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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What is function of this Fourier series?

I'm a beginner in the Fourier series and I can't find the function of the below Fourier series. Any help and hint are much appreciated. after that, I want to calculate this integral and I think ...
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Find $a_n =1/\pi\int_{-\pi}^{\pi}\cos(x)\cos(nx)dx$

I am trying to compute Fourier coefficients for the function $f(x) = \cos x$. I have managed to solve that the coefficients "$c$" and "$b$" for the function are $0$. However, I am ...
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Compute $\sum_{n \in Z} \exp \left(ia*(n^2-2fn)\right) $ with $a>0\ $

I want to Compute $$\sum_{n \in Z} \exp \left(ia*(n^2-2fn)\right)$$ with $a>0\ $ being a real number. My final result should be in the form of $$ \sum_{n=-\infty}^{\infty} \boldsymbol{c}(\...
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Laplace's equation on the plane: how much boundary data must be specified to guarantee existence and uniqueness?

My question stemmed from a specific problem, so let's jump right in. I want to solve Laplace's equation in plane polars $(r, \theta)$ on the domain $(r,\theta) \in [1,\infty)\times[0,2\pi)$ subject to ...
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Proof that $\exp (i2\pi f_{0}t)=\sum_{k \in \Bbb{Z}}\exp \left(i2 \pi f_{0}k\Delta\right) \operatorname{sinc}\left(\frac{t-k \Delta}{\Delta}\right)$

I want to prove using the complex Fourier series that for $f_{0} \in\left(-\frac{1}{2 \Delta}, \frac{1}{2 \Delta}\right)$, $ \Delta >0 $, we have $$ \exp \left(i 2 \pi f_{0} t\right)=\sum_{k=-\...
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Properly Subtracting Minimum Value from Fourier Series

I'm using Fourier series representations of data and I need to subtract the minimum value away from the Fourier series before integrating it. I define my Fourier series $I(\phi)$ as: $$I(\phi) = \...
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Find Fourier transform of e^-2/t sin(t) [closed]

Please solve this qstn. I need to know. Someone give explanation to this one
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Need clarification on the complex form of Fourier series

I wish to ask you guys to fill in a few steps for the derivation of complex form of Fourier series. This is taken from "Fourier series" of Tolstov (Dover publication). $$f(x)\sim c_0+\sum_{n=...
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1answer
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Integrating $x \sin(nx)$

It should be simple but I'm overthinking this so hard, that I can't see what I'm supposed to do anymore. Instead of taking a break from it, I'll ask here. $\int x\sin(nx) \, dx$, $\sin(nx)$ would be $\...
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How piecewise smoothness of a function is related to the Dirichlet conditions?

Do all piecewise smooth functions satisfy Dirichlet conditions for Fourier series representation? In the theorem of Fourier series can we write that being piecewise smooth is the sufficient condition ...
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Bochner's Theorem for Periodic Functions

It's fairly straightforward to prove the following, which seem like special cases of Bochner's Theorem for periodic functions, but don't quite match the statements of Bochner's Theorem that I've seen, ...
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How do we find the fourier coefficiant c_n of the Dirichlet kernel [closed]

I have the Dirichlet kernel: $D_n = \frac{1}{2\pi} \frac{sin(N+\frac{1}{2})\theta}{sin(\frac{\theta}{2})}$ and I want to find the Fourier coefficient $c_n$ of the Dirichlet kernel. Do I use the ...
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Fourier series real and complex form [closed]

Please refer to the attached image Question
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Obtain the Fourier Series [closed]

I have tried nothing yet. A function f(x) is such that: f(x)=(x-pi)/2.....-pi<x<0 f(x)=(x-pi)/2.....0<x<pi f(x)=(x+2pi) Obtain the Fourier Series:
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Finding fourier series of odd square wave [closed]

$$f(t) = -1$$ $$\frac{-T}{2} \ne t<0$$ $$+1, 0<=t<\frac{T}{2}$$ Since it's an odd wave, coefficient involving cos will be zero. I'm trying to calculate coefficient involving sine $b_r = \frac{...
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Cosine Fourier series

Given the following function $$f(x)=\left\{\begin{array}{ll} 0, & 0\leq x\leq1 \\ 1, & 1<x\leq2 \end{array}\right.$$ I have to find its cosine Fourier series. I expanded it in a even way ...
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Sine Fourier coefficients of derivative

Let $f$ be a $C^1$-function on $[0,\pi]$ with $f(0)=f(\pi)=0$. We know that the family $(\sin(nx))_{n\in\Bbb N}$ is a complete orthonormal system on $L^2(0,\pi)$ (maybe up to some rescaling). Let $(...
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Fourier Series: question on the period and terms

