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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19 views

Geometric Interpretation of Fourier Transforms

I'm interested in tsunami wave science and I've already got an engineering degree and a basic knowledge of signal processing. The courses I took were intensively computational and taught some skills ...
1
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0answers
48 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 $h[n]$ ...
4
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1answer
203 views

About Mellin convolution technique

Recently I was studying the Mellin convolution technique to solve definite integrals. I am just wondering is the technique valid only for any definite integrals (with any range $\int^b_a$)? Or only ...
3
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0answers
230 views

An inverse Mellin transform

Is it possible to compute the inverse transform of $$ \frac{1}{a^{-s}\cos( \frac{\pi s}{2})\Gamma (s)} $$ or similarly is it possible to compute the Inverse Mellin transform ?? $$ \frac{ \zeta ...
3
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1answer
183 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 ...
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0answers
2 views

Consistency/range conditions for (integral) transform mapping into higher-dimensional space

I am interested in learning more about what the (formal) implications are when transforming a $n$-dimensional function space (e.g., the space of all $\mathbb{R}^n\to\mathbb{R}$) to a higherdimensional ...
2
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0answers
19 views

Using Mellin transform for a certain function

In short, I want to use the Mellin transform to obtain the asymptotic behavior of the sequence $D_n = \frac{ [z^n] D(z)} {C_n}$ where $$ D(z) = \frac 1{2z}\sum_{p \ge 1}C_p \left( ...
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0answers
18 views

Singularities in the Gauss Hypergeometric Function

I am evaluating the following term in a series: $$I_k = \int\!x^{-3(2k+1)}(1+\lambda x^4)^{-1/2}\,\mathrm dx$$ When I plug this into WolframAlpha, I get the following result: $$I_k = ...
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0answers
10 views

Identify functional form of expectation value

I have the equation: $g(z) = {\int_{\mathbb{R}^n}}\;f(x)\exp(-c(x)^Tz)dx = \mathbb{E}_X[\exp(-c(X)^Tz)]$ where $c:\mathbb{R}^n \to \mathbb{R}_+^m$, $z \in \mathbb{R}_+^m$ and $f(x)$ is a ...
2
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2answers
187 views

What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.

I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
2
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0answers
38 views

A Mellin Transform of a generating function

I am trying to find the Mellin transform of the function $$ G(z) = \sum_{k \ge 1} C_k\left( 1- \exp \left( \frac{-z}{4^k} \right )\right), $$ where $C_k$ denotes the $k$-th Catalan number ($C_k = ...
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1answer
56 views

Is the Gamma Function multivalued??

Consider the definition of the Gamma function $$ \Gamma(s) = \int_{0}^{\infty}\left[x^{s-1}e^{-x} \right] dx $$ Clearly: $x^{s-1}$ may have multiple defined values for $s$ if $s-1$ is rational or ...
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0answers
29 views

Solving initial value problem using Laplace transforms, one other method, and comparing results

So for my solution using characteristic equations I get (fixed a typo for first coefficient) $$\frac{11}{30} e^{-3t} - \frac{21}{20} e^{-2t} + \frac{21}{20} e^{2t} - \frac{11}{30} e^{3t}$$ For the ...
1
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1answer
316 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: ...
0
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1answer
15 views

Graphing Fourier transforms on a frequency versus intensity plot (how to deal with complex numbers)

I am trying to understand how Fourier transforms are used to make HNMR plots. HNMR basically consists of hitting some molecules with some radiation and listening to the radio signal that results. ...
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0answers
15 views

Transforming a function - general methods

I have some rather general questions about transforming a function from one form to another. I have tried reading about integral transforms, but I wish to know more about the theory in general, as ...
8
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1answer
86 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
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1answer
20 views

Quick Question on Zeroes of Transfer Function

Sorry for not providing context here. Suppose I have an output $Y(z)=\frac{z-1}{z}$ and input $X(z)=\frac{z^2+3z+2}{z^2}$ to yield a transfer function ...
16
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1answer
486 views

Is Fourier transform characterized by its diagonalization properties?

Let us fix the following convention for the Fourier transform in $L^1(\mathbb{R})$ space: $$\hat{f}(\xi)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i x\xi}\, dx.$$ We then have the following properties: ...
42
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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 ...
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1answer
19 views

Find a function in terms of another

I need to express an in terms of f(x). I did it but I'm not sure if it is right. Consider \begin{cases} -u_{xx}+u=f(x), &0<x<\ell\\[0.5em] u_x(0)=0,\;u_x(l)=0& \end{cases} ...
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1answer
31 views

Why is this laplace identity true $\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$?

I was wondering why this laplace identity is true? Does it follow from definition? $$\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$ I'm trying to understand the first ...
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1answer
36 views

Transform an Integral bounds from -inf, inf to 0 to 1

Good day, If i have an integral from -infinity to infinity, how do I change the bounds/limits to 0 to 1? I don't want to give exact question since this is part of an assignment. I know how to figure ...
2
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0answers
55 views

About Mellin transform and harmonic series

Let $\mathfrak{M}\left(*\right)$ the Mellin transform. We know that holds this identity$$\mathfrak{M}\left(\underset{k\geq1}{\sum}\lambda_{k}g\left(\mu_{k}x\right),\, ...
4
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3answers
80 views

How to learn Integral Transform?

