# 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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### Extend a function 2pi periodically and calculate fourier

I have the function $$f(x)= \begin{cases} \frac{\pi}{2}+x & x \in (-\pi,0] \\ \frac{\pi}{2}-x & x \in (0,\pi]\\ \end{cases}$$ I need to extend it $2\pi$ periodically and then ...
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### Convergence of a Fourier series to a point

Consider the function $f\left(x\right)=1+x$, $x \in \left[-\pi,\pi\right]$ I have calculated its Fourier series to be $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin nx.$$...
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### Fourier function expansion for extension over a $2\pi$ period

So I am currently looking at a fourier expansion for $$f(x)=\left\{\begin{array}{ccl}\sin x &\text{ if }& x\in[0,\pi]\\0 & \text{ if } & x\in[\pi,2\pi]\end{array}\right.$$ I am ...
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### How to find particular solution of an ODE by fourier series expansion?

I encountered the question for the particular solution of, $$k \frac{d^4y}{dx^4} = m x$$ where m and k are real numbers. I would solve this question with basic methods for ODEs but question ...
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### Find a recurrence relation and the Fourier-Legendre Series

Rodrique's Formula for the $n$th Legendre Polynomial is $$P_n\left(x\right)=\dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}\left(\left(x^2-1\right)^n\right)$$ The Fourier-Legendre series of a function f is ...
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### Converting Fourier Series into elementary expression

If a Fourier series corresponds to an elementary function, is there any algorithm that will produce the elementary expression of this function?
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### Is $\frac{1}{n}\sin (\frac{n\pi}{2})-\frac{\pi}{2n}\cos (\frac{n\pi}{2})=\frac{(-1)^{n+1}}{(2n-1)^2}$, where $n \in \mathbb N$?

Is $\frac{1}{n}\sin (\frac{n\pi}{2})-\frac{\pi}{2n}\cos (\frac{n\pi}{2})=\frac{(-1)^{n+1}}{(2n-1)^2}$, where $n \in \mathbb N$? I am doing Fourier series, and my hand computed solution is the one on ...
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### Proof of $\sum\limits_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}$ using the Fourier series of $|x|$
I'm sure easier proofs exist, but I have to specifically use the method in the picture: This is what my attempt is: First, I did some manipulation to figure out that  \sum_{k=1}^\infty \frac 1{k^...
### Closed form of integral $\int_a^b e^{-ix^2} dx$
Does any know how to find the closed form of integral $\int_a^b e^{-ix^2} dx$ for any real $a$ and $b$. It seems that I need to use the fresnel integrals.