Questions tagged [integral-transforms]
This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine transforms.
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16 views
Extending function from hyperplane segment to ball, estimating integral by integral on manifold
Let $B:=B_R (0) \subset \mathbb{R}^d$ be the ball with radius $R$ at $0$ and $H \subset \mathbb{R}^d$ a hyperplane that satisfies $H \cap B \neq \emptyset$ and $0 \not\in H$. Furthermore, let $f \in C^...
1
vote
1answer
21 views
Is there a name for the square of a function plus the square of its Hilbert transform?
Given a real-valued analytic function $f$ defined on the whole real line, and its Hilbert transform ${\cal H}f$, it seems that the quantity $f(x)^2+{\cal H}f(x)^2$ should have some kind of importance ...
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0answers
32 views
How can I solve this integral equation with the inverse Laplace Transform?
This question is related to Solving an integral equation with inverse Laplace transform.
Let $\alpha,\beta,\mu>0$ with $\alpha/\beta>\mu$ and $X\sim\operatorname{Gamma}(\alpha,\beta)$. I am ...
1
vote
0answers
21 views
Boundary to boundary transformation of an integral
In my textbook "Mathematical analysis I" we saw something called "Boundary to boundary transformation of an integral" (Note that my textbook is a Dutch textbook, I've tried to translate the name the ...
2
votes
0answers
27 views
Binomial sum for an arbitrary function
I'm looking for some known results for sum of this type but I can't find anything.
The sum is defined as:
$$S(x,a,b,n)=\sum_{k=0}^n \binom{n}{k} (-1)^{k} f((a(n-k)+bk)x)$$
where $f$ is an arbitrary ...
2
votes
0answers
65 views
Bessel integral invovling algebraic and hyperbolic functions
I am desperate in evaluating the following Hankel transform
$$
\int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} \frac{\cosh(ky)}{\cosh(k)} k\mathrm{d} k,
$$
where $J_0(kr)$ is the Bessel function of ...
2
votes
0answers
22 views
How can the following be transformed in to a sum of complete elliptic integrals of the first and second kind
I have the following, that I known from a numerical implementation of the problem by a third party should be able to be transformed in to elliptic integrals of the first and second kind however I can'...
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votes
1answer
61 views
Tricky integral relationship
I am trying to prove that
$$\begin{equation}\int_x^{x+1}\left(\int_0^{v} (u-0)f(u)\textrm{d}u+\int_v^{1} (u-1)f(u)\textrm{d}u\right)\textrm{d}v=\\\int_0^x\int_v^{v+1}f(u)\textrm{d}u\textrm{d}v\end{...
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votes
0answers
56 views
Laplace Transform of Complementary Error Function
I need to apply one Laplace transform formula while I have no idea how to prove it:
$$\int_0^\infty e^{-st} e^{a k} e^{a^2 t}
\operatorname{erfc} \left( a \sqrt{t} + \frac{k}{2 \sqrt{t}} \right) dt =
...
9
votes
2answers
150 views
Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $
After weeks of going back and forth I've been able to solve the following definite integral:
$$I = \int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $$
To solve this I employ ...
1
vote
0answers
34 views
Fourier and Mellin transforms of Hilbert Transform
I am reading Hilbert transform recently and meet two questions. The book I am reading is Debnath and Bhatta "Integral Transforms and Their Applications".
If we define the Hilbert transform on the ...
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0answers
19 views
Reference Request: n-dimensional Laplace Transform
I am looking for a reference, where the conditions for the existence of the n-dimensional Laplace transform are proven, i.e. when the laplace transform
\begin{equation}
F(\lambda_1, ..., \lambda_n) = ...
3
votes
0answers
66 views
About $ F(u) = \int_{-\pi/2}^{+\pi/2} \ln(g(x) + u) dx $
We know for $ u > 1 $
$$ \int_{-\pi/2}^{+\pi/2} \ln(\sin(x) + u) dx
= \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right) $$
Usually this is shown by using differentiation under the ...
