# Tagged Questions

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### Simplifying big expression

What to do with this? $$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$ Can it be simplified?
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### Evaluate the integral $\int_{-\infty}^\infty e^{-(t²+2t)/2}e^{-i\omega t}dt$

I have this integral: $$\int_{-\infty}^\infty e^{-(t²+2t)/2}e^{-i\omega t}dt$$ I don't know how to solve it, but I have tried, like this: $$\int_{-\infty}^\infty e^{(-(t²+2t)/2)+(-i\omega t)}dt$$ ...
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### How to develop the Fourier Transform in my mind now that I know the Fourier Seires?

I know that we can represent some function $f$ in this way: $$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right) + \sum_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$$ ...
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### 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 ...
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### 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 ...
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### Check: Find $h(x)$ when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$.

Find h(x) when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(y)=\frac{1}{(1+y^2)^2} =\frac{1}{1+y^2} \times \frac{1}{1+y^2} =\hat{\frac{1}{2}e^{-|x|}} \times \hat{\frac{1}{2}e^{-|x|}}$$ Let ...
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### missing $j*\omega$ in integral

let us consider following integral according to property of delta function,we can write this intgeral as $\int^{t=\infty}_{t=t_0} e^{-j*\omega*t}$ or we can write as ...
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### Solution of definite integral of product of bessel function and exponential

I have an integral $I=\int_{\theta} \int_r J_m(k_1r)e^{-j[P_x r \cos(\theta)+P_y r \sin(\theta)]} r dr d\theta$ $0\leq\theta\leq2\pi; r<\infty$ is there any method to solve this?
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### Fourier Transform-1

I am trying to solve a Fourier transform problem and I am stuck. The problem is: $$f(t)= \frac{\sin(2t)}{e^{|t|}}.$$ I have used integration, but the answer that I come up with is different than ...
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### For every $a>0$, show that $\langle f_a, \psi\rangle= \int_{|x|>a} \frac{\psi(x)}{|x|}dx+\int_{|x|>a} \frac{\psi(x)- \psi(0)}{|x|}dx$

To prove that the inner product is a distribution it must satisfy the following property" $$|T(\phi)|=|\langle T,\psi\rangle| \leq C_N \sum_{|\alpha| \leq N} \|\partial^\alpha \psi\|_\infty$$ Part ...
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### $\lim_{k \rightarrow \infty} k(1+\cos(2k \pi x)) \chi_{(-1/2k,1/2k)}$

How do I calculate the following limit: $$\lim_{k \rightarrow \infty} g_k(x) =\lim_{k \rightarrow \infty} k(1+\cos(2k \pi x)) \chi_{\left(\frac{-1}{2k},\frac{1}{2k}\right)}$$
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### A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$\sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2)$$ with Dirac delta function $\delta$. ...
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### Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
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### Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
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### Can one express $f'(x)$ with the same basis as one uses for $f$?

If I have an orthonormal basis $\{\phi_n\}_1^\infty$ in space $L^2(a,b)$ and the generalized Fourier series expansion for $f$ would be: $$f= \sum \langle f, \phi_n\rangle\phi_n,$$ then can one use ...
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### Show that $f(x)$ is orthogonal to $f'(x)$ in $L^2(-\pi, \pi)$

I have the following problem: Suppose $f$ is of class $C^{(1)}$, $\;2\pi$-periodic, and real-valued. Show that $f'$is orthogonal to $f$ in $L^2(-\pi, \pi)$ by a) expanding $f$ in ...
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### Proving that $\langle f, g\rangle = \sum_n \langle f, \phi_n \rangle \overline{\langle g, \phi_n \rangle}$

I have the following problem to solve: If the set of functions $\{\phi_n \}_1^\infty$ is an orthonormal basis in $L^2(a,b)$ and the functions $f, g \in L^2(a,b)$, then show that: ...
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### Understanding dot product of continuous functions

I'm reading about Fourier analysis and in my book the author speaks about dot product for continuous functions $f, g\in L^2(a,b)$(the set of functions which are square-integrable on the interval ...
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### compute the derivative of a function defined by an integration over the whole real line

Let $h(t)=\int _{\mathbb{R}}e^{-\frac{(x+it)^2}{2}}d\lambda(x)$, where $\lambda$ is the Lebesgue measure. I want to prove that $h$ is differentiable and compute the derivative of $h$: ...
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### Greens function of 1-d forced wave equation

[ORIGINAL PROBLEM] You are given hat the Green's function $g(x,t,\xi, \phi)$ is $\frac{\partial^2g}{\partial t^2} - \frac{\partial^2g}{\partial x^2}=\delta(t-\tau)\delta(x-\xi)$ with ...
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### $\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$ [duplicate]

How we can do this sum? $$\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$$ I know that we could possibly use a Fourier series decomposition however I don't know what function to start with. I ...
If I know that the set of functions $\{\phi_n\}_1^\infty$ forms an orthonormal basis on $L^2(a,b)$ and the set $\{\psi_n\}_1^\infty$ is an orthonormal set on $L^2(\frac{a-d}{c}, \frac{b-d}{c})$, with ...