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 ...

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Clarification on the absolute convergence of Mellin transform

I have a question I haven't been able to find a direct answer to that I presume is true but I am unable to show. We know these two following results on the mellin transform. If $$\int_0^\infty ...
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41 views

Mellin Transform of the floor function

I have from integral transform tables that: $$\operatorname{Mellin}(\lfloor x\rfloor) = -\dfrac{\zeta(-z)}{z},\quad \operatorname{Re}(z)<-1$$ How can this be proved?
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49 views

Inverting probability generating function via mellin transform substitution.

The pgf is defined as: $E(z^k)= \sum_{k=0}^{\infty} p(k)z^k$ which is a discretised version of the transform: $\widetilde{p}(z) =\int_{-\infty}^{\infty} z^k p(k) \, \mathrm{d}k$ The Mellin ...
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66 views

Bessel function integral

How to solve the integral for $J_1{(2x\sin{\frac{\theta}{2}})}$ at $[0,\pi]$? If solving by Matlab, please provide me the source. Thank you!
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33 views

“Reduction of Dirichlet series into power series”

In a paper of Riemann, he states to following formal identity. If $f(s)=\sum\limits_{k=1}^{\infty}\frac{a_k}{k^s}$ and $F(x)=\sum\limits_{k=1}^{\infty}a_kx^k$ then ...
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63 views

Delta function with imaginary argument

We have an integral representation for the Dirac delta function as $\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dk e^{ikx}$ . On the other hand, we have for delta function the property: ...
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1answer
13 views

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 ...
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1answer
66 views

A Parseval-like theorem for Mellin transforms

A particular case of Parseval's theorem for Fourier transforms says that if $f$ is square integrable on $\mathbb{R}$, then $$ \int_{-\infty}^{\infty} |f(t)|^{2} \ dt = \int_{-\infty}^{\infty} ...
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1answer
32 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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91 views

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
132 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
82 views

Inverse of an integral transform

Suppose that in a certain domain of analyticity we're given a function $A(s)$ in terms of the integral : $$A(s)=\int_{0}^{\infty}\frac{a(t)}{t(t^{2}+s^{2})}dt$$ How can we recover $a(t)$?
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1answer
84 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 ...
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31 views

Hankel transform and Laplacian in cylindrical coordinates

My book solved a PDE containing the Laplacian in cylindrical coordinates. It doesn't really explain why the Hankel transform is useful in this case (symmetries etc..); just brute force math. So yeah, ...
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17 views

Abel and Radon Transform

I am learning Radon and Abel transforms. As far as I understood, basically both the transforms are projection of a 3D object onto a 2D plane. Then what is the difference between both transforms? Under ...
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2answers
58 views

how is the inverse Hankel transform defined

The finite Hankel transform in my notes is defined as $$ f^*(k_i) = \int_{0}^{\infty} {r f(r)J_0(rk_i)dr}, $$ where $k_i$ is one of the positive roots of $J_0$. However, my notes don't say anything ...
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32 views

Which function satisfies the conditions?

I'm solving a problem and, in order to run test case, I need a function $ b(x,y) $ that satisfies: $$ \int_0^L \int_0^H b(x,y) \, dx \, dy = 0 $$ and $$ \int_0^L \int_0^H b(x,y) \cos \left(\frac{n ...
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1answer
23 views

Transformation formula for multidimensional integral

Let $A,B$ be positive definite symmetric $n\times n$ matrices. I stumbled upon the following identity and don't see why it should hold: $$\int_{\mathbb{R}^n}\frac{1}{\sqrt{\det A \det ...
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36 views

Fourier transform of a function involving $\sec(\omega)$

The summary of my question is: What should I make of $\mathcal{F}^{-1}\left[\frac{\csc(\omega)}{\omega^2-\beta^2}\right]$ where $\mathcal{F}^{-1}$ is the inverse fourier transform (taking $F(\omega)$ ...
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1answer
41 views

