MHB Calculating Fourier Series of $f(\theta)$ on $[\pi, -\pi]$

[Dustinsfl]

Calculate the Fourier series of the function $f$ defined on the interval $[\pi, -\pi]$ by
$$
f(\theta) =
\begin{cases} 1 & \text{if} \ 0\leq\theta\leq\pi\\
-1 & \text{if} \ -\pi < \theta < 0
\end{cases}.
$$
$f$ is periodic with period $2\pi$ and odd since $f$ is symmetric about the origin.
So $f(-\theta) = -f(\theta)$.
Let $f(\theta) = \sum\limits_{n = -\infty}^{\infty}a_ne^{in\theta}$.
Then $f(-\theta) = \sum\limits_{n = -\infty}^{\infty}a_ne^{-in\theta} = \cdots + a_{-2}e^{2i\theta} + a_{-1}e^{i\theta} + a_0 + a_{1}e^{-i\theta} + a_{2}e^{-2i\theta}+\cdots$
$-f(\theta) = \sum\limits_{n = -\infty}^{\infty}a_ne^{-in\theta} = \cdots - a_{-2}e^{-2i\theta} - a_{-1}e^{-i\theta} - a_0 - a_{1}e^{i\theta} - a_{2}e^{2i\theta}-\cdots$

$a_0 = -a_0 = 0$

I have solved many Fourier coefficients but I can't think today.

What do I need to do next?

 

[Dustinsfl]

$$
a_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(\theta)e^{-im\theta}d\theta
$$
Since my function is defined piecewise, would I write it as
$$
a_n = \frac{1}{2\pi}\left[\int_0^{\pi}e^{-im\theta}d\theta - \int_{-\pi}^0e^{-im\theta}d\theta\right]
$$

 

[Opalg]

$$
a_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(\theta)e^{-im\theta}d\theta
$$
Since my function is defined piecewise, would I write it as
$$
a_n = \frac{1}{2\pi}\left[\int_0^{\pi}e^{-im\theta}d\theta - \int_{-\pi}^0e^{-im\theta}d\theta\right]
$$

Yes. (Yes)

(But since this is an odd function, you might find it easier to use the real rather than the complex Fourier series. The cosine terms will all be zero and you will only have to deal with the sine terms. To evaluate them, do just what you are doing with the complex terms, writing them as the difference between the integrals on the intervals [0,1] and [-1,0].)

 

[Dustinsfl]

Yes. (Yes)

(But since this is an odd function, you might find it easier to use the real rather than the complex Fourier series. The cosine terms will all be zero and you will only have to deal with the sine terms. To evaluate them, do just what you are doing with the complex terms, writing them as the difference between the integrals on the intervals [0,1] and [-1,0].)

When I solve, I have
$$
-\frac{1}{2\pi}\int_0^{\pi}\sin m\theta d\theta = \frac{1}{\pi m}
$$
and
$$
-\frac{1}{2\pi}\int_{-\pi}^0\sin m\theta d\theta = -\frac{1}{\pi m}
$$

I think the integral has to be
$$
-\frac{1}{\pi}\int_0^{\pi}\sin m\theta d\theta = \frac{2}{\pi m}
$$

Now what?

 

[Opalg]

When I solve, I have
$$
-\frac{1}{2\pi}\int_0^{\pi}\sin m\theta d\theta = \frac{1}{\pi m}
$$
and
$$
-\frac{1}{2\pi}\int_{-\pi}^0\sin m\theta d\theta = -\frac{1}{\pi m}
$$

I think the integral has to be
$$
-\frac{1}{\pi}\int_0^{\pi}\sin m\theta d\theta = \frac{2}{\pi m}
$$

Now what?

Try doing those integrals again. $$\begin{aligned}\int_0^\pi\sin m\theta\,d\theta &=\Bigl[-\tfrac1m\cos m\theta\Bigr]_0^\pi \\ &= -\tfrac1m\bigl(\cos m\pi - \cos 0\bigr) \\ &= -\tfrac1m\bigl((-1)^m - 1\bigr) \\ &= \begin{cases}0 &\text{ (if $m$ is even),} \\2/m &\text{ (if $m$ is odd).}\end{cases} \end{aligned}$$

The integral from -1 to 0 is the same but with a minus sign. You should then find that the Fourier series for $f(\theta)$ is $$\sum_{k=0}^\infty\frac4{(2k+1)\pi}\sin(2k+1) \theta.$$