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.
1
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1answer
60 views
Hilbert Transform: limit of xHf(x)
In Terence Tao's notes page 1, cited below, he mentions that it is easy to see that
$\lim_{|x| \to \infty} xHf(x) = \frac{1}{\pi}\int f$
where $f$ is a Schwartz function and $H$ is the Hilbert ...
0
votes
1answer
39 views
How to evaluate the Laplace transform of the square root using Residue theory?
My lecturer mentioned that it is possible to evaluate the Laplace integral transform (definition below) of $\sqrt{t}$ using complex analysis. How is that possible?
$$\hat f (s)=\int^{\infty} _0 {\...
1
vote
0answers
42 views
Is $\int_0^1 \Psi(x)\Psi(1-x)\,dx$ related to any transform?
Is this related to any integral transform?
$$\int_0^1 \Psi(x)\Psi(1-x)\,dx=\int_{0}^{1} e^{{\frac{1}{\log(x)}}+{\frac{1}{\log(1-x)}}} \, dx.$$
The integral, where $K$ is the modified Bessel function ...
1
vote
0answers
46 views
Mellin transform involving $\sinh({A_1}/2)$
So I need to figure out how to take the Mellin transform of
$$ f(x)=\int_2^x \sin(A_1/2)+\sinh(A_1/2)dt,$$
where $A_1=1/\ln(t).$ I'd also like to know how well the Mellin transform of $f(x)$ ...
3
votes
1answer
39 views
How can I prove that the Hilbert transform on the 1-torus doesn't map $L^1(\mathbb{T})$ into itself?
Let $\mathbb{T}$ be the 1-torus. Then, it is well defined the Hilbert transform:
$$\mathcal{H}:L^1(\mathbb{T})\to L^0(\mathbb{T}), \vartheta\mapsto\int_{-\pi}^\pi f(\vartheta-t)\cot\left(\frac{t}{2}\...
0
votes
1answer
25 views
Laplace Transform of an integral function of a convolution
Making suitable assumptions wherever necessary, what is the Laplace Transform $\mathcal{L}(S(t))$ where
$S(t)=\int_{0}^{t}\int_{0}^{t}f(t-s_1,t-s_2)g(s_1)h(s_2)ds_1ds_2$.
I tried using the Double ...
4
votes
2answers
62 views
Laplace transforms with Fresnel(?) integrals
I've come into contact with this two part question, and the latter I'm not too sure how to go about; at least to me upon researching, I can't find anything remotely similar to what I've been asked.
...
0
votes
0answers
9 views
What is purpose of wavelet scaling function and how is it derived for e.g Haar wavelet or Dabuchies wavelet?
Scaling function is also called father wavelet. I understand concept of mother wavelet but not father wavelet. In the continuous wavelet transform there is no concept of scaling function but only when ...
1
vote
2answers
40 views
Why is the Fourier transform called a 'transform', and not a 'transformation'?
Why are the Fourier transform, Laplace Transform, etc called transforms, and not transformations?
This is about linguistics or terminology in mathematics. I feel there should be a reason why the word ...
0
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0answers
27 views
How to prove the validity of this solution?
I am trying to solve the following initial-boundary value problem by using Hankel transformation:
$$ \frac{dT}{dt}= \frac{d^2T}{dr^2} + \frac{1}{r}\frac{dT}{dr} - ζ T + ψ\left(\frac{1}{t+t_{o}} exp\...
6
votes
3answers
68 views
Transforming integral to polar coordinates
By transforming to polar coordinates, show that
$$\int_{0}^{1} \int_{0}^{x}\frac{1}{(1+x^2)(1+y^2)} \,dy\,dx$$
Is equal to
$$ \int_{0}^{\pi/4}\frac{\log(\sqrt{2}\cos(\theta))}{\cos(2\theta)} d\...
0
votes
0answers
12 views
derivative of surface integral over sphere
let $u$ be harmonic in the domain $U \subset \mathbb{R}^n$ and $B_R(0) \subset U$ and $u(0)=0, u\neq 0$. Let $0<r<R$. Define $a(r):= \frac{1}{r^{n-1}} \int_{\partial B_r(0)} u^2dS, b(r):= \frac{...
0
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0answers
19 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
27 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 ...
0
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0answers
41 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
26 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
31 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
75 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
27 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'...
