5
votes
2answers
53 views

Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where ...
1
vote
0answers
24 views

how to solve this inverse mellin transform

We know from the Perron's formula that $$ \int _{c-i\infty}^{c+i\infty}\frac{\zeta(s)}{2\pi i s}x^{s}ds=[x] $$ $$ \int _{c-i\infty}^{c+i\infty}\frac{\zeta(s)^{2}}{2\pi i s}x^{s}ds=\sum_{n\le x}d(n) ...
6
votes
2answers
73 views

Mellin Transform of the floor function

I have from integral transform tables that: $$\operatorname{Mellin}(\lfloor x\rfloor) = -\dfrac{\zeta(-z)}{z},\quad \operatorname{Re}(z)<-1$$ How can this be proved?
4
votes
1answer
151 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} ...
0
votes
1answer
44 views

Transformation formula for multidimensional integral

Let $A,B$ be positive definite symmetric $n\times n$ matrices. I stumbled upon the following identity and don't see why it should hold: $$\int_{\mathbb{R}^n}\frac{1}{\sqrt{\det A \det ...
0
votes
0answers
59 views

Fourier transform of a function involving $\sec(\omega)$

The summary of my question is: What should I make of $\mathcal{F}^{-1}\left[\frac{\csc(\omega)}{\omega^2-\beta^2}\right]$ where $\mathcal{F}^{-1}$ is the inverse fourier transform (taking $F(\omega)$ ...
1
vote
1answer
82 views

Change of variables in double integral - what's wrong?

I have a homework problem, as follows: Evaluate the double integral by making an appropriate change of variables. $\iint_R 9\sin(49x^2+16y^2)\,dA$, where $R$ is the region in the first ...
0
votes
1answer
163 views

Find a transformation from tetrahedron to cube in $R^3$ to calculate a triple integral?

I would like to calculate the triple integral of a function $f(x,y,z)$ over a region given by a tetrahedron with vertices $(0,0,0)$, $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$. I am trying to do this by ...
1
vote
0answers
40 views

inverse of integral transform

given a general integral transform $$ g(x)= \int_{0}^{\infty}dyf(y)K(xy) $$ for a general formula of the kernel $ K(xy) $ is there an inverse of the Integral transform to obtain $ f(x) $ from above ...
3
votes
2answers
57 views

Evaluating Laplace Transform

I have a Laplace transform function of the following form and I'm trying to evaluate it. From my research I think I need to take the Inverse Laplace Transform and then integrate, but I'm having ...
36
votes
1answer
971 views

To evaluate $\int_0^{+\infty} \frac{\;dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;dx}{\sqrt{x^2+(\frac{a+b}{2})^2}\sqrt{x^2+ab}}$$ ...
1
vote
0answers
68 views

Laguerre transform function

For continuous indices 'n' and 'm' is it possible to have $$ \int_{0}^{\infty}dx e^{-x}L_{n}(x)L_{m}(x) = \delta (n-m)? $$ Another question: let $g(m)$ be the function defined via the transform $$ ...
3
votes
2answers
400 views

Continuity of integral function

How to show that the following function is right continuous at $0$ (that is, when $a\to0+$): $I(a) = \int_0^{\infty}\frac{\sin x}{x}e^{-ax}dx$? I know that Lebesgue integral $I(0) = \frac{\pi}{2}$. ...
3
votes
0answers
132 views

Understand integral from Gradshteyn and Ryzhik book “Table of integrals, series, products”

I was checking useful integrals in this book. I have found one (6.298) that is what I need, but I don't understand how every step towards the final result works. ...
0
votes
0answers
176 views

Triple Product Integral on Real Spherical Harmonics Basis Functions

Okay I know that Real Spherical Harmonics are given by If $m \lt 0$ $~$ then $\sqrt{2}$ $~$ $Im(\text{SphericalHarmonicY}[l,|m|])$ If $m=0$ $~$ then $~$ $\text{SphericalHarmonicY}[l,0]$ If $m \gt ...
1
vote
1answer
218 views