Wiki defines the Fourier coefficients as: $a_n = \frac 2 P \int_P s(x) cos(2\pi x \frac n P) dx, b_n = \frac 2 P \int_P s(x) sin(2\pi x \frac n P) dx$ It's my understanding the leading term $\frac 1 P$...
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Uniqness of Fourier expansion in a non-separable Hilbert space

All my life, I have been operating inside separable Hilbert spaces. Everything is all dandy as any orthonormal basis is a sequence. Let $e_n$ be the orthonormal basis of the Hilbert space $H$. I have ...
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31 views

Determination of Fourier coefficient (Euler's Formulae)

Is the integral from -π to π the same with integral from 0 to 2π?
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Fourier series of $sgn(\cos(x))$

Knowing formulas for Fourier series when I'm given interval $(0,2l)$ & $(-l,l)$ I can't seem to find Fourier series of $sgn(\cos(x))$. I know that $a_n = \frac{2}{l} \int_0^{l}f(x)\cos(\frac{n\pi ...
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Fourier series of a piecewise function, is my coefficient $a_n$ correct?

Here is a piecewise function extended periodically with period $2\pi$. $$f(x)= \begin{cases} \pi + x & -\pi<x<-\frac{\pi}{2} \\ x & -\frac{\pi}{2}< x<\frac{\pi}{...
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Is it possible to prove that $\sum_{k=1}^{\infty}\frac{\sin(kx)}{k^a}\geq 0$ for $a\geq 1$ and $x\in [0,\pi]$?

I was trying to prove that $$\sum_{k=1}^{\infty}\frac{\sin(kx)}{k^a}\geq 0,$$ for $0\leq x\leq \pi$, and $a\geq1$. For $a$ being an odd integer, this is not really a problem, as the sums may then be ...
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Laplacian's expression for Fourier series function

Given a generic function f written in polar coordinates: \begin{equation} f(\underline{r})=f(r,\theta)=\sum_{n=-\infty}^{\infty} f_n(r) e^{in\theta} \end{equation} Considering the Laplacian, we have: \...
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Fourier Series Representation of Discontinuous Function

$f(x) = \begin{cases} 1& \text{ if } -1\leq x < 0 \\ 1/2 & \text{ if } x = 0 \\ x& \text{ if } 0 < x \leq 1 \end{cases} \text{for the interval} [-1,1]$ So I understand that you ...
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35 views

Fourier series of $|x|$ on $[-\pi,\pi]$ and sum of series

So I have been solving various problems on Fourier series and this particular one got me struggling a bit. Given a function $f(x) = |x|$ find a Fourier series on $[-\pi,\pi]$ and find the sum of ...
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Convergence of Fourier series at discontinuities

For the Fourier series in this question: Math Question How do I find out where the Fourier series converges, I initially thought I just apply this equation: $ f\left(x\right)=\frac{1}{2}\left\{f\left(...
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Fourier expansion of (small perturbations of constant) loops

In this paper on page 4 the authors write: We would like to apply such a construction to the tangent bundle of a free loopspace. [...]. However, such a splitting does exist in a neighborhood of the ...
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Series for $f(x)=a\ln(1+b\sin(cx+df(x)))$?

This question is inspired by the two questions here and here. The answers to these questions show several ways to obtain approximations respectively give explicit fourier expansions for the functions ...
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How to show that $c_n=\frac{2}{n\pi i}$ for odd $n$? (Fourier series)

This 3blue1brown video on Fourier series leaves the viewer with the following exercise: $$c_n=\int_0^{0.5}e^{-2\pi int}\,\,\mathrm{d}t\,\,+\int_{0.5}^1-e^{-2\pi int}\,\,\mathrm{d}t\tag{1}$$ Show that ...
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Finding the Fourier series of $\sin(x + a)$ for $[-\pi, \pi],$ where $a$ is a constant [closed]

I've been tasked with finding the Fourier series of $\sin(x + a)$ for the range $[-\pi, \pi]$, where $a$ is a constant. When I apply the formulas for $a_0$, $a_n$ and $b_n,$ I find all of them to be $...
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Are there any exotic examples of series expansions?

The idea of a series expansion of a function is to express the function as an infinite series of simpler terms, which follow a pattern. Truncating such an expansion allows us to approximate the ...
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1answer
46 views

Elementary proof of the Fourier series of $f(x)=x$

Is there an elementary proof or an intuitive explanation of the following formula that can be understood if you know the definition of sin? $$x = -2\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\sin(nx)$$
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Calculate $\sum_{k=1}^{\infty}\frac{|\sin(k x)|}{1+k^2}$

In https://math.stackexchange.com/a/4015346/198592 it was shown that the sum $$s(x) = \sum_{k=1}^{\infty}\frac{\sin(k x)}{1+k^2}$$ is exactly expressible in terms of the hypergeometric function. I ...
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Why does continuity on $\mathbb{T}$ imply that $f(-\pi) = f(\pi)$?

