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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118 views

An integral transform.

Let's consider a complex function that can be represented in the following form: $$ K(z)=\int_{-\infty}^{\infty}A(\alpha)z^\alpha d\alpha $$ Writing $z=re^{i\theta}$, we get: $$ ...
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1answer
68 views

How to establish the smoothness class

Consider a smooth function $f(x)$ on $\mathbb{R}^{n}$ from some smoothness class $S_1$ and define $$ F(y) = \int\limits_{g(x,y)\leqslant 0}f(x)dx $$ where $y \in \Omega$, $\Omega$ is some domain in ...
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1answer
1k views

Fourier transform of $\log x$ $ |x|^{s} $ and $\log|x| $

Can anyone provide or give an expression in the sense of distribution theory for the functions $|x|^{s} , \log|x| $? I mean I would like to evaluate the Fourier transform $ ...
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3k 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 ...
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1answer
90 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 $ ...
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2answers
240 views

Stueckelberg Feynman propagator computation

On page 35 of Itzykson-Zuber's textbook on quantum field theory, I am having trouble deriving equation (1-180): $\displaystyle G_F(0,r) = \frac{i}{(2\pi)^2 r} \int_m^\infty dp ...
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190 views

Mellin inverse transform

would be possible to evaluate the Mellin inverse transform $ \int_{c-i\infty}^{c+i\infty}ds \frac{\zeta(s)}{2i \pi s}$ in terms of the zeros of the RIemann Zeta ??? i know how to compute the invers ...
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107 views

Different proofs of support theorem for Radon Transform

Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...
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1answer
967 views

Bessel function integral and Mellin transform

Gradshteyn&Ryzhik 6.635.3 provides the following integral, with the usual constraints on $\nu,\alpha,\beta$, $$\int\limits_0^\infty \exp\left(-\frac{\alpha}{x}-\beta x\right)J_\nu(\gamma ...
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393 views

Fourier Transform of Bessel function with square root argument

Fourier Transform of the following function: ...
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169 views

Inverting an integral transform

This question is either very obvious or by nature unsolvable in a general case. I Google'd around for a solution to no avail. Given an integral transform of kernel K across some interval I as a ...
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179 views

Interchanging the order of integration.

Can someone explain why the following works please? $\int^t_0 f(s) \int^s_0 f(u) \,du\,ds = \int^t_0 f(s) \int^t_s f(u) \,du\,ds $ EDIT: All I know is that $f(x)$ is an integrable function. This is ...
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257 views

Integral Transform

I am having some trouble with knowing the best route to go about evaluating this integral for the I.F.T. It states the following. $w(t)$ has the Fourier Transform $W(f)= \dfrac{(j\pi f)}{(1+j2\pi ...
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1answer
89 views

Domain rotations in the Mellin integral transform

Let's consider the following form of the Mellin integral transform: $$m_{pq} =\iint\limits_{D_R} \! x^p y^q f(x,y) \, dx\; dy, \, D_R={\{(x,y)\,|\,x^2 + y^2 \le R^2\}}$$ If we scale the domain of the ...
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1answer
531 views

Commutativity of Convolution in higher dimensions

I have a basic question about how to show that convolution in dimension $n$ is commutative - or maybe it is rather a question about change of variables .. So on $\mathbb{R}$ I know how to show ...
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173 views

What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
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1answer
711 views

Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this ...
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229 views

How to apply the solution of $y(n) = (0.85)y(n-1) + x(n)$ to data

I learned how to solve difference equation $y(n) = (0.85)y(n-1) + x(n)$ using z Transform, and inverse z Transform, I get $h(n) = 0.85^n u(n)$ where $u(n)$ is unit step sequence. Now my ...
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4answers
363 views

Image of closed ball under degenerate integral operator is a closed set

I will be putting a bounty on this problem as soon as it lets me. For those who want to understand where the problem came from I encourage reading the edits, as I cut out several failed attempts and ...
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1answer
194 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 ...
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360 views

What is the difference between resolvent kernel and iterative kernel of an integral equation?

As we have different methods to find resolvent kernel, which is more suitable among all those methods? And what is the difference between resolvent and iterative kernels?
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1answer
119 views

Series around $s=1$ for an integral

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
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2answers
158 views

Fourier transform in $\mathbb R^3$

I try to show that $$ \int\limits_{R^3} \frac{e^{i\xi x} d\xi}{\xi^2 - k^2 - i0} = e^{ikx} \int\limits_{R^3} \frac{e^{i\xi x}d\xi}{\xi^2 + 2(k + i0\frac{k}{|k|})\xi}, \;\;\; k,x \in \mathbb R^3 $$ ...
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1answer
5k views

Fourier transform of Bessel functions

I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being ...
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76 views

Evaluating the limit $y \to 0^+$

Given $t \in \mathbb{R}$ and $z = x + iy$ and $y>0$. $\lim_{y\to0^+} \frac{1}{t - z} = \frac{1}{t-x} + \pi i \delta(t-x)$ This limit is given in the book Integral Transforms and Their ...
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1answer
237 views

Stieltjes Transform

Stieltjes transform for a distribution $F(x)$ is defined as $$m(z)=\int \frac{dF(x)}{x−z}$$ where z is complex with positive imaginary parts and $F(x)$ is a distribution function. Basically, I am ...
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1answer
178 views

Mellin transforms with zeros on the critical line

Are there examples of Mellin transforms of functions $ \int_{0}^{\infty}f(x)x^{s-1}\mathrm dx$ that have only real zeroes or have only zeroes on the critical line? For example, the Mellin transform ...
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1answer
121 views

Fourier transform integral

I'm trying to calculate the 3D fourier transform of this function: $$\frac{1}{(x^2+y^2+z^2)^{1/2}}$$ Any help would be appreciated, thanks.
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1answer
279 views

Evaluate $\int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle}$

The task is to evalute $$ \int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle}, \;\;\; \theta \in \mathbb{C}^2 \setminus ( \mathbb{S}^1 \cup \left\{ 0 ...
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507 views

Are Laplace Transforms a Special Case of Fourier Transforms?

