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[SOLVED] Fourier series |\theta|

dwsmith

Well-known member
Feb 1, 2012
1,673
Suppose $f(\theta) = |\theta|$ for $-\pi < \theta < \pi$.
Find the formal series solution of the corresponding heat problem in the disk.
How many terms of the series will give $u(r,\theta)$ with an error $< 0.1$ throughout the disk?
Evaluate $u\left(\frac{1}{2},\pi\right)$ to two decimals.
Show that $u\left(r,\pm\frac{\pi}{2}\right) = \frac{\pi}{2}$.
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I know from previous that
$$
f(\theta) = \frac{\pi}{2} - \frac{4}{\pi}\sum_{n = 1}^{\infty}\frac{1}{(2n - 1)^2}\cos (2n - 1)\theta.
$$

I am not sure what I am supposed to do though.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
We know from previous that
$$
f(\theta) = \frac{\pi}{2} - \frac{4}{\pi}\sum_{n = 1}^{\infty}\frac{1}{(2n - 1)^2}\cos (2n - 1)\theta.
$$
The polar form of $f$ is
$$
u(r,\theta) = \frac{\pi}{2} - \frac{4}{\pi}\sum_{n = 1}^{\infty}\frac{r^{2n - 1}}{(2n - 1)^2}\cos(2n - 1)\theta.
$$
Take $r < 1$ and evaluate the partial sum to $k$.
Then $\frac{\pi}{2} - \frac{4}{\pi}\sum\limits_{n = 1}^k\frac{r^{2k - 1}}{(2k - 1)^2}\cos(2k - 1)\theta$.

How can I use the integral test now?
 
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