# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 $e^{3t}$. ...
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### 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: ...
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### 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 $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$ ...
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### 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 $c_n = \frac{1}{2\pi} \int _{-\pi}^\pi f(x) e^{-inx}\,dx.$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
### 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 ...