Solving a Fourier sine transform equation

Suppose we have the following Fourier sine transform equation $$\int_0^\infty f(x;p)\sin(\lambda x)dx \equiv 0,$$ where $f(x;p)$ has some parameters $p\in\mathbb{R}$ we can choose freely. Does this ...
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23 views

Conditions on $F(y)$ for the existence of $f(x)$ where $F(y) = \int_{0}^{\infty} y \exp{[(-\frac{1}{2}(y^2 + x^2)]} I_{0}(xy) f(x) \; \mathrm{d}x$

I've been working with the following integral transform: $$F(y) = \int_{0}^{\infty} y \exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_{0}(xy) f(x) \; \mathrm{d}x$$ where: $x,y$ are positive-definite ...
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31 views

Legendre transformation of the square of a function

I take the Legendre transform of a function $f(x)$ by $$ f_l = \frac{1}{2(-i)^l}\int_{-1}^1 P_l(x)f(x) dx$$ I am interested in possible relationships between $f_l$ and $$\frac{1}{2(-i)^l}\int_{-1}^1 ...
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51 views

Fourier Transform of a Gaussian Signal?

As far as I know this is the formula for FT : On this question on part b) I fint on the answer the part with e^-jwt is changed with cos(wt) I have no idea how cos(wt) came in ... would you please ...
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1answer
37 views

Fourier transform of angular part of function in spherical coordinates.

This is the maths part of a physics problem - that of the solution of Schrodinger equation for a 3D harmonic oscillator in spherical coordinates. The solution is a product of associated Laguerre ...
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46 views

Integrating the Fourier Transform

I am self-studying the Fourier Transform using this book. One of the exercises asks the following: Show that $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{- \infty}^w \hat{f}(w') \, d ...
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16 views

Laplace Transform of $L^{2}$ Function.

I know the Fourier transform is an isometry of $L^2$ functions. I've read that the Laplace Transform of an $L^2$ function is $L^2$ but cannot prove it nor can I find a proof. Does anyone know of a ...
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51 views

What is a transform?

I've been working in vain to find a way to find the integral of an intractable function. It's great practice anyway. I thought about using intergration by parts with three functions to solve it and ...
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30 views

Mellin-Barnes transform of $\frac{1}{\Gamma}$

Let $\varphi \in L_1(c+i\mathbb R)$, where $c > 0$. Then we can define the Mellin-Barnes transform (or the inverse Mellin transform) of the function $\varphi$ by the formula $$ \mathcal ...
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1answer
30 views

Misunderstanding with Fourier sine transform…

In my copy of Table of Integrals, Series, and Products (Gradshteyn & Ryzhik) on p.1121, it says that the Fourier sine transform is defined $$F_s(\xi) = \sqrt{\frac{2}{\pi}}\int_0^\infty ...
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29 views

The Laplace transform - does it have an associated differential operator, if the kernel is to be viewed as a Green's function?

I've begun learning about Green's functions, and if I understand correctly, the Green's function for a linear differential operator $L$ with appropriate boundary conditions is the kernel for the ...
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3answers
183 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
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1answer
136 views

How fast does the function $\displaystyle f(x)=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $ grow?

Let $x$ be a positive real number and $f(x):=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $. How fast does this function grow ? In other words can we find a good asymptote for ...
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65 views

Where to submit a small work of applied math written by a physicist?

Working in physics, I recently discovered a mathematical identity useful to solve a particular partial differential equation. Using the same idea, I found several other identities but I do not know ...
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1answer
38 views

Dirichlet Problem on the Disk - Question about this proof of solution using Poisson transform

I'm looking at a proof for the Dirichlet problem on the disk. The problem is as follows. Let $D=U(w,\rho)$ and$\phi : \partial B(w,\rho) \to \mathbb{R}$ continuous. Then $$g = \begin{cases} P_D ...
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1answer
42 views

Need help with integral related to Mellin transform

I need help solving the following integral: $$I = \frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} \mathrm{d}p \hspace{2pt} m^{d-2p} \Gamma(-p)\Gamma(p-\frac{5}{2})A(p)$$ where $A(p)$ is an analytic ...
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26 views

Integral transforms with interesting pointwise multiplications?

