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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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8 views

Time Settings of the $z$ and Laplace Transforms.

I'm aware that the $z$-transform and the Laplace Transform have an analogous relationship but I want to be doubly-sure that the $z$-transform only works in discrete-time and that the Laplace transform ...
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1answer
46 views

Mellin transform of $e^{iat}$

When doing the change of variables $v=-iat$, shouldn't the limits be reversed? Or is it because its the same as $v=\frac{at}{i}$ I cant see why this is not the case
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29 views

Integral transform with reciprocal complex exponential functions?

I tried answering a question that ended up with an expression $$\mathcal F\left\{e^{\left(\frac{2\pi j} {t}\right)}\right\}$$ Now this function we know from famous identity is $$e^{ai} = \cos(a)+i\...
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0answers
21 views

Laplace transform $\mathcal{L}[t^\alpha f(t)]$

I am interested in the Laplace transform $\mathcal{L}[t^\alpha f(t)]$ where $f(t)$ is an arbitrary function and $\alpha$ is non integer. I know that for $\alpha=n$ integer, this is the n-th ...
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1answer
23 views

Inverse integral transform of $\cos(t-u)$

I have the following integral transform $$ f(u) = \int_0^{2\pi} g(t)\cos(u-t)\,dt $$ where I know what $f(u)$ is (I have raw data rather than an analytical form) and I need to reconstruct $g(t)$. ...
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33 views

Proof that $\frac{e^{st}}{2\pi i}$ is an orthogonal basis.

I was studying the Linear Algebra perspective about the Laplace Transform. We know that the Laplace Transform is given by: $$ F(s) = \int_{0}^{\infty}f(t)e^{-st}dt $$ Where $e^{-st}$ is the integral ...
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1answer
70 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 ...
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1answer
40 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 {\...
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43 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 ...
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47 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 $f(x)$ ...
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1answer
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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}\...
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1answer
26 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 ...
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2answers
67 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. ...
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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 ...
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2answers
42 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 ...
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31 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\...
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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\...
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0answers
13 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{...
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20 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^...
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1answer
35 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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44 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 ...
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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 ...
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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 ...
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0answers
80 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 ...
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28 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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1answer
72 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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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 = ...
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2answers
170 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 ...
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0answers
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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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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) = ...
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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 ...
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2answers
308 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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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 ...
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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 $\tau$ (...
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1answer
355 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 ...
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40 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 ...
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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{...
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1answer
63 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 ...
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2answers
156 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)~...
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1answer
89 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 ...
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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 ...
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1answer
42 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 ...
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1answer
83 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 ...
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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 ...
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2answers
193 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,\...
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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 ...
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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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0answers
137 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}^{...
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1answer
52 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 ...
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1answer
125 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 $$f(t) = \frac{A^2}{2\sqrt{\pi}t^\frac{3}{2}}e^{-\frac{A^2}{4t}}$$ ...