Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.
2
votes
0answers
30 views
What are the Fourier series of the function?
What are the Fourier series of $f(x)$ where $x \in [-\pi, \pi]$ defined by
$$f(n) =
\begin{cases}
1, & \text{if $x \in$ [0,$\pi$)} \\
0, & \text{if $x \in$ [$-\pi$,0)} \\
\end{cases}
$$
...
1
vote
1answer
45 views
Fourier representation for $\tan(x)$
Q: Which Fourier representation is suitable for $f(x) = \tan(x)$: Fourier trigonometric series, Fourier half-range expansion, or Fourier integral and why?
Well I searched and found that:
$\tan(x)$ ...
1
vote
1answer
39 views
summation of this series as $ x \to \infty $ ??
given the series for the Mangoldt function $ \Lambda (n) $
$$ f(x)= \sum_{n=1}^{\infty}\frac{\Lambda (n)}{\sqrt{n}}\cos(\sqrt{x} \log n+\pi /4) $$
if we truncate the series, can we say that
...
4
votes
3answers
145 views
Finding the Fourier Series of $\sin(x)^2\cos(x)^3$
I'm currently struggling at calculation the Fourier series of the given function
$$\sin(x)^2 \cos(x)^3$$
Given Euler's identity, I thought that using the exponential approach would be the easiest ...
3
votes
1answer
50 views
Using Fourier series techniques to solve $x'' + 3x = 7$ with $x'(0) = x'(5) = 0$
$$x'' + 3x= 7$$
Given conditions $x'(0)=x'(5)=0$.
I checked the list and I went through three books. I am doing intro to differential equations. I just don't know how to get the extensions... I was ...
2
votes
1answer
140 views
+50
Fourier analysis questions
Can anyone give me a hand with the proof of this properties?
Prove that:
a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$. ...
1
vote
1answer
37 views
Fourier analysis question, orthonormal basis.
I need some help with this exercise:
Given $A>0$, let $L_{A}^2(\mathbb{R})$ the subspace of $L^2(\mathbb{R})$ of the functions $f$ that satisfy $\hat{f}=\chi_{[\frac{-A}{2},\frac{A}{2}]}\hat{f}$. ...
3
votes
1answer
54 views
Fourier series $\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}$
Does anyone know the sum of Fourier series $$\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}?$$
I tried WA; it does not return a function.
1
vote
1answer
35 views
Parseval's identity
How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
41
votes
3answers
639 views
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.
I found a solution by myself 10 hours after I posted it, here it is:
...
6
votes
2answers
156 views
How to expand the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} $?
My Question: My Goal is to determine the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} \quad$ for $x \in [-\pi, \pi ]$ This function is $2\pi$-periodic.
My Approach: i found ...
0
votes
1answer
25 views
Determining Fourier series for $\lvert \sin{x}\rvert$ for building sums
My math problem is a bit more tricky than it sounds in the caption.
I have the following Task (which i in fact do not understand):
"Determine the Fourier series for $f(x)=\lvert \sin{x}\rvert$ in ...
2
votes
1answer
70 views
In my Fourier text book, there are the following exercises to prove. why do some of them have the same left side but have different right sides?
In my Fourier text book, there are the following exercises to prove.why do some of them have the same left side but have different right sides? The demand of these question is to prove these ...
2
votes
1answer
27 views
Need to find a Fourier Series…
I am to find a Fourier Series for the following function:
$$
y(x)=\sqrt {R^{2}-x^{2}}
$$
about
$$
-R \leq x \leq R
$$
with the recursion
$$
y(x+2R)=y(x)
$$
Do I let$\sqrt {R^{2}-x^{2}}$equal $y$ ...
3
votes
2answers
163 views
A Fourier series exercise
Can anyone give me a hand with this exercise about Fourier series?
Let $f(x)=-\log|2\sin(\frac{x}{2})|\,\,\,$ $0\lt|x|\leq\pi$
1) Prove that f is integrable in $[-\pi,\pi]$.
2) Calculate the ...
