# Tagged Questions

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### how to compute this integral for fourier series

I am trying to find the Fourier sine and cosine series of $\frac{1}{(1+x^2)}$ from $0$ to $2$, and do not know where to even begin to evaluate this integral: $\int \frac{sin(nx)}{(1+x^2)} dx$ (and ...
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### fourier series representation

Find the Fourier series with period $2$ of $$f(x) = -x,\qquad-1<x<1$$ so I find that $a_0$ and $a_n$ both are $0$ since odd functions so the Fourier series is on the form: ...
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### Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
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### Fourier coefficients assume a maximum and minimum?

Let $f:\mathbb R\to\mathbb R$ be continuously differentiable and periodic with period $2\pi$. The Fourier coefficients are defined by $$\hat f_n=\int_{-\pi}^\pi f(x)\exp(-inx)dx$$ My questions: Is ...
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I have the following problem from my Fourier analysis book I would need some guidance with. I have tried it, but apparently I made some mistakes...here is my problem: We have: $$\sin \theta ... 2answers 94 views ### Proof of Wirtinger inequality Quoting from Ana Cannas da Silva's book on Symplectic Geometry: "As an exercise in Fourier series, show the Wirtinger inequality: for f\in C^1([a,b]), with f(a)=f(b)=0 we have$$ ...
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I'm studying Fourier analysis and my book gives the following definitions for the Fourier series and Fourier coefficients: Fourier series of $2\pi$-periodic function $f(\theta)$ is defined as: ...
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### Evaluate $\int_{-\pi}^\pi \big|\sum^\infty_{n=1} \frac{1}{2^n} e^{inx}\big|^2 \operatorname d\!x$

I am trying to solve exercises for the coming exam, and I am stuck on this exercise: Evaluate $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$ ...
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### Proving periodicity of sine and cosine [duplicate]

If we define the sine and cosine functions by their Maclaurin expansions, how do we prove they are periodic with period $2\pi$?
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### Find complex Fourier coefficients

let $f(x) = \sum^{10}_{m=1}(-1)^m \sin(2^m x)$. denote complex Fourier coefficients of $f(x)$ over $[-\pi, \pi]$ as $c_n = \frac{1}{2\pi} \int _{-\pi}^\pi f(x) e^{-inx}\,dx.$ ...
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### Finding Fourier coefficients?

The question is as follows: $f(x) = \cos(\pi x)$, $g(x) = f(x+2010)$. I need to find the sum of all of $g$'s Fourier coefficients from $-\infty$ to $\infty$. I know that $f=g$. Therefore $g$'s $n$th ...
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### Evaluate $\sum^\infty_{n=1} \frac{1}{n^4}$using Parseval's theorem (Fourier series)

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4}$using Parseval's theorem (Fourier series). I have , somehow, to find the sum of $\sum_{n=1}^\infty \frac{1}{n^4}$ using Parseval's theorem. I tried ...
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### Proving uniform convergence with some kernel

Question Given $K_n=\cases 0$ elsewhere , $n- n^2|x|$ for $x<|\frac 1n|$ , $f$ is continuous, $2\pi$ periodic $\Bbb R \to \Bbb C$ . $f_n(x)=\int _{-\pi}^ \pi f(t)Kn(x-t)$ prove that ...
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### Fourier series verification

Question: $$f(x)= \sum_{n=0}^\infty \frac {e^{inx}}{1+n^2}$$ if $x\ne 2\pi k$ and $f(x)=0$ if $x=0 , x=2\pi k$ Find $\hat f(n)$ Find the Fourier series of $\displaystyle g(x)=\int _0^xf(t)dt$ ...
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### Convergence of a fourier series of $f(x)=1+\sin \frac {\pi^2}x$

Question: let $f:\Bbb R \to \Bbb R, f(0)=1 \forall x\in[-\pi,\pi] \setminus \lbrace0\rbrace , f(x)=1+\sin \frac {\pi^2}x$ Does the fourier series of this function converge at zero? If it does what is ...
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### Finding Fourier series of $\sin^2 x$ (STILL not clear - read comments)

I am attempting to do some sums on Fourier series, but need help with one calculation, after which I can proceed on my own. The question is: Find the Fourier series for f(x) = $\sin^2x$, on ...
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### How can we prove that a Fourier Series exists?

How does one show that an arbitrary periodic function, so long as it is reasonably well behaved, can always be represented as a sum of sine and cosine functions? It sounds like the first thing you ...
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### These questions are all about Fourier analysis.

Please prove these equalities,these questions appear in the chapter of Fourier series. If you can use other methods,please tell me more about it, and I am glad to know how to solve the questions: ...
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### Fourier integral representations using only cosine functions.

Hi I have a question about Fourier integrals. Can Fourier cosine integrals represent any function, or just even functions?
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### Find Fourier Series of the function $f(x)= \sin x \cos(2x)$ [duplicate]

Find Fourier Series of the function $f(x)= \sin x \cos(2x)$ in the range $-\pi \leq x \leq \pi$ any help much appreciated I need find out $a_0$ and $a_1$ and $b_1$ I can find $a_0$ which is ...