In Vretblad's Fourier Analysis and its Applications, it says: Suppose that $f \in C^1(\mathbb{T})$ , which means that both $f$ and its derivative $f'$ are continuous >on $\mathbb{T}$. We compute ...
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Fourier Sine transform of $\arctan(x/a)$.

How can I find the Fourier sine transform of $arctan(x/a) ;a>0$? I am solving it as, $$F_{s}\left[\arctan\left(\frac{x}{a}\right)\right]=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\arctan\left(\frac{x}{...
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Solving general non-homogenous wave equation with homogenous boundary conditions

I have been given the following PDE to solve: $$ u_{tt}-u_{xx}=g(t)\sin x \;\;\; (t,x)\in (0,\infty)\times(0,\pi)\\u(0,x)=u_t(0,x) \;\;\;x\in(0,\pi)\\ u(t,0) = u(t,\pi) =0 \;\;\; t>0 $$ This ...
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Show that $f(t) \star g(t) = \sum_{n = -\infty}^{\infty} {c_n d_n e^{j n w_0 t}}$

I am looking at a question which states that if $f(t) = \sum_{n = -\infty}^{\infty} {c_n e^{j n w_0 t}}$ and $g(t) = \sum_{n = -\infty}^{\infty} {d_n e^{j n w_0 t}}$ where $w_0 = \frac{2 \pi}{T}$ then ...
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How to do Fourier series, deduce question?

Hello, This is a question using Fourier series to show that $t^2$ = $f(t)$ in Fourier series. I could do the Fourier series part but I am not sure how to do the ...
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Orthogonality in the context of Fourier coefficients

Is the following interpretation correct? I have a $T-$ periodic function $f(t)$ such that $f(t)=f(t+T)$ according to definition. In the trigonometric Fourier series, it is known that the coefficients ...
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Modeling using Fourier Series

If I have a set of discrete data points x = 1,2,3 etc. that are roughly periodic in nature and I want to model them using a Fourier series, how would I go about doing this. I tried researching this ...
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Fourier series of a complex exponential of a periodic function

There is a well-known relation known as Jacobi-Anger expansion: $$ e^{ix \sin\theta} = \sum_{n=-\infty}^{\infty}J_{n}(x)e^{in \theta}, $$ where $J_{n}$ are the Bessel functions of the first kind. Do ...
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Characterisation for real analytic function vis Fourier coefficient

A $C^{\infty}$ function on $\mathbb R$ is real analytic if for every $x\in \mathbb R$, $f$ is the sum of its Taylor series expansion bases at $x$ in some neighbourhood of $x$. If $f$ is periodic we ...
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A continuous function whose Fourier series is divergent at 0

Let $\{p_n\}$ be a sequence of orthogonal trigonometric polynomials satisfying : (1) $\{p_n\}$ is uniformly bounded (2) $\lim_{n\to \infty}|\sum_{-n}^{n}\hat{p}_n(k)|\to \infty$. Does there exist ...
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107 views

Limit of fourier series

I want to compute the limit $f(x)$ of $$ f(n,x)=\sum_{k=1}^n\frac{1}{k^2+1}\sin(kx) $$ and didn't succeed. I tried with the means used to compute the >>easy<< limits of trigonometric ...
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1answer
70 views

How can one calculate $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$?

I am working through a textbook for my Fourier Series class and I came across the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$. I know from the context of the question that the answer has to be $-\...
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24 views

What is the canonical orthonormal basis of the space $L^2(\mathbb{S}^2)$

I'm going to express a function in Fourier series. From Fourier Analysis we know that $L^2(\mathbb{S}^1)=\overline{\bigoplus_{n\in \mathbb{Z}} \langle e^{in\theta}\rangle}$ , and for $L^2(\mathbb{S}^3)...
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1answer
52 views

Need help evaluating this infinite sum involving a sine function and squared denominator

I need major help in evaluating the following infinite sum: $$\sum_{k=1}^{\infty} \frac{k\sin(kx)}{(k^2+a^2)^2}\tag{1}\label{whatiwant}$$ where a is a constant. I know that (from Gradshteĭn et al, ...
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1answer
19 views

Given specific real Fourier coefficients, does there have to be a function that matches that?

We were taught this in class: Given the real numbers b1, b2,..., bn, there exists a cyclic function such that its non-zero Fourier coefficients are b1, b2, ... bn. Can someone please explain why this ...
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36 views

Solution verification for this Fourier series problem

I'm told to find the following signal as just a sum of cosines. The only Fourier coefficients that aren't zero are Where $X[k]$ comes from $x(t) = \sum_{k = -\infty}^{\infty}X[k]e^{i \omega tk}$ and $...

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