0
votes
0answers
43 views
Fourier series for $e^x$ over $[0,\pi)$
I am trying to solve the following,
Find the Fourier series of $h(x) = \text{e}^x, x \in [0,\pi)$.
I'm not sure how to approach it since the question does not specify whether to use an even or ...
0
votes
1answer
14 views
Question about a small part of this Fourier series problem
Calculate the Fourier series expansion for the following function of period 2:
$f(t)=2+2t^2$ for $-1<t<1$
I just have a small question for this problem.
I've already gotten $A_0$ to be ...
9
votes
4answers
278 views
Singular asymptotics of Gaussian integrals with periodic perturbations
At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$,
$$
\int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
0
votes
0answers
26 views
I can't go on to compute the Fourier coefficients.
I can't go on to compute the Fourier coefficients.How to continue?
1
vote
0answers
50 views
Intervals where the function is similar to the Fourier series
$$f(x)=\left\{\begin{array}{l l}
0,\quad x \in [-L,0[\\
1,\quad x \in [0,L]
\end{array}\right.$$
I need to know in which intervals the sum of the Fourier series is "equal to the function $f(x)$".
...
-1
votes
1answer
38 views
How to prove this Fourier question?
How to prove this Fourier question? I hope for a procedure in detail.
1
vote
0answers
63 views
Fourier Analysis of Prime Counting Function
I was thinking about the following:
Denote $\pi(x)$ as the prime counting function such that:
$$
\pi(x) = \#\text{ of prime numbers}\leq x
$$
It is well known from the prime number theorem that
$$
...
1
vote
1answer
52 views
What is the odd fourier extention of sin x cos(2x)
odd half range extension of
f(x) = sin x cos(2x) with limits 0 to pi
2
votes
1answer
26 views
Convergence of Fourier series $\frac 1 {2i} \sum_{n \neq 0} \frac { \exp (inx)} n$
Let
$$
f(x) := \begin{cases}
-\frac \pi 2 - \frac x 2 && x \in (-\pi,0) \\
\frac \pi 2 - \frac x 2 && x \in (0, \pi) \\
0 && x = 0
\end{cases}
$$ I have to show that $\frac 1 ...
0
votes
1answer
67 views
What is the Fourier transform for $f(x)=e^{-x^2}$
I remember their being a special rule for this kind of function but I cant remember what it was.
Anyone know how ?
thanks
1
vote
0answers
22 views
Understanding the indices in a Fourier series
Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written
$$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$
which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
0
votes
1answer
33 views
Prove that the given sum is not Fourier series
Book browsing Banach spaces of Analytic function of the author Kenneth Hoffman on page 74 is one example. This example is compiled in this way:
Prove that
$$
\sum_{n=1}^{\infty}\frac{1}{\log ...
16
votes
1answer
277 views
Seeking Fourier series solution on Laplace equation…still looking, am I on track?
Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps.
The problem as given says:
Consider the BVP for $u=u(x,y)$:
...
2
votes
1answer
69 views
Computation of standard series
I am stuck on the computation of the following sum:
$\sum_{k=1}^\infty e^{-n^2}\cos(n)$. Simple tricks fail and also i have no idea how to fit it for Fourier series. Are there any other ways?
1
vote
2answers
58 views
fourier series by lebesgue integral
hw:
anyone knows how to find fourier series over the function
$$
f(x)=
\begin{cases}
1 & \text{if $x$ is irrational}\\
0 & \text{if $x$ is rational}
\end{cases}
$$
by lebesgue integral?
...
0
votes
1answer
30 views
Show that Fourier coefficients approach zero uniformly
Let $f(t)$, $g(t)$ be piecewise continuous functons on $[-\pi,\pi]$, periodically continued on $\mathbb R$. I want to show that
$$
a_n(x) = \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x+t)g(t) ...
-1
votes
0answers
41 views
Laurent expansion of this funtion
What is the Laurent expansion of this.
$$f(x)=\frac{2−a(z+z^{-1})}{2(1−az)(1−az^{-1})}=\frac{a}{2(z−a)}+\frac{1}{2(1−az)}+\frac{1}{2}$$
Please help me on this so I can calculate next the fourier ...
