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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3answers
62 views
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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1answer
22 views
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
$$
\sum_{n=-\infty}^{\infty} \boldsymbol{c}(\...
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1answer
22 views
Laplace's equation on the plane: how much boundary data must be specified to guarantee existence and uniqueness?
My question stemmed from a specific problem, so let's jump right in. I want to solve Laplace's equation in plane polars $(r, \theta)$ on the domain $(r,\theta) \in [1,\infty)\times[0,2\pi)$ subject to ...
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28 views
Proof that $\exp (i2\pi f_{0}t)=\sum_{k \in \Bbb{Z}}\exp \left(i2 \pi f_{0}k\Delta\right) \operatorname{sinc}\left(\frac{t-k \Delta}{\Delta}\right)$
I want to prove using the complex Fourier series that for $f_{0} \in\left(-\frac{1}{2 \Delta}, \frac{1}{2 \Delta}\right)$, $ \Delta >0 $, we have
$$
\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:
$$I(\phi) = \...
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23 views
Find Fourier transform of e^-2/t sin(t) [closed]
Please solve this qstn. I need to know. Someone give explanation to this one
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1answer
20 views
Need clarification on the complex form of Fourier series
I wish to ask you guys to fill in a few steps for the derivation of complex form of Fourier series.
This is taken from "Fourier series" of Tolstov (Dover publication).
$$f(x)\sim c_0+\sum_{n=...
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1answer
45 views
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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1answer
9 views
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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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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1answer
31 views
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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1answer
43 views
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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1answer
30 views
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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0answers
22 views
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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1answer
31 views
Determination of Fourier coefficient (Euler's Formulae)
Is the integral from -Ļ to Ļ the same with integral from 0 to 2Ļ?
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1answer
51 views
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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1answer
22 views
Fourier series of a piecewise function, is my coefficient $a_n$ correct?
Here is a piecewise function extended periodically with period $2\pi$.
$$f(x)=
\begin{cases}
\pi + x & -\pi<x<-\frac{\pi}{2} \\
x & -\frac{\pi}{2}< x<\frac{\pi}{...
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1answer
67 views
Is it possible to prove that $\sum_{k=1}^{\infty}\frac{\sin(kx)}{k^a}\geq 0$ for $a\geq 1$ and $x\in [0,\pi]$?
I was trying to prove that
$$\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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18 views
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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0answers
30 views
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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1answer
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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1answer
30 views
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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40 views
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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1answer
35 views
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}$$
Show that
...
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1answer
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Finding the Fourier series of $\sin(x + a)$ for $[-\pi, \pi],$ where $a$ is a constant [closed]
I've been tasked with finding the Fourier series of $\sin(x + a)$ for the range $[-\pi, \pi]$, where $a$ is a constant. When I apply the formulas for $a_0$, $a_n$ and $b_n,$ I find all of them to be $...
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Are there any exotic examples of series expansions?
The idea of a series expansion of a function is to express the function as an infinite series of simpler terms, which follow a pattern. Truncating such an expansion allows us to approximate the ...
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1answer
46 views
Elementary proof of the Fourier series of $f(x)=x$
Is there an elementary proof or an intuitive explanation of the following formula that can be understood if you know the definition of sin?
$$x = -2\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\sin(nx)$$
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261 views
Calculate $\sum_{k=1}^{\infty}\frac{|\sin(k x)|}{1+k^2}$
In https://math.stackexchange.com/a/4015346/198592 it was shown that the sum
$$s(x) = \sum_{k=1}^{\infty}\frac{\sin(k x)}{1+k^2}$$
is exactly expressible in terms of the hypergeometric function.
I ...
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1answer
49 views
Why does continuity on $\mathbb{T}$ imply that $f(-\pi) = f(\pi)$?
In Vretblad's Fourier Analysis and its Applications, it says:
Suppose that $f \in C^1(\mathbb{T})$ , which means that both $f$ and its derivative $f'$ are continuous >on $\mathbb{T}$. We compute ...
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60 views
Fourier Sine transform of $\arctan(x/a)$.
How can I find the Fourier sine transform of $arctan(x/a) ;a>0$? I am solving it as,
$$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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1answer
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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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37 views
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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1answer
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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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0answers
14 views
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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0answers
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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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9 views
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 ...
3
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3answers
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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0answers
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)...
2
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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 GradshteiĢn 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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0answers
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 $...
