# Questions tagged [fourier-series]

A Fourier series is a decomposition of a periodic function as a linear combination of sines and cosines, or complex exponentials.

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### Non-negative trigonometric polynomials to exponential form

I've been working on an exercise and have gotten stuck: Suppose that $T(x)=\sum_{n=0}^N a_n\cos(2\pi nx)+b_n\sin(2\pi nx)$ is non-negative on $[0,1]$. Show that there exist $c_0,...,c_N\in \mathbb{C}$...
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### Compare R Programming with Sagemath for Fourier analysis? [on hold]

For Fourier mathematical simulation harmonics which one we will use R or Sagemath.
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### Fourier series representation of piecewise function

$${Expand} \; f(x)= \begin{cases} 2{A\over L}x & 0\leq x\leq {L\over 2} \\ \\ 2{A\over L}\left(L-x\right) & {L\over 2}\leq x\leq L \end{cases}$$ I have determined $A_0$...
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### How do i get fourier series of signals given below

how can i calculate and plot $Ae^{-at}$ amplitude and phase spectrum? how can i determine the trigonometric fourier series of the signal given below ? Your help is appreciated !! Thanks
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### Prove that a function is equal to its Fourier Series

Find the Fourier Series of the function $p(x)=\frac{\cos(5\pi x)+e^{-i\pi x}+1}{2}$ and justify why $p(x)$ must equal its Fourier series. This is a review problem for my Analysis exam. My idea was ...
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### How to find the coefficient of heat equation solution with non-odd initial condition

I've got a situation where I'm supposed to solve the 1-D heat equation $v_t = c^2 u_{xx}$ under boundary conditions and an initial condition, and so after some run-of-the-mill process, I attained the ...
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### Calculate the given sum

Problem: $$\sum_{k=1}^{\infty} \frac{1}{k^2(k+1)^2}$$ This is a problem from a course of Fourier Series and the only hint was to use partial fraction decomposition. I'm sure that this is meant to be ...
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### How to Sketch a graph of a extended function from one intervel to another intervel

I have already done part a,b,c. But in d part they ask to extend the function. I can't understand how to extend it and sketch the extended function.
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### Find the fourier transform of $e^{-ax^2}\cos(bx)$

I am trying to find the Fourier transform of the function $e^{-ax^2}\cos(bx)$. I am changing $\cos(bx)$ to its exponential form but after that I am getting stuck: can you guide me showing how to find ...
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### Fourier Series $f(x)=-1$ for $-\pi<x<0$, $f(x)=x$ for $0<x<\pi$ and expansion help

Apologies in advance as I'm sure you get many similar questions. So I have to expand the Fourier Series $f(x)=-1$ for $-\pi<x<0$, $f(x)=x$ for $0<x<\pi$ I end up with $\frac {a_0}{2}$(ie ...
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### Finding Fourier series of cosh

I have to find an even Fourier series (i.e. only cosine-terms) of $$f(x)=\cosh(x-1)$$ in the interval $0 \leq x \leq 1$ with periodicity 2. (EDIT: With my calculated Fourier coefficients I end up ...
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