1
vote
0answers
30 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
41 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
106 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
90 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
75 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
32 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
162 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
0answers
51 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
58 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
49 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 ...
4
votes
2answers
254 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
25 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
44 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
35 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
84 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
51 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
43 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
38 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
75 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
96 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
99 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
545 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
98 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
57 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
96 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
47 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
61 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
79 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
115 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
1k 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
127 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
196 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
68 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
74 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
99 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
38 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
190 views

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

odd half range extension of $$f(x) = \sin x \cos(2x)\text{ with limits $0$ to $\pi$}$$
1
vote
1answer
2k 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
71 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
56 views

Calculating value of the fourier series

In the following fourier series, how the red marked numbers are calculated?
3
votes
1answer
47 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
150 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 ...
0
votes
1answer
102 views

A small question on fourier series

Why the series is divergent, but the equation holds? $$\sum\limits_{i=1}^{\infty }{\sin kx}=\frac{1}{2}\cot \left( \frac{x}{2} \right)$$
0
votes
1answer
84 views

Sum of series & sequences

I dont know how to evaluate the first one, for second one I can only show the sum is less than 2. $$\begin{align} & \prod\limits_{k=4}^{\infty }{\left( 1-{{\left( \frac{3}{k} \right)}^{3}} ...
5
votes
2answers
369 views

log sin and log cos integral, maybe relate to fourier series

I try to use the method of differentiation under integral sign for the first one And integrate it back, but I failed to find the constant $c$ .... Anyone hav other method? $$\begin{align} & ...
2
votes
1answer
112 views

Two log trig integral

$$\begin{align*} & \int_{0}^{\frac{\pi }{2}}{{{\ln }^{n}}\sin x\text{d}x} \\ & \int_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\ln \left( \ln \tan x \right)}\text{d}x \\ \end{align*}$$
9
votes
7answers
458 views

Why does this Fourier series have a finite number of terms?

I am learning about Fourier series in class and the basic form of a Fourier Series is $$a_{0}+\sum_{n=1}^{\infty} [a_{n}\cos(nx)+b_{n}\sin(nx)]$$ so a fourier series should have an infinity number ...
2
votes
1answer
99 views

A integral equation

Prove that : $$\displaystyle \int_0^{\frac{\pi}{2}}p(x)\cot x\text{d}x=2\sum_{k=1}^{\infty}\int_0^{\frac{\pi}{2}}p(x)\sin (2kx)\text{d}x$$ where $\displaystyle p(x)=x^n$