0
votes
1answer
27 views

How to prove that this equality is the development of a fourier series?

how can I show that this identity is a development of a fourier series? $$f(x)=\sin^3 x=\frac{3}4 \sin x-\frac{1}4 \sin 3x$$ I tried this: obtain the Fourier coefficients whih $$b_n=\frac{2}\pi ...
0
votes
2answers
56 views

$\int_0^\pi\sin(2t)e^{-in2t}dt$ complex number integral for integer values of n

$$\int_0^\pi\sin(2t)e^{-in2t} \, dt$$ wolfram alpha say the answer is $$\frac{1-e^{-2 i n π}}{2-2 n^2}$$ although using the integral trig identity $$\int ...
4
votes
1answer
74 views

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 ...
1
vote
1answer
39 views

Theoretical question about Fourier Series, I'm confused!

If I have a function f(x) defined on $[0,L)$, said to be periodic of period $L$ and such that $f(0)\neq0$, how should I get the Fourier coefficients? I'm hesitating between taking the even extension ...
0
votes
1answer
32 views

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: ...
2
votes
0answers
23 views

A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
1
vote
0answers
83 views

Fourier Series; odd and even half-range expansion

I have some standard Fourier series questions which I cannot solve. My fourier series is defined like this: $$s(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos (nx) + b_n \sin (nx))$$ For $f(t) = ...
1
vote
1answer
122 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
0
votes
0answers
16 views

Obtain the complex Fourier Series of the following function:

$$f(t)=t^3 \;\;\;\;\;\;\;\;\;\;\;\; -3/2<t\leq 3/2 $$ $$f(t)=f(t+3)$$ I've tried setting up an integral for $C_n$ coefficients using the formula $$C_n = \frac{1}{L} \int^{L/2}_{-L/2} f(t) ...
2
votes
1answer
36 views

How can I find this integral for a fourier series?

I have to calculate the following integral $$ b_n = \dfrac{1}{\pi} \int_{-\pi}^{\pi} \dfrac{1}{2}x \sin nx dx$$ The correct answer is apparently $$\dfrac{(-1)^{n-1}}{n}$$ But I have no idea how I ...
0
votes
2answers
35 views

Prove $\frac1T \int_0^T\left(\sum_{k=-\infty}^{\infty}c_ke^{j{\frac{2\pi kt}{T}}}\right)^2dt= \sum_{k=-\infty}^{\infty}|c_k|^2$

This question relate to fourier series in electrical engineering but I post it here as it's only mathematical concern. I cannot prove this $$\frac1T ...
0
votes
1answer
42 views

Calculating Fourier expansion using Legendre Polynomials

I'm trying to write any function of the type $t^m$ using Legendre polynomials $P_n(t)$ . That means: $$t^m=\sum_{n=0}^\infty\langle P_n,t_m\rangle P_n =\sum_{n=0}^\infty a_{mn}P_n$$ Where I have to ...
3
votes
2answers
104 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 $$ ...
0
votes
0answers
18 views

Relating Fourier Transform to an Integral involving Sin(vt)

I have data for a function $S(Q)$ and I'm trying to find values for a different function $g(r)$ Now I know $g(r) = \int_0^{\infty} Q(S(Q)-1) \sin(Qr)\, dQ$ This is closely related to the sine ...
0
votes
0answers
42 views

fourier series and correlation coefficients question?

We have the signal in the figure. I must do the trigonometric fourier series of the signal and also the exponential fourier series.Also,find the correlation coefficients between $f(t)$ and $e^{3t}$. ...
0
votes
1answer
48 views

Fourier transform real and imaginary part question?

I have to find the fourier transform of $f(t)=e^{-a^*t}*u(t)$ For a>0 the signal has an infinite value therefore doesnt have a Fourier transform.For a>0 we have: ...
7
votes
4answers
190 views

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 $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$ ...
1
vote
2answers
63 views

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 $c_n = \frac{1}{2\pi} \int _{-\pi}^\pi f(x) e^{-inx}\,dx.$ ...
1
vote
4answers
382 views

What is $\int_{-\pi}^\pi \cos(nx)\cos(mx)\,dx$?

I'm pretty sure that there's a theorem that says that the Fourier coefficients of a sum of $\cos(nx)$ and $\sin(nx)$ 's are the coefficients of the sum itself. I tried to prove that in the specific ...
1
vote
1answer
39 views

Proving a claim $|c_n e^{in\theta}| = |c_n|$

I'm studying about Fourier series from a book called "Fourier series and its applications" by Folland and on page 40, the author makes the claim that: $$|c_n e^{in\theta}| = |c_n|,$$ where $n$ is an ...
2
votes
2answers
104 views

Finding the complex fourier series of the function $x^2sin(x)$ in the interval $[{-\pi}, \pi]$?

This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ...
0
votes
3answers
87 views

Calculating this integral?

