6
votes
0answers
61 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
1
vote
3answers
30 views

What are the concepts that I need to understand before studying Fourier Analysis?

Background ( Long Story Short ) : For some reasons, I am taking a class in my university that focus on Fourier Analysis Laplace Transform, and Partial Diffiential Equations Problem : I have done ...
4
votes
1answer
72 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 ...
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
3answers
180 views

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 ...
1
vote
2answers
21 views

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 ...
1
vote
0answers
34 views

Finding the number of derivatives for series problems

I have the following problem: How smooth are the following functions? That is, how many derivatives can you guarantee them to have? $$a)\;\;\;\;\; ...
0
votes
1answer
48 views

Complex Fourier series of $f(\theta) = e^{\theta}$

I have the following Fourier series problem: Let $f(\theta)$ be the periodic function such that $f(\theta) = e^\theta$ for $-\pi<\theta\leq\pi\;$, and let ...
4
votes
1answer
170 views

Using the Parseval Identity to compute $ \sum_{n=1}^{+ \infty} \frac{1}{(4n^2-1)^2}$

Parseval's Identity: For continuous $f: [- \pi , \pi] \to \mathbb{R}$ $$ \sum_{n=- \infty}^{+ \infty} |c_n|^2 = \frac{1}{2 \pi} \int_{ - \pi}^{ \pi} |f(x)|^2dx, \text{ where } c_n = ...
1
vote
2answers
109 views

Trigonometric series problem

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 ...
3
votes
2answers
98 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 $$ ...
1
vote
2answers
44 views

About Fourier coefficient definitions

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: ...
7
votes
4answers
187 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 $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$ ...
2
votes
0answers
53 views

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 $2\pi$?
1
vote
2answers
62 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 $c_n = \frac{1}{2\pi} \int _{-\pi}^\pi f(x) e^{-inx}\,dx.$ ...
2
votes
1answer
54 views

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 ...
5
votes
2answers
886 views

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 ...
1
vote
1answer
33 views

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 ...
0
votes
1answer
50 views

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 $\displaystyle g(x)=\int _0^xf(t)dt$ ...
0
votes
1answer
41 views

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 ...
1
vote
1answer
86 views

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 ...
0
votes
1answer
52 views

Integral computation involving $\sin^2 x$

I am attempting to find the Fourier series for $\sin^2x$ but am getting stuck. For the value of $a_0$, I am trying to do it as follows: $$ \frac 1{2\pi}\int_{-\pi/2}^{\pi/2}(\frac 12 -\frac ...
1
vote
1answer
54 views

Fourier Series doubt

I have a doubt regarding the Fourier series usage in terms of the Fourier series formula, which has multiple variants and is quite complicated. EDIT: I would like to mention that this question (of ...
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
95 views

Expansion and convergence of $\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}$

Consider the series: $$\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}\;\;\;\;n\in\mathbb{N}$$ Other than formal manipulation of the Taylor series of the $\sin$ function, is there a way to expand ...
2
votes
2answers
102 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 ...
2
votes
2answers
123 views

Prove Parseval Identity for $f \in C(\Bbb T) 2\pi$ periodic continuous functions

Question: Prove Parseval Identity for $f \in C(\Bbb T) $ $2\pi$ periodic continuous functions $$ \frac{1}{2 \pi} \int_{-\pi}^\pi |f(x)|^2 dx =\sum_{n=-\infty}^\infty |\hat f(n)|^2 $$ Thoughts: We ...
1
vote
1answer
976 views

Fourier series coefficients proof

Can somebody help me understanding the fouries series coefficients? I know that if we have: $$f(n) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase}$$ And because of the ...
2
votes
2answers
128 views

Inner Product vs. Integrals with Fourier Series, When to include 1/2pi?

