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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3
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1answer
91 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 $a(t)$?
1
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1answer
281 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 ...
1
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2answers
333 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 ...
1
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2answers
34 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 ...
0
votes
1answer
74 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 ...
0
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1answer
69 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 ...
1
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0answers
26 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 ...
1
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0answers
88 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 ...
0
votes
1answer
108 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 ...
2
votes
1answer
75 views

Integrating the Fourier Transform

I am trying to show that $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{- \infty}^w \hat{f}(w') \, d w'.$$ Shouldn't it be $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{w}^{+ ...
2
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0answers
26 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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0answers
52 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 ...
2
votes
1answer
63 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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0answers
42 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 ...
4
votes
3answers
224 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 ...
6
votes
1answer
165 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 ...
7
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1answer
77 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 ...
0
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1answer
71 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 ...
1
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1answer
114 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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0answers
39 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\} = ...
3
votes
1answer
95 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 ...
1
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1answer
120 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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2answers
114 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 ...
0
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2answers
312 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 ...
1
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1answer
69 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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0answers
47 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 ...
1
vote
2answers
200 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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0answers
18 views

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

Here is the transformation pair I've been working with. $\hat{f}(n)=\displaystyle\lim_{a\to1}\sum_{j=0}^{\lfloor\log_a n\rfloor}(-1)^j\binom{k}{j}a^j f( a^{-j} n)$ ...
4
votes
1answer
97 views

Continuous version of the Möbius inversion theorem

Is there a continuous version of Möbius Inversion. Essentially, using integrals instead of sums.
1
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1answer
48 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? ...
0
votes
1answer
57 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?
4
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0answers
106 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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0answers
40 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 ...
1
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1answer
36 views

Transforms in cyclic group $\mathbb{Z}/n\mathbb{Z}$

Are there any elementary examples of transforms whose "time-domain" is the cyclic group $\mathbb{Z}/n\mathbb{Z}$?
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0answers
49 views

The Fourier Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
0
votes
1answer
247 views

Find a transformation from tetrahedron to cube in $R^3$ to calculate a triple integral?

I would like to calculate the triple integral of a function $f(x,y,z)$ over a region given by a tetrahedron with vertices $(0,0,0)$, $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$. I am trying to do this by ...
1
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0answers
67 views

inverse of integral transform

given a general integral transform $$ g(x)= \int_{0}^{\infty}dyf(y)K(xy) $$ for a general formula of the kernel $ K(xy) $ is there an inverse of the Integral transform to obtain $ f(x) $ from above ...
2
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0answers
1k views

Probability Integral Transformation

I just attended an introductory course on Statistics and we came across the following: I know what random variables, the uniform distribution, etc. are but the notation from the proposition ...
3
votes
1answer
184 views

How to find the inverse Mellin transform?

On the wikipedia page http://en.wikipedia.org/wiki/Mellin_transform The second formula is an integral transformation for the inverse Mellin transform. Being new to integral transforms, I wonder how ...
3
votes
0answers
55 views

Analytic Continuation of a nowhere existing Mellin Transform

I'm trying to give sense to the (self-made) statement: The analytic continuation of $\int_0^\infty e^t\;t^{s-1}\;dt$ is holomorphic in $s=0$. At first sight this could seem completely ...
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0answers
49 views

Can someone explain this z transformation to me?

I have a signal $h[n]=\frac{1}{z+3}$ and the solution is $H(z) = (-3)^{n-1}\delta[n-1]$. Looking the solution up in a transformation table, I come to the conclusion that I need to transform $h[n]$ ...
1
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0answers
145 views

Inverse Fourier-cosine transfrom

Suppose we have a function $F(x)$ given by the integral: $$F(x)=\int_{0}^{\infty}f(t)\frac{\cos(t\log x)}{t}dt\;\;\;\;\;(x>1)$$ This looks tantalizingly like a Fourier-cosine transform of ...
3
votes
2answers
62 views

Evaluating Laplace Transform

I have a Laplace transform function of the following form and I'm trying to evaluate it. From my research I think I need to take the Inverse Laplace Transform and then integrate, but I'm having ...
1
vote
1answer
140 views

Evaluating part of a triple integral by changing from rectangular coordinates to cylindrical coordinates

Seeing that this is my first time posting, I hope I'm following the rules correctly. Anyways, the question I'm stuck on is: Evaluate: $$ \int_{-3}^3\int_{-2}^2\int_{-\sqrt{9-y^2}}^\sqrt{9+y^2} ...
1
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2answers
195 views

Laplace Transformation Applications

In one of our Mathematics lecture our Prof told us that similar to Logarithmic Transformations we can use Laplace Transformations to solve difficult equations. What kind of equations do Laplace ...
4
votes
1answer
132 views

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be ...
43
votes
1answer
1k views

To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm ...
4
votes
2answers
170 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
2
votes
1answer
54 views

Integral transform Laguerre function

Given the following integral transform $$ g(m)= \int_{0}^{\infty}dxe^{-x}f(x)L_{m}(x), $$ then how could we obtain $ f(x) $ from $ g(m) $ ?? I have thought that for a continuum '$m$' like in our ...
1
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0answers
122 views

Laguerre transform function

For continuous indices 'n' and 'm' is it possible to have $$ \int_{0}^{\infty}dx e^{-x}L_{n}(x)L_{m}(x) = \delta (n-m)? $$ Another question: let $g(m)$ be the function defined via the transform $$ ...