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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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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
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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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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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1answer
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Compatibility Condition of the Poisson Equation with Neumann Boundary Conditions

I am trying to solve the following general Poisson equation with homogeneous Neumann boundary conditions in a rectangular domain ($0 \le x \le L$ and $0 \le y \le H$). $$ \frac{\partial^2 p(x,y)}{\...
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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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Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
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1answer
5k views

Fourier Transform for triangular wave

Could someone tell me if I've worked this out right? I'm unsure of the process, especially the final parts where I convert it to a sinc function. Please let me know if I've made mistakes anywhere ...
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Fourier Transform Help Needed

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one. The only way I could think of to solve this was by considering it ...
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1answer
212 views

Need help with a Fourier Transform Question

I need an way to solve this Fourier transform problem. $$ f(t)= \begin{cases} \cosh(t) & \text{ For } |t|<1\\ 0 & \text{ For }|t|>1 \end{cases} $$ The given answer for the ...
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1answer
356 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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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
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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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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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What exactly is an integral kernel?

I am not sure if I have seen integral transforms in the right way, but given a transform like Fourier transform - it's actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
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1answer
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Dirichlet Series and Asymptotic Expansions: $\tilde{f}(s)= \sum_{n=1}^{\infty} f(n) n^{-s}$

Consider the Dirichlet series $\tilde{f}(s)= \sum_{n=1}^{\infty} f(n) n^{-s}$. In the page Zeta Function Regularization I found a relation among an asymptotic expansion of $\tilde{f}(s)$ and an ...
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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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1answer
961 views

A problem about Laplace transform and Parseval–Plancherel theorem

I am reading a paper about fractional differential equation. One of the piece said as follow: By applying the Parseval–Plancherel theorem we may show: \begin{equation} \int_0^\infty v(t)B_\alpha ...
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1answer
66 views

A specific partial differential equation using Fourier Transform

I have the following PDE problem which I think sounds like a job for the Fourier transform: $ u_t + 2u_x = u_{xx} \space \space \space -\infty < x < \infty \space \space \space t>0 $ $...
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1answer
769 views

Fourier COSINE Transform (solving PDE - Laplace Equation)

I'm trying to solve Laplace equation using Fourier Cosine Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't ...
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2answers
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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
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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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1answer
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Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?

the Fourier transformation of a scalar function with respect to one variable might be defined as $\mathcal{F}\left[w\right](\omega )\equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}w(t)e^{-\mathrm{...
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How 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=...
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1answer
278 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. $$\int_0^{+\infty}\,\left[2\cosh(ab)-...
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3answers
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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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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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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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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
351 views

How to use steepest descent method to approximate $\int_0^{1}s^{1/4+i b x}e^{sx}ds$ as $x\to+\infty$?

Let $0< b\leqslant 1$. I am interested in using the steepest descent method to calculate the asymptotic approximation (as $x\to+\infty$) to the following integral that is related to function ${_1}...
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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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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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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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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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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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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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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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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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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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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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2answers
7k views

Why do engineers use the Z-transform and mathematicians use generating functions?

For a (complex valued) sequence $(a_n)_{n\in\mathbb{N}}$ there is the associated generating function $$ f(z) = \sum_{n=0}^\infty a_nz^n$$ and the $z$-Transform $$ Z(a)(z) = \sum_{n=0}^\infty a_nz^{-n}$...
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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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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) = ...
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3answers
13k views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
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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 ...