1
vote
1answer
67 views
Tricky integral relationship
I am trying to prove that
$$\begin{equation}\int_x^{x+1}\left(\int_0^{v} (u-0)f(u)\textrm{d}u+\int_v^{1} (u-1)f(u)\textrm{d}u\right)\textrm{d}v=\\\int_0^x\int_v^{v+1}f(u)\textrm{d}u\textrm{d}v\end{...
0
votes
0answers
79 views
Laplace Transform of Complementary Error Function
I need to apply one Laplace transform formula while I have no idea how to prove it:
$$\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
162 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
39 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 ...
0
votes
0answers
20 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
75 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
283 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:
$$I=\int_0^\...
0
votes
0answers
18 views
Integral Transformation from circle to unit sphere
I want to show that
$\displaystyle \frac{1}{2\pi r}\int_{\partial B(x,r)}u(y)\,\mathrm{d}s(y)=\frac{1}{|S^1|}\int_{S^1}u(x+r\theta)\,\mathrm{d}s(\theta)$
This is essentially a shift and dilation ...
1
vote
0answers
9 views
Decomposition of time series into weighted (weights known) stretched exponentials with unknown offset function
Looking for solution to decomposing function $f(t)$ into stretched exponential functions all with same meta-exponent and decay constants, but with different amplitudes $p$ (known) and offsets $\tau$ (...
2
votes
1answer
201 views
Proof of inverse Laplace transform
Why is
$$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 \ ?$$
I tried to find out myself, or searched ...
3
votes
0answers
37 views
“Bilateral Mellin convolution”
The Mellin convolution of two functions, when it exists, is of the form
$$
(f \ast_M g)(t)
= \int_0^\infty
f\left( \frac{t}{\tau} \right)
g(\tau)
\frac{\mathrm{d}\tau}{\tau}
$$
and ...
0
votes
0answers
26 views
What are the easiest inverse Mellin transforms?
I'm looking to familiarize myself with the inverse Mellin transform
\begin{align*}
\mathcal{M}^{-1}[\varphi](t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-s}\varphi(s)\, \mathrm{d}s
\end{...
1
vote
1answer
62 views
On the variety of Integral Transforms
Recently I have come across this book of Integral Transforms. At this point I realized for myself that there are so many transforms with different kernel out there. Now to provide a little bit of ...
2
votes
2answers
131 views
Laplace Transform of the incomplete Gamma Function
While looking through this ($178$,$(30)$) Table of Integral Transforms I have come across the Laplace Transform of the Incomplete Gamma Function which is given by
$$\mathcal{L}\{\Gamma(\nu,at\}(p)~...
0
votes
1answer
83 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
30 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
41 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
81 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
178 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:
$$\mathcal M [_2F_1(\alpha,\beta;\gamma;-x)] = \frac {B(s,\alpha-s)B(s,\...
1
vote
0answers
31 views
Can an inverse Mellin transform converge conditionally?
Let $F(x)$ satisfy
$$\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$$
Real constant $c$ is from the strip $\Re(s)\in(a,b)$.
$\textbf{...
0
votes
0answers
120 views
Division by t of an Inverse Laplace Transform
My problem is to find a function $g(s)$, solution of the equation:
$$ \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
50 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
117 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
186 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
72 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
$$\mathcal{L}[...
4
votes
1answer
143 views
Laplace convolution with the Bessel function
The Question:
(i) Find the Laplace Transform of the Bessel Function $J_0(x)$
(ii) Hence, show that if $f(x)$ satisfies the differential equation
$$f''(x)+f(x)=J_0(x) \qquad f(0)=f'(0)=0$$
then $f$ ...
2
votes
1answer
204 views
Solving Bessel Equation using Laplace Transform
The Question:
Given that $J_0(x)$ satisfies
$$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$$
Show that the Laplace Transform $\bar{J_0}(p)$ of $J_0$ is ...
2
votes
0answers
48 views
Can integration contour of an inverse Mellin transform be deformed at will in fundamental strip?
Let $m(x)$ be inverse Mellin transform of $M(s)$:
$m(x)=\frac{1}{2\pi i} \int\limits_{c-i\infty}^{c+i\infty}x^{-s}M(s)ds$
Mellin transform $M(s)$ is analytic on fundamental strip $a<\Re(s)<b$ ...
2
votes
1answer
50 views
Fourier Transform of $\dfrac{1}{x^2+2x+2}$
The Question:
(i) Determine the Fourier Transform of
$$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 $...