I am confused about when to include a prefactor of $\frac{1}{2\pi}$ when dealing with integrals of functions that are expressed as fourier series. This is what I understand (please correct me if I'm ...
0
votes
0answers
63 views

Representing real function as integral over trigonometric functions

Since one can clearly express any function g(x) as integral from 0 to infinity of A(k)cos(kx)dk + integral from 0 to infinity of B(k)sin(kx)dk, how would G(k) relate to A(k) and B(k)? In other words, ...
4
votes
2answers
109 views

Origin of coefficients of fourier series?

I was wondering how we derive these formulas, and why we have a separate formula for $a_0$? All I know from advanced engineering mathematics text book are following formulas but where do they come ...
2
votes
0answers
54 views

Fourier's Method Question

I've been asked to use Fourier's method to obtain the following solution; $$u(x,t) = \sum_{n=1}^{\infty} B_n e^{-(n \pi C / L)^2 t} \sin(\frac{n \pi x}{L})$$ $$B_n = \frac{2}{L} \int_0^L \sin(\frac{n ...
3
votes
1answer
74 views

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 ...
1
vote
1answer
81 views

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: ...
0
votes
1answer
118 views

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?
0
votes
3answers
2k views

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 ...
0
votes
2answers
150 views

Conceptual question about Discrete Fourier Transform

On the wikipedia page for the discrete Fourier transform, the first sentence says: In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a ...
1
vote
1answer
242 views

Even and odd functions using integrals

If $f: [-r, r] \to\mathbb{R}$ is an even function, show that $g(x) = \cos(nx)$ is an even function and $h(x) = \sin(nx)$ is an odd function. Consider: $\int_{-r}^{r} f(x)\cos(nx)dx = 2\int_{0}^{r} ...
1
vote
3answers
70 views

Help to compute the following coefficient in Fourier series $\int_{(2n-1)\pi}^{(2n+1)\pi}\left|x-2n\pi\right|\cos(k x)\mathrm dx$

$$\int_{(2n-1)\pi}^{(2n+1)\pi}\left|x-2n\pi\right|\cos(k x)\mathrm dx$$ where $k\geq 0$, $k\in\mathbb{N} $ and $n\in\mathbb{R} $. it is a $a_k$ coefficient in a Fourier series.
1
vote
1answer
85 views

Fourier Series for $|x|$

How can I calculate the Fourier series for |x| (where $x\in[−\pi,\pi]$) in the complex form? Thanks.
1
vote
1answer
112 views

Fourier series representation of even and odd functions

I'm not sure where to begin on showing that a Fourier series of a periodic function that is neither odd or even can be represented by the sum of the cosine fourier series and sine fourier series. I ...
2
votes
1answer
40 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 $y$ ...
1
vote
1answer
207 views

What is the odd Fourier extension of $\sin x \cos(2x)$?

odd half range extension of $$f(x) = \sin x \cos(2x)\text{ with limits $0$ to $\pi$}$$
2
votes
1answer
3k 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
1answer
84 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
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
58 views

Calculating value of the fourier series

In the following fourier series, how the red marked numbers are calculated?
3
votes
1answer
52 views

How to calculate $\sum_{k=1}^{\infty}\frac{1}{(a^2+k^2)^2}$ after calculating $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Parseval identity?

The task is to calculate sum $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Fourier coefficients of $f(x)=e^{ax}$. First of all I calculated Fourier coefficients of the sum: ...
1
vote
1answer
53 views

Why does $\sin{\alpha}\cdot i\sin{\alpha x}$ disappear from this integral?

In a section on fourier transforms, my textbook contains these steps for an example: $$f(x) = \int_{-\infty}^\infty \frac{\sin{\alpha}}{\pi \alpha}e^{i\alpha x}d\alpha$$ $$= ...
3
votes
2answers
160 views

Another integral with Catalan

Show that: $$\int_0^1\frac{\arcsin^3 x}{x^2}\text{d}x=6\pi G-\frac{\pi^3}{8}-\frac{21}{2}\zeta(3)$$ I evaluated this by some Fourier series. Is there any other method? Start with substitution of ...