-2
votes
2answers
292 views
Calculate the Fourier transform of $b(x)=1/(x^2+a^2)$
I need help to calculate the Fourier transform of this funcion
$$b(x)=\frac{1}{x^{2}+a^{2}}$$
where $$a>0$$
Thanks
0
votes
2answers
83 views
Trapezoid rule over trigonometric polynomials
The question is regarding trapezoid rule applied on trigonometric polynomials
Here is the question
Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
0
votes
1answer
32 views
Wave-Function Series?
So I was basically exploring the function:
$\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is:
...
0
votes
1answer
54 views
Fourier transforms - don't understand this concept!!! Please help me on this
I have two Fourier transforms to solve, but the problem is that a I have a characteristic bijection or some etching that I don't know what it is and I don't know how to solve this... Please help
...
1
vote
1answer
54 views
Sum over cosines = dirac delta - how to get the coefficients?
Given this formula:
$$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$
Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$?
I googled and searched all kinds of ...
1
vote
0answers
26 views
Complex Fourier series of a function [duplicate]
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
2
votes
2answers
187 views
Complex Fourier series
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
3
votes
2answers
122 views
Fourier Series of $f(x) = x$
I am having trouble finding the complex Fourier series of $f(x) = x$ and using that complex series to find 1)the real Fourier series of $f(x)$ and 2) the complex and real Fourier series of $h(x) = ...
0
votes
1answer
71 views
Parseval's Theorem Q
I have this question:
I know Parseval's theorem is given by $2a_0^2 + \sum_1^{\infty} (a_n^2 + b_n^2) = \frac {2}{T} \int_{-T/2}^{T/2} f(x)^2 dx$, where T is the period.
$f(x)$ is even, so I ...
0
votes
1answer
36 views
A function whose derivatives always have a convergent fourier series
I am looking for a solid example that such a function that its derivatives can always be found by taking derivatives component-wisely in its Fourier series. A function with finitely many Fourier terms ...
1
vote
1answer
34 views
Steady Temperature Distribution Pipe
I was wondering if anyone can show me what approach to finding the steady state temperature distribution in this problem. The image is in the link below.
...
2
votes
2answers
64 views
How to solve this equation by Fourier series?
$$ y''+3y=\sin ^4 x ,\quad y=\frac{1}{8} +\frac{\cos2x}{2}-\frac{\cos4x}{104}.$$
Now the text book states the solution, but I don't know the process of solving this equation. I need your help!
1
vote
0answers
28 views
Upper bound on truncation error of a fourier series approximation of a pdf?
Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation:
$$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$
...
2
votes
1answer
106 views
Pointwise convergence of double Fourier series
I'm looking for theorems that deal with the pointwise convergence of double Fourier series expansions for a special class of functions.
Let $D \subset [-\pi, +\pi]^2$ be an arbitrary set of finite ...
0
votes
1answer
43 views
Exponential Form of Fourier Series
Problem Suppose $f$ is a continuous function on interval $[-\pi,\pi]$ such that $\sum_{n\in\mathbb{Z}} |c_n| < \infty$ where $c_n = \dfrac {1}{2\pi} \int_{-\pi}^\pi f(x)\cdot \exp(-inx)~dx$, the ...
1
vote
1answer
55 views
Sine Fourier Series? How do I get to this answer?
Calculate the
Sine Fourier series expansion for $\displaystyle f(t) = t^2 $ in $\displaystyle 0 < t < 2.$
I know I need to use $\displaystyle ∑ B_n \sin\left(\frac{nπt}{2}\right)$
I know the ...
1
vote
1answer
49 views
Roots of a finite Fourier series?
In general, are there any clever tricks to help find the roots of a finite Fourier series? Presumably there aren't analytic methods, but can we use the fact that our function is a finite Fourier ...
0
votes
0answers
22 views
Pointwise convergence of sine series of $x^{-2}$
I was wondering if the sine series of $x^{-2}$ converges pointwise on the open interval $(0,1)$.
What is the most general criterion to ensure pointwise convergence of a Fourier series?