8
votes
2answers
235 views
Double integral with Hankel transform
Let's say we have a double integral in the following form:
$$I=\int_0^\infty \int_0^\infty f(x) g(y) J_0(xy) x y dx dy $$
Using the definition of the Hankel transform, we can write:
$$I=\int_0^\...
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votes
0answers
17 views
Integral Transformation from circle to unit sphere
I want to show that
$\displaystyle \frac{1}{2\pi r}\int_{\partial B(x,r)}u(y)\,\mathrm{d}s(y)=\frac{1}{|S^1|}\int_{S^1}u(x+r\theta)\,\mathrm{d}s(\theta)$
This is essentially a shift and dilation ...
1
vote
0answers
9 views
Decomposition of time series into weighted (weights known) stretched exponentials with unknown offset function
Looking for solution to decomposing function $f(t)$ into stretched exponential functions all with same meta-exponent and decay constants, but with different amplitudes $p$ (known) and offsets $\tau$ (...
2
votes
1answer
78 views
Proof of inverse Laplace transform
Why is
$$f(t) = \frac{1}{2πj}\int_{\sigma-j\infty}^{\sigma+j\infty} F(s) e^{st} \, ds,$$
provided that
$$F(s) = \int_{0}^{\infty} f(t) e^{-st} \, dt \ ?$$
I tried to find out myself, or searched ...
3
votes
0answers
32 views
“Bilateral Mellin convolution”
The Mellin convolution of two functions, when it exists, is of the form
$$
(f \ast_M g)(t)
= \int_0^\infty
f\left( \frac{t}{\tau} \right)
g(\tau)
\frac{\mathrm{d}\tau}{\tau}
$$
and ...
0
votes
0answers
24 views
What are the easiest inverse Mellin transforms?
I'm looking to familiarize myself with the inverse Mellin transform
\begin{align*}
\mathcal{M}^{-1}[\varphi](t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-s}\varphi(s)\, \mathrm{d}s
\end{...
1
vote
1answer
62 views
On the variety of Integral Transforms
Recently I have come across this book of Integral Transforms. At this point I realized for myself that there are so many transforms with different kernel out there. Now to provide a little bit of ...
2
votes
2answers
100 views
Laplace Transform of the incomplete Gamma Function
While looking through this ($178$,$(30)$) Table of Integral Transforms I have come across the Laplace Transform of the Incomplete Gamma Function which is given by
$$\mathcal{L}\{\Gamma(\nu,at\}(p)~...
0
votes
1answer
64 views
Kernel of Hankel Transform
I need to solve a cylindrical diffusion problem that is defined in $[1,\infty]$. I would like to use Hankel Transform that has is defined on $[0,\infty]$. So in order to apply Hankel transform in my ...
0
votes
0answers
26 views
Building an Integral Transform from an Orthonormal Basis on the L2 Circle
Background
It is well known that the orthonormal basis for $L_{2}[-\pi, \pi]$ is $\Omega = \{ e^{-jmt} \}_{m \in \mathbb{Z}}$. We extend this to $L_{2}(\mathbb{R})$ via the Fourier Transform, which ...
1
vote
1answer
39 views
Which transform makes correllation a multiplication?
In Fourier analysis, a central theorem for the Fourier Transform states:
$$\mathcal F\{(f*g)(t)\}(\omega)=\mathcal F \{f(t)\}(\omega)\cdot \mathcal F\{g(t)\}(\omega)$$
In other words, convolution ...
3
votes
1answer
72 views
Asymptotics of Hilbert transform from asymptotics of original function
Suppose we have a locally integrable function $f: \mathbb R^+ \to \mathbb R$ and we consider the 'Hilbert transformed' function
$$ h(t) := \int_0^\infty \frac{f(\tau)}{t -\tau} \mathrm d \tau. $$
We ...
0
votes
0answers
51 views
Fix points of integral transfroms
After I have worked a little bit with different integral transform, especially with the Laplace transform, I was confronted with the fact that there are some functions which remain of the same type ...
