# Tagged Questions

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### 2 similar question about how to find the $a_n$s and $b_n$ of a Fourier series

Find the terms $b_n,\ n\geq 1$ so that $$x-\frac{\pi}{2}=\sum_{n=1}^{\infty}b_n \sin nx$$ for all $x\in (0,\pi)$. A similar one: Find the term $a_n, \ n \geq 0$ so that ...
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### Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation $$\sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2}$$ defined for $0\le x\le 2\pi$. Here is ...
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### Complex Fourier series of $f(\theta) = e^{\theta}$

I have the following Fourier series problem: Let $f(\theta)$ be the periodic function such that $f(\theta) = e^\theta$ for $-\pi<\theta\leq\pi\;$, and let ...
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### Fourier Series doubt

I have a doubt regarding the Fourier series usage in terms of the Fourier series formula, which has multiple variants and is quite complicated. EDIT: I would like to mention that this question (of ...
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### Proving a claim $|c_n e^{in\theta}| = |c_n|$

I'm studying about Fourier series from a book called "Fourier series and its applications" by Folland and on page 40, the author makes the claim that: $$|c_n e^{in\theta}| = |c_n|,$$ where $n$ is an ...
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### Expansion and convergence of $\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}$

Consider the series: $$\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}\;\;\;\;n\in\mathbb{N}$$ Other than formal manipulation of the Taylor series of the $\sin$ function, is there a way to expand ...
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### Finding the complex fourier series of the function $x^2sin(x)$ in the interval $[{-\pi}, \pi]$?

This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ...
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### Prove Parseval Identity for $f \in C(\Bbb T) 2\pi$ periodic continuous functions

Question: Prove Parseval Identity for $f \in C(\Bbb T)$ $2\pi$ periodic continuous functions $$\frac{1}{2 \pi} \int_{-\pi}^\pi |f(x)|^2 dx =\sum_{n=-\infty}^\infty |\hat f(n)|^2$$ Thoughts: We ...
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### Fourier series coefficients proof

Can somebody help me understanding the fouries series coefficients? I know that if we have: $$f(n) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where Ph_n = phase}$$ And because of the ...
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### Inner Product vs. Integrals with Fourier Series, When to include 1/2pi?

I am confused about when to include a prefactor of $\frac{1}{2\pi}$ when dealing with integrals of functions that are expressed as fourier series. This is what I understand (please correct me if I'm ...
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### Representing real function as integral over trigonometric functions

Since one can clearly express any function g(x) as integral from 0 to infinity of A(k)cos(kx)dk + integral from 0 to infinity of B(k)sin(kx)dk, how would G(k) relate to A(k) and B(k)? In other words, ...
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### Origin of coefficients of fourier series?

I was wondering how we derive these formulas, and why we have a separate formula for $a_0$? All I know from advanced engineering mathematics text book are following formulas but where do they come ...
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### Fourier Series for $|x|$
How can I calculate the Fourier series for |x| (where $x\in[−\pi,\pi]$) in the complex form? Thanks.