# Tagged Questions

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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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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 $f(t) = ... 1answer 122 views ### Numerical approximation of trigonometric polynomial I have the following problem: Let$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). $... 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 $$C_n = \frac{1}{L} \int^{L/2}_{-L/2} f(t) ... 1answer 36 views ### How can I find this integral for a fourier series? I have to calculate the following integral$$ 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}$$But I have no idea how I ... 2answers 35 views ### Prove \frac1T \int_0^T\left(\sum_{k=-\infty}^{\infty}c_ke^{j{\frac{2\pi kt}{T}}}\right)^2dt= \sum_{k=-\infty}^{\infty}|c_k|^2 This question relate to fourier series in electrical engineering but I post it here as it's only mathematical concern. I cannot prove this$$\frac1T ... 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 ... 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 $$... 0answers 18 views ### Relating Fourier Transform to an Integral involving Sin(vt) I have data for a function S(Q) and I'm trying to find values for a different function g(r) Now I know g(r) = \int_0^{\infty} Q(S(Q)-1) \sin(Qr)\, dQ This is closely related to the sine ... 0answers 42 views ### fourier series and correlation coefficients question? We have the signal in the figure. I must do the trigonometric fourier series of the signal and also the exponential fourier series.Also,find the correlation coefficients between f(t) and e^{3t}. ... 1answer 48 views ### Fourier transform real and imaginary part question? I have to find the fourier transform of f(t)=e^{-a^*t}*u(t) For a>0 the signal has an infinite value therefore doesnt have a Fourier transform.For a>0 we have: ... 4answers 190 views ### Evaluate \int_{-\pi}^\pi \big|\sum^\infty_{n=1} \frac{1}{2^n} e^{inx}\big|^2 \operatorname d\!x I am trying to solve exercises for the coming exam, and I am stuck on this exercise: Evaluate$$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$... 2answers 63 views ### Find complex Fourier coefficients let f(x) = \sum^{10}_{m=1}(-1)^m \sin(2^m x). denote complex Fourier coefficients of f(x) over [-\pi, \pi] as c_n = \frac{1}{2\pi} \int _{-\pi}^\pi f(x) e^{-inx}\,dx. ... 4answers 382 views ### What is \int_{-\pi}^\pi \cos(nx)\cos(mx)\,dx? I'm pretty sure that there's a theorem that says that the Fourier coefficients of a sum of \cos(nx) and \sin(nx) 's are the coefficients of the sum itself. I tried to prove that in the specific ... 1answer 39 views ### Proving a claim |c_n e^{in\theta}| = |c_n| I'm studying about Fourier series from a book called "Fourier series and its applications" by Folland and on page 40, the author makes the claim that:$$|c_n e^{in\theta}| = |c_n|,$$where n is an ... 2answers 104 views ### Finding the complex fourier series of the function x^2sin(x) in the interval [{-\pi}, \pi]? This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ... 3answers 87 views ### Calculating this integral? I'm trying to calculate$$\int\limits_{-\pi}^0e^{-x}\cos(nx)\,\mathrm{d}x$$as part of a Fourier series calculation. My problem is the calculations seem to loop endlessly - I'm integrating by parts ... 2answers 59 views ### Fourier series - Integral Let f be a complex-valued piecewise continuous function defined on the interval [-\pi,\pi] and let \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos(nx)+b_{n}\sin(nx) ... 1answer 142 views ### Parseval's Identity (Integral) Calculate the integral: $$\int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^{n}}e^{inx}\right|^{2}dx$$ I'm familiar with Parseval's identity which states that for ... 2answers 119 views ### Fourier Series Coefficient of a given signal$$ {\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] $$I need to find the Fourier series coefficient of x(t). I ... 1answer 223 views ### Computing Fourier transform for L^2 function For a function f\in L^1(\mathbb{R}), its Fourier transform is defined as$$\hat{f}(y)=\int_{-\infty}^\infty f(x)e^{-ixy}dx$$For a function f\in L^2(\mathbb{R}), its Fourier transform is ... 1answer 722 views ### fourier series of |\sin x| I need to find the fourier series of$$|\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$$but how do I calculate: ... 2answers 93 views ### Can I calculate approximately a definite integral of a function by integrating its Fourier Sine Series term-by-term? I'm not sure how to put fancy formulae here because I'm a fairly new user. So bear with me for a moment as we go through a formulae-less reasoning. 1) I have a function f(x). 2) I want to ... 1answer 315 views ### fourier expansion of \coth and justifying an identity The problem: Justify the following equalities:$$\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}$$I am trying to figure ... 1answer 108 views ### Checking work on Fourier series for 10 \cos t Well, I am checking this out because even though I know the problem (a 2nd order differential) can be solved more easily, I want to try this out. So we have 10 \cos t and want the Fourier ... 1answer 73 views ### Computing The Fourier Sine Series. Compute the Fourier Sine series of the odd function: f(x) = x^3 - 4x, -2 \leq x \leq 2 . (Periodically extended with period 4) I know how to compute this of course where: b_n = ... 2answers 100 views ### Bessel function to \sin(kr) J_{\frac{1}{2}}(kr)=\frac{\sqrt{\frac{2}{\pi }} \text{Sin}[\text{kr}]}{\sqrt{\text{kr}}}) This can be easily obtained by Mathematica, How to do the details? 1answer 143 views ### Inverse Fourier transform to find out \hat c_1 If we have an integration which is need to solve inversely$$a_0 e^{-r^2/R^2} = \int_0^\infty \hat{c}_1(k) \frac{\sin(k r)}{r} dk,$$If I transform the \sin(kr), then we get imaginary part. Please ... 0answers 234 views ### Fourier-Bessel series coefficients When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for k_1and k_2 both zeroes of J_n(t), the orthogonality relation given by:$$\int_0^1 ... 1answer 133 views ### Complex form of Fourier Series So, the last part of the university syllabus in the chapter of Fourier Series is: ... 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 ... 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 ... 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 ... 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 ... 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 \\ ...
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? ...