3
votes
2answers
154 views
Mellin transform of a Gaussian Hypergeometric Function with negative $x$-argument
I am quite fascinated by the formula for the Mellin transform of the Gaussian Hypergeometric Function, which is given by:
$$\mathcal M [_2F_1(\alpha,\beta;\gamma;-x)] = \frac {B(s,\alpha-s)B(s,\...
1
vote
0answers
30 views
Can an inverse Mellin transform converge conditionally?
Let $F(x)$ satisfy
$$\begin{equation}\int\limits_0^\infty \vert F(x)\vert x^{c-1}dx<\infty\end{equation}\tag{1}$$
for some $c\in(a,b)$ with $a,b\in \mathbb{R}$ and $a<b$.
Then, the Mellin ...
0
votes
0answers
15 views
Are there inverse Mellin transforms with two distinct strips of convergence?
Let
$$F(x)=\frac{1}{2\pi i}\int\limits_{c-i\infty}^{c+i\infty}f(s)x^{-s}ds$$
$$f(s)=\int\limits_{0}^{\infty}F(x)x^{s-1}dx$$
Real constant $c$ is from the strip $\Re(s)\in(a,b)$.
$\textbf{...
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votes
0answers
85 views
Division by t of an Inverse Laplace Transform
My problem is to find a function $g(s)$, solution of the equation:
$$ \frac{1}{t} \mathcal{L}^{-1} \left\{ f(s)\right\}(t)=\mathcal{L}^{-1} \{ g(s)\}(t)$$
I know the general property:
$$ \mathcal{L}^{...
-1
votes
1answer
40 views
Conditions for uniqueness of a Mellin transform
Let $f(x)$ and $F(s)$ be a Mellin pair, such that one is the Mellin inversion of the other in the fundamental strip $S_f$.
Let $g(x)$ and $G(s)$ be a Mellin pair, such that one is the Mellin ...
3
votes
1answer
101 views
Laplace transform of a “heat kernel”
This question is closely releted to this question:
How do we solve the laplace transform of the Heat Kernel?
Let $A>0$ and
$$f(t) = \frac{A^2}{2\sqrt{\pi}t^\frac{3}{2}}e^{-\frac{A^2}{4t}}$$
...
1
vote
1answer
151 views
Solving the Heat Equation using the Fourier Transform
The Question:
Solve the Heat Equation (for $u = u(x,t)$)
$$\frac{\partial u}{\partial t} = \frac{\partial^2u}{\partial x^2} \qquad u(x,0)=T(x)$$
by applying the Fourier Transform in the $x$ ...
1
vote
1answer
64 views
Holomorphic extension of the Laplace transform
Let $u \in L^1_{\mbox{loc}}((0,+\infty))$ be such that $e^{-\lambda t} u(t) \in L^1((0,+\infty))$ for some $\lambda >0$. Let $\mathcal{L}[u]$ be the Laplace transform of $u$, that is
$$\mathcal{L}[...
3
votes
1answer
93 views
Laplace convolution with the Bessel function
The Question:
(i) Find the Laplace Transform of the Bessel Function $J_0(x)$
(ii) Hence, show that if $f(x)$ satisfies the differential equation
$$f''(x)+f(x)=J_0(x) \qquad f(0)=f'(0)=0$$
then $f$ ...
2
votes
1answer
150 views
Solving Bessel Equation using Laplace Transform
The Question:
Given that $J_0(x)$ satisfies
$$x\frac{d^2J_0}{dx^2}+\frac{dJ_0}{dx}+xJ_0=0 \qquad J_0(0)=1 \qquad \frac{dJ_0}{dx}(0)=0$$
Show that the Laplace Transform $\bar{J_0}(p)$ of $J_0$ is ...
2
votes
0answers
42 views
Can integration contour of an inverse Mellin transform be deformed at will in fundamental strip?
Let $m(x)$ be inverse Mellin transform of $M(s)$:
$m(x)=\frac{1}{2\pi i} \int\limits_{c-i\infty}^{c+i\infty}x^{-s}M(s)ds$
Mellin transform $M(s)$ is analytic on fundamental strip $a<\Re(s)<b$ ...