I major in Electronic Engineering when back in college. I learned the Fourier Transform, Laplace Transform, Z Transform and wavelet Transform. But I always feel a lack of thorough understanding of ...
1
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1answer
35 views

Fourier transform of a random variable

During my research i'm dealing with a stochastic partial differential equation. The random term appearing in my equation is a tensorial random variable: $\boldsymbol{\sigma}(\boldsymbol{x},t)$ Which ...
0
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1answer
95 views

Can asymptotic of a Mellin (or laplace inverse ) be evaluated?

I mean, given the Mellin inverse integral $ \int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $, can we evaluate this integral, at least as $ x \rightarrow \infty $? Can the same be made for $ ...
56
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6answers
9k views

Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
2
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1answer
67 views

Inverse transform of a modified Abel transform

I have been struggling for 6 months on finding the analytical inverse transform of a transformation below: $$F(y,k) = 2 \int_y^{\infty}\cos\left(ka\sqrt{r^2-y^2}\right) f(r,k) ...
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3answers
33 views

2-Dimensional FOURIER TRANSFORM

How can I do to calculate the Inverse Fourier Transform of: $$G(w,y)=e^{-|w|y}$$ where w is real (w is the transform of x). I want to have $g(x,y)$, where $G$ is the Fourier Transform of $g$ Thanks
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1answer
22 views

Positivity of a Sine transform of a positive function

Consider a function $f(t)$ with $f(t>0)>0$ and $f(-t)=-f(t)$. Can I make any statement about the positivity of the Sine transform $$\hat{f}(\omega) = \int_{0}^{\infty} \sin(\omega t) f(t) ...
2
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1answer
133 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 ...
0
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2answers
308 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 ...
0
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1answer
31 views

Fourier Transform - Laplace Transform - Which variable transform?

I need to know when do I have to transform $x$ and when $y$ in a PDE in Fourier Transform and Laplace Transform. In an exercise of Fourier Transform, I have to solve a Laplace Equation, where ...
3
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0answers
28 views

Mellin transform defined for function on group $(\mathbb{R}^{+}, \times)$

In the article on the $\Gamma$ function in the Princeton Companion to Mathematics, the author states The Mellin transform is a type of Fourier transform, but it is defined for functions on the ...
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0answers
40 views

Check my Solution: Possible kernels for an integral transformation

I solved a problem and would like to know if my solution is correct. Feel free to comment. Problem: Given the integral equation $$\frac{1}{2\pi}\int dk e^{-ikx'}\int dx e^{ikx} u(k,x) f(x)=f(x')$$ ...
8
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2answers
2k views

What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
0
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1answer
35 views

Fourier cosine and sine transform of $\exp{(-ax)}(1+bx)^{-1}$ and $\exp{(-ax)}(1+bx)^{-2}$

As stated in the title I should calculate the cosine and sine Fourier transform of: $$f_1(x)=\exp{(-ax)}(1+bx)^{-1}$$ and $$f_2(x)=\exp{(-ax)}(1+bx)^{-2}$$ That obviously means calculating: ...
3
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1answer
128 views

Relation between Hankel transform and Fourier transform

As a physics student, I ran into the following problem. I left out a lot of context, if anything is unclear please ask me. I quote: The statistic that is observable is the angular correlation ...
2
votes
2answers
51 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
3
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1answer
37 views

Prove that the Laplace transform of $I_0\left(2\sqrt t\right)$ is $\exp\left(1/s\right)/s$

Wolfram Alpha gave me the answer to this, but unfortunately Wolfram Alpha doesn't show its work, I can't find a proof anywhere else, and my feeble attempts to show it myself went nowhere. How can it ...
7
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2answers
292 views

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi - 5\log(2))$$ The identity follows ...
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0answers
24 views

Can wavelets be used for texture discrimination?

I've recently been studying wavelet analysis with a view to differentiating certain areas of texture images where the texture differs from the background pattern (which is quite random); for example a ...
1
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1answer
52 views

Connection between $\nu=\frac{N}{2}-1^\text{th}$ order Hankel transforms and hyperspherically symmetric functions?

In The Transforms And Applications Handbook 2nd edition chapter 9 (Hankel Transforms), Piessens briefly mentions that the Fourier transform of an $N$-dimensional hyperspherically symmetric function ...
0
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1answer
19 views

Why is the Z-transform of $e^{at}$, t = kT, different from Laplace transform of $e^{at}$

The Laplace transform of $e^{at}$ takes a well known form of $\frac{1}{s-a}$ The Z transform of $e^{at} = e^{akT} $ T is the sampling period takes the form of $\frac{z}{z-e^{aT}}$ Does anyone know ...
0
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1answer
37 views

Laplace-Fourier transform issue

Given a function $f:\mathbb{R}\rightarrow\mathbb{R}$ we take the generalised Fourier transform $\hat{f}(w)=\int_{-\infty}^{+\infty}e^{iwx}f(x)dx$ where $w\in \mathbb{C}$. Now assume, this transform ...
5
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1answer
44 views

Can $f(x+1) = f(x)^{\ln(x)}$ be expressed as integral transform $\int g(x,t) dt $?

Let $x$ be a real number. Can some real-analytic function $f$ that satisfies for $x>3$ :$f(x+1) = f(x)^{\ln(x)}$ be expressed by standard functions as an integral transform : $$f(x) = ...
0
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1answer
48 views

Kernels of integral transform and linear transformation

Is there any relation between the $kernel$ of an $integral \ transform$ and the $kernel$ of a $linear \ transformation$?
0
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1answer
68 views

Good recommendations for solving PDE's by integral transforms

I look for good books on solving partial diffrential equations (PDE's) using integral transforms specially Fourier and laplace transforms. Do you have any recommendations for such books? I don't ...
4
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2answers
2k views

Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...