The convolution theorem states that the Fourier transform of the convolution of functions equals the pointwise multiplication of Fourier-transformed functions, i.e.: $$\mathcal{F}\{f*g\} = ...
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1answer
78 views

Fourier Integral evaluation

We're doing fourier integrals in class, but unfortunately I have no idea how to even begin to tackle this one. The examples we have done in class were way simpler than this one: $$ \int_0^\infty ...
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1answer
69 views

Change of variables in double integral - what's wrong?

I have a homework problem, as follows: Evaluate the double integral by making an appropriate change of variables. $\iint_R 9\sin(49x^2+16y^2)\,dA$, where $R$ is the region in the first ...
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76 views

Convolution theorem for other transforms

The Fourier transform is an integral transform with turns any function into a superposition of sinusoidal waves. The convolution theorem states the astonishing property that if you convolve two ...
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95 views

Fourier transform convention: $\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{\pm ikx}dx $?

I've come across the Fourier transform being defined as: $$\tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{ikx}dx$$ But this convention is not present in the Wikipedia article. The ...
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1answer
42 views

Take Laplace Transform of the integral J_0

I was just wondering how to use tables from Spiegal to solve $\int_0^\infty J_0(2\sqrt{ut}) J_0(u) du$ At the moment, I see similar transforms on page 244, but I don't actually know how to combine the ...
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22 views

Steepest descent?

Here I would like to see the behavior of a function as an integral when its argument (which is a parameter in the integral) goes to zero. If I try to evaluate an integral ...
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35 views

Problem with the Cauchy-Green transform

Let $u \in C_c(\mathbb C)$. Then it's Cauchy-Green transform $$ \tilde u(z) = -\frac{1}{\pi} \int\limits_{\mathbb C} \frac{u(\xi)}{\xi-z} d\xi_R d\xi_I, \quad \xi = \xi_R + i \xi_I, $$ is ...
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80 views

Calculating integral with standard normal distribution.

I have a problem to solving this, Because I think that for solving this problem, I need to calculate cdf of standard normal distribution and plug Y value and calculate. However, at the bottom I ...
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16 views

Can this transform be rewritten as a more standard integral transformation?

Here is the transformation pair I've been working with. $\hat{f}(n)=\displaystyle\lim_{a\to1}\sum_{j=0}^{\lfloor\log_a n\rfloor}(-1)^j\binom{k}{j}a^j f( a^{-j} n)$ ...
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1answer
73 views

Continuous version of the Möbius inversion theorem

Is there a continuous version of Möbius Inversion. Essentially, using integrals instead of sums.
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1answer
30 views

Does the transformation $x=Pabc$, $y=Qab(1-c)$, $z=Ra(1-b)$ map a unit cube to a tetrahedron?

Does the transformation $x=Pabc$, $y=Qab(1-c)$, $z=Ra(1-b)$ map a unit cube in $abc$ coordinates to the tetrahedron with vertices $(P,0,0)$, $(0,Q,0)$, $(0,0,R)$ and $(0,0,0)$ in xyz coordinates? ...
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52 views

Integral solve only using

How can I solve $$\int \sqrt{1+\cos(6x)} \,dx$$ only using algebraic, trigonometric methods, immediate integrals and integral properties?
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96 views

Involutive fourier transform

The writer here states I am introducing a viewpoint (the involutive convention) which makes the Fourier transform its own inverse (i.e., the Fourier transform so defined is an involution). ...
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38 views

Property of Laplace transforms

I was looking at this answer to the question asked and I am curious about the $$\int_0^\infty F(u)g(u) du = \int_0^\infty f(u)G(u) du $$ relationship being used. I referred to the link provided in ...