2
votes
1answer
49 views
Fourier Transform of $\dfrac{1}{x^2+2x+2}$
The Question:
(i) Determine the Fourier Transform of
$$f(x) = \frac{1}{a^2+x^2} \qquad a>0$$
(ii) Hence determine the Fourier Transform of
$$g(x) = \frac{1}{x^2+2x+2}$$
My Attempt:
(i) I got $...
0
votes
0answers
9 views
Error function and its associated operation
Consider a manifold which may or may not be bounded, being the possibility space of functions for which all possible polynomial integrals exist. One could consider the integral transform as a change ...
2
votes
0answers
36 views
Fredholm integral equation of the 2nd kind with a weakly singular convolution kernel
I've reviewed the literature but could not find a solution of the integral equation
$$\int_{0}^{1} dx\frac{\phi(x)}{|x-y|^{\alpha}}+a\phi(y)=b.$$
for the function $\phi$ where $\alpha\in(0,1)$ and ...
0
votes
0answers
23 views
What overarching term would include use of a Fourier transform, other similar integral transforms, and their discrete counterparts?
Question: What overarching terms could be used to refer to the use of a Fourier transform, Hankel transform, etc. as well as discrete versions of those, such as a Fast Fourier Transform (FFT), etc.?
...
0
votes
1answer
71 views
The tauberian final value Theorem
I have found many formulations of the FVT (final value theorem), but none of this with the precise assumptions that one needs. As far I understood, one needs to assume that the
$\lim_{t\to+\infty}f(t)...
2
votes
0answers
80 views
When an Hilbert–Schmidt operator is invertible? How to construct the kernel of the inverse?
This question is motivated by the following example.
I have the following integral operator defined on $L^2(\mathbb{T}^n)\to L^2(\mathbb{T}^n) $
$$ T f(x) = \sum_{k \in \mathbb{Z}^n} \sigma(k,x) \ \...
1
vote
0answers
63 views
Maximum/Minimum of a function defined as Laplace-Transform
Consider a function $f(t)$ that is only defined via its Laplace transform $F(s)$
$f(t) := \int\limits_{c - i\infty}^{c + i \infty} d s \; e^{st} F(s)$,
where $c$ is chosen appropriately (i.e. to ...
0
votes
0answers
32 views
I want to disprove an equality involving a double integral
I want to show that the following equality does not hold:
\begin{equation}\label{at3}
\frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
0
votes
1answer
66 views
Solve ODE by integral transform
$$g’’(x) + f^2*g’(x) + 4 g(x) = 0$$$$ g(0) = 0$$$$ g’(\pi/2) = 0$$ where $x \in\mathbb{R}$, $f$ is frequency.
If we use Laplace transform, the initial value of g'(pi/2) makes me cannot go further, ...
2
votes
0answers
39 views
Transform $\cosh(x)$ into $\exp(x^2/2)$
I'm looking for an integral transform that maps $\cosh(x)$ into $\exp(x^2/2)$. More specifically I'm looking for a kernel $K(y,x)$ and a domain $\Omega \subset \mathbb{R}$ such that
$$
\int_\Omega dx ...
0
votes
0answers
84 views
two dimensional fourier transform of a bessel function multiplied by a second bessel function
$\frac{J_1 \bigl(\sqrt {x^2+y^2}\bigr)}{\sqrt {x^2+y^2}}$ and $\frac{J_1 \bigl(\sqrt {(x-a)^2+(y-b)^2}\bigr)}{\sqrt {(x-a)^2+(y-b)^2}}$ are two Bessel functions defined in the x-y coordinate plane, ...
0
votes
0answers
25 views
Master equation for a Multiplicative random process
I would like some assistance to derive a master equation for a Multiplicative random process. I suppose it should take the form of a Mellin convolution.
It´s well known that for an additive random ...
1
vote
2answers
27 views
Can the Laplace transform be inverted?
Consider the Laplace transform $\tilde{f}(s)$ of a function $f(t)$ defined as $$\tilde{f}(s)=\int_0^\infty f(t)e^{-st} \, dt.$$ Can this relation be inverted to obtain $f(t)$ in terms of $\tilde{f}(s)$...
