# Tagged Questions

Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials or sines and cosines are used.

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### Fourier transform for spectral method equation explanation.

Consider the advection equation: $$u_t+u_x =0$$ Using the spectral method compute the Fourier transform of $U(x_j,t)$ which will give us an approximation for the spacial derivative. The Fourier ...
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### Laplace equation with boundary conditions in polar coordinates

Show that the problem with this boundary conditions $u_{rr}+1/ru_{r}+1/r^2u_{\theta\theta}=0$, $\quad 0 < r < 1, \quad 0 < \theta < \pi$ $u(r,0)=0$ $u(r,\pi) =T_0$ $u(1,\theta) =T_0$ ...
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### Quick Fourier Series help?

I was given a graph (shown above) and was asked to represent this as a Fourier Series. I was able to solve $a_0$ with no problem. However, when I was integrating for $a_n$ and $b_n$, I was having a ...
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### Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots = \frac{1}{2}\cot\frac{x}{2}$ for $x \neq 2k\pi, k \in \mathbb{Z}$

I'm learning about Fourier series (specifically Cesàro summation) and need help with the following problem: Show that the Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots$ is equal to ...
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### Help solving a Fourier Series for a function with P=2L

I have a periodic function $$f(x) = \begin{cases} -2x-x^2, & -2 \leq x < 0 \\ 2x-x^2, & 0 \leq x < 2 \\ f(x)=f(x+4) & otherwise \end{cases}$$ and this period is repeated. I ...
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### Application of Abel's Method to Summation of Fourier Series Question

The series $$f(x, r) = \frac{a_0}{2} + \sum_{n=1}^\infty r^n (a_n \cos nx + b_n \sin nx)$$ where $0 \le r \lt 1$ clearly converges, as the terms monotonically decrease. My question: My textbook ...
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