18
votes
1answer
309 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 ...
0
votes
1answer
16 views

Connection between Expected power and Expected Energy over Frequency - Dirac Delta Squared?

I know math people don't like the Dirac delta, so feel free to answer with your measure theory - I'll try my best to understand. Suppose $x$ is a WSS stochastic process $\{x[n] : n \in ...
4
votes
2answers
130 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.
1
vote
0answers
64 views

construction of a special series of functions

Here is the problem: Let $A$ be the set of positive integers greater than 1. For each $L\in A$, we want to construct a smooth function $f_L$ with compact support such that ...
2
votes
0answers
121 views

Fourier transforms of Marcum Q function

I cannot find in the literature the following: $$\int_0^{\infty}\,Q_1(a,bx)\,\cos(\omega x)\,\,dx$$ and $$\int_0^{\infty}\,Q_1(a,bx)\,\sin(\omega x)\,\,dx$$ with $a,\omega>0$. $Q_1(a,x)$ is ...
2
votes
1answer
304 views

Fourier transform of the indicator of the unit ball

What is the Fourier transform of the indicator of the unit ball in $\mathbb R^n$? I think it is known as one of special functions, so I would be happy to know which one.
3
votes
2answers
124 views

Fourier transform of 3D Sinc function

What is the Fourier trasnform of the function $$\frac{\sin(P|\mathbf{x-y}|)}{|\mathbf{x-y}|}$$ where $P$ is a real parameter and $\mathbf{y}$ is a fixed point in three-dimensional space?
1
vote
0answers
109 views

Fourier transform of projection of spherical cap

I am currently trying to derive an analytical expression for the Fourier transform of the projection of a spherical cap of the unit sphere onto the xy-plane. Setting up the integration in cylindrical ...
0
votes
0answers
32 views

Fourier transform of $f(r,r',\theta,\theta')$

How can I calculate the FT of: $$\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,f_n(r,r')=\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,\frac{J_n(\alpha r)J_n(\alpha r')}{[(\alpha ...
2
votes
1answer
458 views

Fourier Transforms of shifted sinc funtions

I would like to calculate the Fourier transform of the following functions: $$\left(\dfrac{\sin(\pi x\pm\pi n/2)}{\pi x\pm\pi n/2}\right)^2$$ $$\dfrac{\sin(\pi x+\pi n/2)}{\pi x+\pi ...
0
votes
0answers
38 views

Square equivalent of $circ(r)$

I would like to know if there is a similar function to $$circ(\sqrt{x^2+y^2})=1 , 0\leq \sqrt{x^2+y^2}\leq 1$$ but with a square domain $0\leq x\leq 1$ and $0\leq y\leq 1$. If yes, which is its ...
11
votes
1answer
344 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
3
votes
1answer
211 views

Non-normalized sinc function

In looking for an efficient way to show the relation $$\int_{-\infty}^\infty \frac{e^{inx}}{\Gamma(\alpha+x)\Gamma(\beta-x)} \, dx = \frac{(2 \cos n/2)^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)} e^{ ...
1
vote
2answers
476 views
4
votes
0answers
251 views

How to find the inverse Fourier Transform of the product of two bessel functions of the first kind and a complex exponential function?

I am attempting to find a closed form or symbolic expression of the inverse Fourier transform of the product of two Bessel functions of the first kind and a complex exponential, e.g. $P(t) = IFT_w \{ ...
0
votes
1answer
103 views

Approximation of the Fourier Transform of General Functions in a Box

I'm trying to get a general approach for the Fourier Transform of functions $f$, only in a restricted area $-\frac M2\le x \le \frac M2$, where ${\frak F}_{f(x)}(\omega)$ exists. My idea was the ...
2
votes
1answer
175 views

Convergence of the Fourier Transform of the Prime $\zeta$ Functions

I think I found a way to write the truncated Prime $\zeta$ function like this: $$ P_x(s)=\sum_{p<x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} ...
3
votes
0answers
218 views

Questions about the Fourier expansion of $e^{iz\cot(x)}$

By analogy with Jacobi‚ÄďAnger expansion, one expects that $e^{iz\cot(x)}$ has a Fourier expansion of the form : $$e^{iz\cot(\theta)}=\sum_{n=-\infty}^{\infty}\Lambda_{n}(z)e^{in\theta}$$ ...
0
votes
1answer
77 views

Integral of Scaled Bessel Function With Linear Phase

I am trying to solve a problem part of which includes the following integral ($j=\sqrt{-1}$): $$\int_{k_1}^{k_2} k e^{-jk\sigma} J_n(\rho k) \, \mathrm{d}k$$ The $e^{-jk\sigma}$ term is making my ...
0
votes
0answers
85 views

question on the expansion of the function

For a given real number $c>0$ define functions $\left(\psi_k^c(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...
1
vote
1answer
241 views

How to show integral of different order Hankel transformed functions are equal?

Say I have a function $p_v(r) \in L^2(\mathbb{R})$ given by $$p_v(r) = \int_0^\infty P(k) J_v(rk)\,k\,dk$$ From mucking around in MATLAB it seems the following is true: $$\int_{r=0}^\infty ...
0
votes
1answer
45 views

Bessels and initial conditions

I'd like to know if I have got the following ideas right: 1) $f(r,\theta,t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)\exp[in\theta-j^2_{nk}t]$ subjected to initial ...
1
vote
0answers
80 views

Fourier analysis confusion

I think I may have misinterpreted this question, anyhow I am very confused. Here it is in its full glory: Let $f(r,\theta, t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty ...
2
votes
1answer
4k views

Fourier transform of Bessel functions

I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being ...
3
votes
0answers
221 views

2 dimensional Fourier transform integral

I'm trying to calculate the two dimensional Fourier integral $$\iint \mathrm d\vec{R} \; (x^2 + y^2) \; e^{-2 \sqrt{ x^2 + y^2 + z^2}} \; e^{i\vec{K}\vec{R}} \;,$$ with $\vec{R}=(x,y)$. Switching to ...
2
votes
1answer
312 views

How do I find the inverse Hankel transform of $k^2e^{-k^2}$?

I am trying to solve: $f_l(r)=\int_0^{\infty}e^{-k^2}k^4j_l(kr)dk$, where $j_l$ is the spherical Bessel function of the first kind, for any integer l >= 0. Thanks in advance for any answers!
3
votes
3answers
681 views

Hint on how to prove $\zeta ( 2) =\pi ^{2}/6$ using the complex Fourier series of $f(x)=x$

I know how to prove $\zeta (2)=\pi ^{2}/6$ by using the trigonometric Fourier series expansion of $x^{2}/4$. How can one prove the same result using the complex Fourier series of $f(x)=x$ for $0\leq ...