Tagged Questions

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Complex-valued Fourier integral: $\int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx}$

I'm working on the Fourier transform, but I don't know how to evaluate the integral: $$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx}$$
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Fourier transform of $\frac{1}{1+x^2}$

we know that the Fourier transform of $\frac{1}{1+x^2}$ is $f(y) = \int_{-\infty}^{\infty} \frac{1}{1+x^2} e^{-2\pi i x y} dx$ . Here is the idea used in my textbook, for y<0 : We calculate the ...
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Calculating Inverse Fourier Transform

I can't quite get an inverse Fourier Transform to match up with a statement in my textbook. At one point, my textbook writes: "If $g$ is a function that is one on the interval $(- \pi, \pi]$ and ...
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Highly Oscillating Integrals

I'd like to know the behavior of integrals of the form: $$\int_0^1 f(x) \cos(k x) dx$$ as $k \rightarrow \infty$ where f is a smooth function. It is easy to see, by expanding f in power series, ...
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Why aren't these two question equal?

Firstly I doubt whether the 12 is right in Q1.If it is right,please give a proof. Secondly why (1) is not equal to (2) in Q2?
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I can't understand the last second step.

It is known that $f(x)=\sum_{n=-\infty}^{\infty}c_{n}e^{inx}$, with $c_{n}:=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\:dx$, for $n\in\mathbb{Z}$. To prove: ...
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Complex Fourier series of a function [duplicate]

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
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Complex Fourier series

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
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Arnold's Trivium problem 51

Calculate $$f(k) = \int_{-\infty}^{+\infty} e^{ikx}\frac{1 - e^x}{1+e^x}dx.$$ As far as I know, this is not a function but rather the Fourier transform in tempered distributions. 1) What is ...
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Fourier transform of given characteristic function

I have a function $$g(l) = E [ e^{iuX}|X>l ] - Prob (X>l)$$ and i need to derive how its Fourier transform is: $$F_{l,v}(g(l)) = \frac{\phi_X(u+v)-\phi_X(v)}{iv}$$. This gets down to ...
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How to recover a measure from its Fourier transform?

Let $f$ be the complex function defined on $\mathbb{R}$ by $$f(t)=\frac{1-it}{1+it}.$$ 1) Does there exist a complex bounded measure $\mu \in M(\mathbb{R})$ such that $\hat{\mu}=f$ (where $\hat{}$ ...
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Fourier transform of $\mathrm{rec}(x) =\begin{cases} 1 & \text{if }|x| < 0.5,\\ 0.5& \text{if }|x| = 0.5,\\ 0& \text{if }|x| > 0.5 \end{cases}$

$$\mathrm{rec}(x) =\begin{cases} 1 & \text{if }|x| < 0.5,\\ 0.5& \text{if }|x| = 0.5,\\ 0& \text{if }|x| > 0.5 \end{cases}$$ The Fourier transform of this function is ...
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Fourier's method for PDE

We have $$U_{tt}=U_{xx} \quad 0<x<{\pi \over 2}, \quad t>0$$ $$U(x,0)=U(0,t)=U_x({\pi \over 2},t)=0$$ $$U_t(x,0)=cos(5x)\cdot sin(x)$$ We are looking for a solution in the form ...
332 views

A log improper integral

Evaluate : $$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$ I found it can be simplified to $$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$ I found the exact value in the ...
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3D Fourier Transform

I'm trying to calculate the inverse of the following 3D Fourier transform. $$\widetilde{f}= \frac{1}{(k^6-\alpha*k^2-\alpha*k_3^2)}$$ where $k = (k_1^2+k_2^2+k_3^2)^{1/2}$ the fourier transform is ...
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how to prove the convolution formular?

let $\overset{\backsim} {g}(x)=g(-x)$; suppose $u,\phi,\psi$ always make the integral significant,$E_n$ is the n-dimensional euclidean space. Then how to prove ...
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Path integrals using Fourier transformation

While going through a book named Mirror Symmetry, I came across a path integral, $$Z(\beta) = \int\limits_{X(t+\beta) = X(t)} DX(t) \exp\left(-\int\frac{1}{2}( \dot{X}^2 + X^2)dt\right)dt$$ where ...
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is there a nice way to find the fourier transform of…

I am looking for a nice way to calculate the FT of the following function $f(x)=\biggl(\sum_{n=1}^{c}~a_n~e^{-\frac{i}{2}~x~b_n}\biggr)^d$, where $d,c>0$, $a_n$ and $b_n$ are real coefficients, ...
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Bounds on integral

I am calculating Fourier coefficients for certain functions and have come across an integral of the form $$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$ where ...
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How to find this moment generating function

I am trying to find the moment generating function of a random variable $X$, which has probability density function given by $$f_{X}\left( x\right) =\dfrac {\lambda ^{2}x} {e^{\lambda x}}$$ Where ...
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equivalence in Fourier space

I have a comprehension problem regarding Fourier transforms. So far I know, the Fourier transform can be defined on the whole Schwartz space $\mathcal{S}(\mathbb{R})$ and is bijective on it. So I have ...
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Computing the Gaussian integral with Fourier methods?

There are many proofs that $$\int_{-\infty}^\infty e^{-x^2} \, \mathrm dx = \sqrt{\pi}.$$ For example, using a change to polar coordinates, differentiation under the integral sign, and the theory ...
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Boundedness of supremum of an Integral operator

I am trying to find an $L_2$ - bound on a certain class of operators, and on my way I produced an estimate for which I need to show that \sup_{x \in \mathbb{R}^n} \, ...
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the integral of the inverse of a Fourier series

Let $\{a_h\}$ be a double-sided complex sequence such that $\sum_{h=-\infty}^{\infty} |a_i| <\infty$ with $a_{0}\neq0$. Set $f(x) := \sum_{h=-\infty}^{\infty} a_h \exp(ixh)$ and assume that ...
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Do Fourier transforms of $\min$ and $\max$ exist (in closed form)?

I am wondering if there are Fourier transforms of $\min(x,a)$ and $\max(x,a)$ functions. Please forgive me if this is a dumb question, I don't normally use Fourier transforms. I attempted to simply ...
Fourier transform of function involving $\log$
I found the following problem which I am unable to solve. Calculate the following integral $$\int_{\mathbb{R}} \frac{d\omega}{2\pi} \log (1 + i a/\omega ) e^{-i \omega t}$$ for $a>0$ and ...