt[/itex]

## The Attempt at a Solution

for k = 0

[itex] a_{0} = \int_{0}^{1} t dt[/itex]

[itex]a_{0} = \frac{1}{2} t^{2} [/itex] from 0 to 1 = 1/2

for k != 0

[itex]\int_{0}^{1} te^{-j(2\pi) kt}dt[/itex]

u = t

du = dt

dv = [itex]e^(-j2\pi kt)[/itex]

[itex] v = \frac{-1}{j2\pi k}e^{-j2\pi kt} [/itex]

[itex] t * \frac{-1}{j2\pi k}e^{-j2\pi kt} - \int \frac{-1}{j2\pi k}e^{-j2\pi kt} dt[/itex]

[itex] t * \frac{-1}{j2\pi k}e^{-j2\pi kt} - \frac{e^{-j2\pi kt}}{4\pi^2k^2} [/itex]

-1/j = j

[itex]t * \frac{j}{2\pi k}e^{-j2\pi kt} - \frac{e^{-j2\pi kt}}{4\pi^2k^2}[/itex]

[itex]e^{-j2\pi kt} (t \frac{j}{2\pi k} - \frac{1}{4\pi^2 k^2})[/itex]

getting close but not seeing where to go from here.

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