Tagged Questions
1
vote
0answers
65 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
49 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 ...
2
votes
1answer
60 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
174 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
128 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 ...
3
votes
1answer
110 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
72 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 ...
5
votes
1answer
201 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 ...
1
vote
1answer
128 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
242 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
201 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
$$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
3
votes
2answers
304 views
Series of nested integrals
I'm trying to calculate the following series of nested integrals with $\varepsilon(t)$ being a real function.
$$\sigma = 1 + \int_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\mathrm dt_1 ...
0
votes
1answer
103 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 ...