I'm trying to calculate $$\int\limits_{-\pi}^0e^{-x}\cos(nx)\,\mathrm{d}x$$ as part of a Fourier series calculation. My problem is the calculations seem to loop endlessly - I'm integrating by parts ...
1
vote
2answers
59 views

Fourier series - Integral

Let $f$ be a complex-valued piecewise continuous function defined on the interval $[-\pi,\pi]$ and let \begin{equation} \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos(nx)+b_{n}\sin(nx) ...
3
votes
1answer
142 views

Parseval's Identity (Integral)

Calculate the integral: \begin{equation} \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^{n}}e^{inx}\right|^{2}dx\end{equation} I'm familiar with Parseval's identity which states that for ...
0
votes
2answers
119 views

Fourier Series Coefficient of a given signal

$$ {\rm x}\left(t\right) = \sum_{k = -\infty}^{\infty}\left[\delta\left(t-\dfrac{k}{3}\right) + \delta\left(t-\dfrac{2k}{3}\right)\right] $$ I need to find the Fourier series coefficient of x(t). I ...
7
votes
1answer
223 views

Computing Fourier transform for $L^2$ function

For a function $f\in L^1(\mathbb{R})$, its Fourier transform is defined as $$\hat{f}(y)=\int_{-\infty}^\infty f(x)e^{-ixy}dx$$ For a function $f\in L^2(\mathbb{R})$, its Fourier transform is ...
1
vote
1answer
722 views

fourier series of $|\sin x|$

I need to find the fourier series of $$|\sin x|$$. Im not sure my way is right, would be happy if someone fix me. I found $$a_0=4/\pi$$, the function is even, so $$b_n=0$$ but how do I calculate: ...
2
votes
2answers
93 views

Can I calculate approximately a definite integral of a function by integrating its Fourier Sine Series term-by-term?

I'm not sure how to put fancy formulae here because I'm a fairly new user. So bear with me for a moment as we go through a formulae-less reasoning. 1) I have a function $f(x)$. 2) I want to ...
4
votes
1answer
315 views

fourier expansion of $\coth$ and justifying an identity

The problem: Justify the following equalities: $$\cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2}$$ I am trying to figure ...
0
votes
1answer
108 views

Checking work on Fourier series for $10 \cos t$

Well, I am checking this out because even though I know the problem (a 2nd order differential) can be solved more easily, I want to try this out. So we have $10 \cos t$ and want the Fourier ...
1
vote
1answer
73 views

Computing The Fourier Sine Series.

Compute the Fourier Sine series of the odd function: $f(x) = x^3 - 4x, -2 \leq x \leq 2 $. (Periodically extended with period 4) I know how to compute this of course where: $b_n = ...
1
vote
2answers
100 views

Bessel function to $\sin(kr)$

$J_{\frac{1}{2}}(kr)=\frac{\sqrt{\frac{2}{\pi }} \text{Sin}[\text{kr}]}{\sqrt{\text{kr}}})$ This can be easily obtained by Mathematica, How to do the details?
0
votes
1answer
143 views

Inverse Fourier transform to find out $\hat c_1$

If we have an integration which is need to solve inversely $$a_0 e^{-r^2/R^2} = \int_0^\infty \hat{c}_1(k) \frac{\sin(k r)}{r} dk,$$ If I transform the $\sin(kr)$, then we get imaginary part. Please ...
1
vote
0answers
234 views

Fourier-Bessel series coefficients

When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for $k_1$and $k_2$ both zeroes of $J_n(t)$, the orthogonality relation given by: $$\int_0^1 ...
0
votes
1answer
133 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
0
votes
1answer
32 views

Am I understanding this integration right?

This is the snippet of a problem from this PDF here. What I dont understand is why they retain the $Sin$ part for evaluation after integration when all that it is going to evaluate to is 0. If I ...
0
votes
1answer
56 views

Can we estimate the lower bound in this way?

This post is aimed to find a lower bound of $\sum_{k=1}^{n}\frac{\cos(kx)}{k}$ for arbitrary $n \geq 1$ ================================= My approach: Let $S_n(x)$ denote the partial sum of the ...
1
vote
2answers
67 views

Calculating $a_0$ in Fourier Series

I am using this YouTube video to learn Fourier Series. The question can be clearly seen in the picture. The instructor calculates $a_0$ as the area under the triangle which is fine. Nothing wrong ...
-1
votes
4answers
10k views

How does knowing a function as even or odd help in integration ??

So, I am learning Fourier Series and it involves integration. I am not too good at integration. Now, the resource I use is videos by Dr. Chris Tisdell. In the ...
2
votes
1answer
41 views

Integration question verifying piecewise

I have the following question: from direct integration show $\displaystyle \int \limits_{-L}^{L} \cos({m πx\over L})\cos({nπx\over L}) \ dx = \begin{cases}0 & m \neq n \\ L & m = n \\ ...
1
vote
2answers
142 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
2answers
355 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
66 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
0answers
28 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: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
3
votes
2answers
368 views

Complex Fourier series

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
1
vote
1answer
72 views

Computation of $\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy$

Find the value $$\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy,\qquad k\in N^{+}$$ My idea: \begin{align} ...
0
votes
1answer
66 views

How to calculate integral of fourier series?

I know how to integrate, but I can't understand how the integral of this fourier serie is calculated. my problem is with integral of the sigma. fourier: integral: Can anyone say me how this ...
2
votes
3answers
221 views

Calculating the Fourier series of $x^{3}$

I was given as homework to calculate the Fourier series of $x^{3}$. I know, in general, how to obtain the coefficients of the series using integration with $$\sin(nx),\cos(nx)$$ multiplied by the ...
1
vote
0answers
218 views

Double Fourier Series $\cos(nx)\cos(my)$

Let $f(x,y) = xy$ on the square $[0, \pi]^2$. Find the Fourier cosine-cosine series of $f$. I am working on this question with a group and one of us gets all the coefficients as zero. Is this correct ...
2
votes
0answers
268 views

Parseval's formula application [closed]

I need to show that $$1+\frac{1}{3^2}+\frac{1}{5^2}+\cdots=\frac{\pi^2}{8}$$ by evaluating $\|Sq\|^2$ such that $Sq(t)=\frac{4}{\pi} \sum \limits_{n \text{ odd} \geq 1} \dfrac{\sin nt}{n}$ We know ...