A Laplace Transform is based on the integral: $F(\xi) = \int_0^{\infty} f(x) e^ {-\xi x}\,dx.$ In a roundabout way, a Fourier transform can get to $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- ...
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1answer
231 views

Does this Laplace transform have a closed form?

I want to know if the Laplace transform of $$x^\alpha (1+ax)^\beta$$ has any closed form? I really appreciate your help.
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1k views

2 dimensional Fourier transform integral

I'm trying to calculate the 2D fourier transform of this function: $$\frac{1}{(x^2+y^2+z^2)^{3/2}}$$ I only want to do the fourier transform for x and y (and leave z as it is). So far, I've tried ...
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2answers
178 views

Z transform of a complex convolution

I found this paper on Hilbert Transform, which is a very nice read. I've studied signal processing, but from a more practical than mathematical perspective. Can someone explain to me how we arrive at ...
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2answers
187 views

Why does it seem I can't apply the Radon transform to the Helmholtz equation?

Say we a function $u$ and a bounded region $\Omega \subset \mathbb{R}^2$, such that $(\Delta+\lambda)u = 0$ everywhere, and $u=0$ on the boundary. We extend it to the entire plane by defining $u=0$ ...
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1answer
356 views

Qualitative interpretation of Hilbert transform

the well-known Kramers-Kronig relations state that for a function satisfying certain conditions, its imaginary part is the Hilbert transform of its real part. This often comes up in physics, where ...
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168 views

Limits in Double Integration Question

I’m really having problems getting suitable limits when using double integration. For example: Let E be the region defined by {(x,y): y-x <= 2, x + y >= 4, 2x + y <= 8} Sketch the region E. ...
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1answer
168 views

Having such integral, how to optimize it in maple?

So we have : (1/3)*sig0*h^3*(int(int(sin((1/3)*arctan(y, x)), x = 0 .. r), y = 0 .. 2*Pi)) Is it possible to optimise it? (in maple or any other way...) How I ...
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1answer
105 views

Can anyone help me get this Laplace transform of

$f(t) = a + b \exp(-c \cdot t ^ d) $, where $a,b,c,d$ are constants, and $d$ is power of $t$.
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514 views

integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?

In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain: ...
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1answer
176 views

minus sign in the exponent of kernels of integral transformations

From planetmath and wolfram, the Fourier-Stieltjes transform of a function $\alpha$ is defined as $\displaystyle \int_{\mathbb{R}} e^{itx} d(\alpha(t)).$ The kernel $\displaystyle e^{itx}$ is unlike ...
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3answers
740 views

Comparison of different types of integral transforms

I was wondering why we have both Laplace transform and Fourier transform, instead of just one? why we have both generating function and Z transform, instead of just one? In other words, in each ...
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1answer
353 views

How do I find the inverse Hankel transform of $k^2e^{-k^2}$?

I am trying to solve: $f_l(r)=\int_0^{\infty}e^{-k^2}k^4j_l(kr)dk$, where $j_l$ is the spherical Bessel function of the first kind, for any integer l >= 0. Thanks in advance for any answers!
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1answer
4k views

What exactly is the Probability Integral Transform?

I've been going back over my notes from Stats class and came across the Probability Integral Transform. From my limited understanding, the basic idea is that from a cdf in terms of one variable, can ...
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227 views

Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
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2answers
422 views

Series of nested integrals

I'm trying to calculate the following series of nested integrals with $\varepsilon(t)$ being a real function. $$\sigma = 1 + \int_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\mathrm dt_1 ...
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1answer
294 views

Inverse Laplace Transform

What's the inverse laplace transform of s/((s-.5)^2+1)?
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851 views

Inverse Laplace Transform with squared irreducible quadratic in denominator

Okay, the equation is $$\frac{2s^3-2s}{(4s^2-4s+5)^2}$$ So I use partial fractions with $$\frac{As+B}{4s^2-4s+5} + \frac{Cs+D}{(4s^2-4s+5)^2} = 2s^3-2s$$ and square that quadratic get ...
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2answers
189 views

Laplace transform of $[f''[x]]^n$

Can anyone help me get this Laplace transform, $$ L[(f''(x))^n] $$ where $f'(0)=0$ and $f''(0)=0$ and $n$ is power of $$f''(x)$$?
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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 ...
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252 views

A Projection Problem in Functional Analysis - Uniqueness of a Solution

I have the following system of equations: $$\alpha f_1(x) = \int_\mathbb{R} g(k) h_1(k) e^{\mathrm{i}kx} dk$$ $$\beta f_2(x) = \int_\mathbb{R} g(k) h_2(k) e^{\mathrm{i}kx} dk$$ with known functions ...