Gelfand-Levitan-Marchenko equation

how can one solve the integral $$ f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$ (1) so $$ q(x)= 2\frac{d}{dx}K(x,x) $$ (2) $$ -y''(x)+q(x)y(x)=0 $$ (3) $$ y(0)=0=y(\infty) $$ $ q(x) $ here is ...
1
vote
2answers
65 views

Help solving $\frac{1}{{2\pi}}\int_{-\infty}^{+\infty}{{e^{-{{\left({\frac{t}{2}} \right)}^2}}}{e^{-i\omega t}}dt}$

I need help with what seems like a pretty simple integral for a Fourier Transformation. I need to transform $\psi \left( {0,t} \right) = {\exp^{ - {{\left( {\frac{t}{2}} \right)}^2}}}$ into ...
0
votes
1answer
113 views

Tonelli's theorem using in mean residual life definition

If X is a nonnegative random variable representing the life of a component having distribution function F,the mean residual life ...
2
votes
1answer
93 views

Proof of the Direct mapping Theorem for Mellin transform.

I cannot understand an integration in the proof of the Direct mapping Theorem for the Mellin transform. A statement of the Theorem, together with an outline of the standard proof, can be found at page ...
3
votes
0answers
59 views

Can I solve for a unique integral kernel?

Consider, for $\mathbf{v},\mathbf{w} \in \mathbb{R^3}$, $$ f(\mathbf{w}) := \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) \, d\mathbf{v} \, .$$ Is it possible to solve for the integral kernel, ...
4
votes
0answers
508 views

Do kernel functions of integral transforms have any special properties?

From the Wikipedia page on integral transforms, it states that: ...an integral transform is any transform $T$ of the following form: $$ (Tf)(u)=\int^{t_2}_{t_1}K(t,u)f(t)dt $$ ...There are ...
1
vote
0answers
167 views

Inverting an integral transform

This question is either very obvious or by nature unsolvable in a general case. I Google'd around for a solution to no avail. Given an integral transform of kernel K across some interval I as a ...
0
votes
1answer
173 views

Interchanging the order of integration.

Can someone explain why the following works please? $\int^t_0 f(s) \int^s_0 f(u) \,du\,ds = \int^t_0 f(s) \int^t_s f(u) \,du\,ds $ EDIT: All I know is that $f(x)$ is an integrable function. This is ...
2
votes
1answer
470 views

Commutativity of Convolution in higher dimensions

I have a basic question about how to show that convolution in dimension $n$ is commutative - or maybe it is rather a question about change of variables .. So on $\mathbb{R}$ I know how to show ...
3
votes
1answer
118 views

Series around $s=1$ for an integral

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
2
votes
0answers
76 views

Evaluating the limit $y \to 0^+$

Given $t \in \mathbb{R}$ and $z = x + iy$ and $y>0$. $\lim_{y\to0^+} \frac{1}{t - z} = \frac{1}{t-x} + \pi i \delta(t-x)$ This limit is given in the book Integral Transforms and Their ...
6
votes
1answer
232 views

Stieltjes Transform

Stieltjes transform for a distribution $F(x)$ is defined as $$m(z)=\int \frac{dF(x)}{x−z}$$ where z is complex with positive imaginary parts and $F(x)$ is a distribution function. Basically, I am ...
7
votes
1answer
275 views

Evaluate $\int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle}$

The task is to evalute $$ \int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle}, \;\;\; \theta \in \mathbb{C}^2 \setminus ( \mathbb{S}^1 \cup \left\{ 0 ...
1
vote
1answer
163 views

Having such integral, how to optimize it in maple?

So we have : (1/3)*sig0*h^3*(int(int(sin((1/3)*arctan(y, x)), x = 0 .. r), y = 0 .. 2*Pi)) Is it possible to optimise it? (in maple or any other way...) How I ...
2
votes
1answer
328 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
0answers
224 views

Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
3
votes
2answers
396 views

Series of nested integrals

I'm trying to calculate the following series of nested integrals with $\varepsilon(t)$ being a real function. $$\sigma = 1 + \int_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\mathrm dt_1 ...
0
votes
1answer
116 views

When to use the Functional Determinant in Polar Coordinate Transformation

I am currently learning about polar coordinate transformation, especially for integrating over certain regions. Let's say we have to calculate $\int_{n}{xy \; dx dy}$ Then I think the correct ...