# Tagged Questions

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### Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
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### How to evaluate the integral $\int e^{ipx}e^{ipx} d^{3}x = 0$

I am embarrassed to ask this question. But I came across the following in a physics book: $$\int e^{ipx}e^{ipx} d^{3}x = 0$$ $d^{3}x = dydydz$, as @Semiclassical shows below. This came up in the ...
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### fourier transform of scaled function

let us consider following example one thing which i did not understand is where absolute value of $a$ came from?ok if we have $\int^{\infty}_{-\infty} x(a*t)*e^{-j\omega*t}dt$ then we may have ...
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I am working on image recontruction and I try to find out how the radon transformation works. I have benn using mainly Natterer, F. and Wubbeling, F.: Mathematical Methods in Image ...
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### Problem about average of cos square (nt) where n is arbitrary

I often see people just say time average of cos^2(nwt) is 1/2, I want to know in what cases this is not valid? w is just the frequency, can be assumed as a constant. Assuming you are always ...
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### Easiest way to prove integral of $e^{ikx}$ is $\delta(k)$

What is the easiest way to to derive the following equation: $$\int_{-\infty}^{\infty}e^{ikx}dx = 2\pi\delta(k)$$ I understand the equation can be derived by assuming the Fourier integral theorem, ...
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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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### Integral is equal to $0$

Let be $f \in L^1[0,1]$, then it applies $\int_0^1 \exp(2i\pi xk)f(x n)\,dx=0$ for $n,k\in \mathbb{N}$ with $0<k<n$. Ideas: f can be extended to a function on $\mathbb{R}$ with period $1$, ...
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### The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
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I want to solve the following integral: $$\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$$ I did the following: Substitute $\gamma(k) = k-k_0 \Leftrightarrow k = \gamma + k_0;~\gamma(\pm\infty) = ... 1answer 30 views ### Apply the Fourier Transform to$A\cdot e^{-a|k - k_0|}$I have the following problem: The task is to show that $$f^*(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(k) e^{ik(x-vt)} dk$$ with$f(k) = A\cdot e^{-a|k - k_0|}$equals $$f^*(k) = ... 2answers 163 views ### The Fourier transform of a power of the absolute value function (and a related integral) What (Fourier-analytic?) methods would I use to compute the following two integrals? \displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ... 0answers 16 views ### Obtain the complex Fourier Series of the following function:$$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 47 views ### Integral equation, Fourier transform Find all functions$ f : \mathbb{R} \rightarrow\mathbb{R} $, that solve$\int_{-\infty}^{\infty} f(t-x)f(x) dx =e^{-t^2}$,$ t\in \mathbb{R}$How do I solve this? I know that the left part is the ... 1answer 37 views ### Integral-Fourier sum I am trying to prove the following relation in (3) where$\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$I=\frac{1}{2}\int_0^\alpha dx \left( \beta ... 2answers 50 views ### How to calculate this basic Fourier Transform? I am trying to calculate the Fourier Transform of g(t)=e^{-\alpha|t|}, where \alpha > 0. Because there's an absolute value around t, that makes g(t) an even function, correct? If that's ... 2answers 98 views ### fourier transform of sinc function let us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i want ... 1answer 42 views ### Integral from inverse Fouriertransform of 1/(1+p^2)^2 In a calculation I end up with the following integral$$\int_0 ^\infty \frac{p \sin (pr)}{(1+p^2)^2}dp , $$could someone give me a hint how to evaluate this one? (This integral comes from the ... 1answer 22 views ### Existence of a subsequence and convergence to 0 of function solving heat equation Let f^j(x) be a sequence of integrable functions on the circle such that$$\int_{-\pi}^{\pi}|f^j (x) |^2 dx = 1.$$ALso, let u^j(x,t) solve the heat equation on the circle with initial data ... 0answers 60 views ### Prove the converse of convolution theorem I am trying to prove the converse of convolution theorem:$$ \mathscr{F}[f(x)g(x)]=\frac{1}{\sqrt{2\pi}}\,\widetilde{f}(\omega)*\widetilde{g}(\omega)$$I try to apply the definition of convolution ... 0answers 18 views ### How to find the inverse fourier transform of the fourier transform of \delta(x-x_0) [duplicate] I know that F(\delta (x-x_0 )) = e^{j\omega x_0} and so therefore F^{-1}(e^{j\omega x_0 }) = \delta(x-x_0) but when I try to do the integral for the inverse fourier transform: I get ... 2answers 74 views ### Fourier transform of function What is Fourier transform of$$f(x)=\frac{e^{-|x|}}{\sqrt{|x|}}?$$I tried to calculate it using$$F(e^{-|x|})=\sqrt{\frac{\pi}{2}}e^{-|a|}$$and$$F(\frac{1}{\sqrt{|x|}})=\frac{1}{\sqrt{|a|}}$$and ... 1answer 331 views ### Fourier transform of \operatorname{erfc}^3\left|x\right| (this is a follow-up on my another question) Could you please help me to find the Fourier transform of$$f(x)=\operatorname{erfc}^3\left|x\right|,$$where \operatorname{erfc}z denotes the the ... 0answers 199 views ### What's the Fourier Transform of an Error Function? What is the Fourier transform of \displaystyle \operatorname{Erf}\left[a+bx^{2}\right] ? I need this in order to evaluate$$ \int_{-\infty}^{\infty}e^{-\beta x^{2}}Erf\left[a+bx^{2}\right]dx $$... 0answers 37 views ### Invariant functions under integral transforms We all know Fourier transform has invariants such as e^{-x^2}, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ... 0answers 63 views ### On \int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t} How to count this?$$ \int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t} $$Can we use residue formula? 1answer 104 views ### Integral \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx Hi I'm trying to solve this integral Fourier Transform$$ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k)) $$where sgn(k)=1 for k>1 and -1 for k<1. I am ... 2answers 61 views ### How can this integral be rewritten with convolutions? I've got f:\mathbb{R}\rightarrow\mathbb{R} bounded and I'm trying to write `\mathtt{f},' a discrete version of f, where each element in the domain takes on the average of the corresponding ... 2answers 145 views ### Fourier transform of \operatorname{erfc}^2\left|x\right| Could you please help me to find the Fourier transform of$$f(x)=\operatorname{erfc}^2\left|x\right|,$$where \operatorname{erfc}z denotes the the complementary error function. 1answer 150 views ### A Parseval-like theorem for Mellin transforms A particular case of Parseval's theorem for Fourier transforms says that if f is square integrable on \mathbb{R}, then$$ \int_{-\infty}^{\infty} |f(t)|^{2} \ dt = \int_{-\infty}^{\infty} ... 1answer 33 views ### time integration property of fourier transform I am having some trouble with some Fourier transform, Suppose that$F(\omega)$is the Fourier transform of$f(x)$, i.e. where $$F(\omega)=\int_{-\infty}^{\infty}f(x)e^{-i\omega x}\,dx.$$ What is ... 1answer 44 views ### Riemann-Lebesgue application By the Riemann-Lebesgue lemma, I have shown that for any finite interval measurable set$I$of finite measure, any$h \in \mathbb{R}$, $$\lim_{n \to\infty}\int_I \cos (n(x+h)) \mathop{dx} = 0.$$ I ... 0answers 16 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 ...
Can anybody please guide me how to compute the following inverse Fourier Transform ? $$p(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{1}{(1-j\omega\bar{x})^K}e^{-j\omega x}d\omega$$ I shall ...