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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10 views

Hankel transform of oscillating function

I'm looking for the Hankel transform of the following function $f(x,\rho_0)= \left[\frac{4 \sinh ^2(x)}{\cosh (2 \rho_0)+\cosh (2 x ...
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1answer
72 views

Exercise 2 , chapter 5 , Stein & Shakarchi real analysis

Consider the Mellin transform defined initially for continuous function $f$ of compact support in $R^+=${$t\in R:t>0$} and $x\in R$ by $Mf(x)=\int_0^\infty f(t)t^{ix-1}dt$ Prove that ...
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64 views

If derivative of f(x) is f'(x), then what is the integral of (f'(x))^2 in terms of f(x)? [duplicate]

Is there some procedure for figuring this out or am I venturing into unsolved territory here?
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17 views

non square transformation of random variables

Let $x_0$ and $w_0$ be independent random variables and let $x_1$ be related to them by $x_1 = f(x_0, w_0)$. I want to find the joint density of $x_1, x_0, w_0$. The transformation I am interested ...
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1answer
24 views

Sine Curve Circular Transform - Parametric Equations

Is there a way to transform a sine curve so that the x-axis of the sine curve would become a circle, with the sine wave oscillating around the now-circular x-axis? What would be the parametric ...
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1answer
20 views

Weierstrass transform on the Riemannian manifold

I've read on this Wikipedia article that Weierstrass transform (WT) can be defined on any Riemannian manifold $(M,g)$, but it seems a bit complicated to me. I'm not sure but I guess one can write the ...
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1answer
18 views

Expressing the Weierstrass transform in terms of the unilateral Laplace transform

I was looking for a way to express the Weiestrass transform of $f(t)$, $$\mathcal{W}\{f(t)\}(s)=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(t)\exp\left(-\frac{(s-t)^2}{4}\right)dt$$ in terms of the ...
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28 views

Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where ...
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2answers
80 views

Find “$g(x)$” knowing that “$x=\int_{0}^{\infty} g(tx) dt$”???

The entire question states what I am looking for. I'm looking for a function $g(x)$ in terms of $x$ which satisfies the condition that follows. This seems like it's related to "integral ...
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0answers
6 views

How can I obtain the inverse transform?

The inverse Fourier transform is defined as: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$ I can't get an inverse Fourier Transform to ...
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26 views

Approximate solution for integration of two Bessel Functions

I am an engineering student. I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$ which is a function of $(n-1)$. degree polynomial. My statement as follows: ...
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11 views

Abel transformation - sources

I'd like to study the Abel transformation, that is, $$Af(x) = \int\limits_x^\infty\frac{f(t)t}{\sqrt{t^2 - x^2}}\ \mathrm{dt},\quad x\in(0,\infty).$$ I'm especially interested in ...
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11 views

Transforming a PDE to an ODE by discretizing the size-axis of the integro-partial differential equations

I have this issue on modelling a process. It is a system of ODEs except for one equation related to the population balance of crystals in a mixture that is a PDE. The author of the publication ...
4
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1answer
54 views

Differentiation under integral sign?

I am trying to understand the following argument given in a text book: Suppose $f \in L^1(\mathbb R^n)$, consider the function $\hat{f}(\zeta)= \int_{\mathbb R^n} \exp(-2\pi i X.\zeta)f(X)dX$. ...
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18 views

Fourier transform of integral with isotropic kernel

The textbook I'm reading claims that this integral: $$ A = \int_V \,d\mathbf{r} \int_V\,d\mathbf{r}' f(\mathbf{r}) K (| \mathbf{r} - \mathbf{r}'| ) f(\mathbf{r}')$$ can be written in Fourier ...
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1answer
29 views

Using Fourier Transform to solve an ODE

Consider the differential equation $$f^{iv}+3f^{''}-f=g$$ I have read that taking the Fourier Transform of both sides gives ...
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1answer
107 views

Riesz transform does not preserve continuity

I've read somewhere that the Riesz operator $R_j$ defined by $$R_j f(t) := c(n) \, \text{pv} \int_{\mathbb{R}^n} \frac{x_j}{|x|^{n+1}} f(t-x) \, dx$$doesn't preserve the continuity, but I can't ...
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35 views

Complex inversion of a function

I am trying to find the function whose laplace transform is below using the complex inversion formula: $$ f(s)= \frac{se^s}{(s-2)^3}$$ My attempt below seems to be giving me the wrong answer but I'm ...
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2answers
47 views

Is this kernel invertible ? $K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}$

Is the following Kernel invertible? $K(x,y)=\frac{e^{-\frac{xy}{x+y}}}{x+y}, x\in[0,1],y\in [0, \infty)$ i.e. if $\int_0^1 K(x,y) f(x) dx=0 ,\forall y\in [0, \infty)$ can we conclude $f(x)=0,x\in ...
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9 views

Transform and domain such that phase shift of the original function results in shift in transformed domain variable

It is known that the Fourier transform of a phase-shifted function results in a constant shift of the dependent variable of the phase spectrum: If $ F(x(t)) = X(w) = |A(w)| \cdot e^{-i \cdot ...
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22 views

Why M.G.F transform is injective a.s.?

We always use the theorem that If we know a random variable's MGF, we can determine its Pdf, which means the map from Pdf to Mgf is injective almost surely. And I just wanna know why this is ture.
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2answers
72 views

Inverse Laplace Transform of $e^{\frac{1}{s}-s}$

doing some work on a PDE system I have stumbled across a Laplace transform which I'm not sure how to invert: $$ F(s) = e^{\frac{1}{s}-s} $$ I can't find it in any table and the strong singular growth ...
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31 views

Solving differential equation by transform with contour integral

In my notes, I have an example for solving for the Airy function in the equation: $$\frac{d^2y}{dx^2}-xy=0$$ So he uses the contour integral to represent the solution (with $C$ as a yet unspecified ...
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1answer
118 views

Is it possible to calculate the inverse Laplace transform of the following?

I have a Laplace tranform in the form given below $\mathcal{L}_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$ Can some one help me to find ...
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35 views

Fourier sine/cosine transforms of :1) derivatives raised to power & 2)derivatives in exponential

I need help to solve the Fourier finite sine and cosine transforms: First, reciprocal of derivative: $$ F_s\left(\frac{1}{\frac{\partial u}{\partial x}}\right)= \int_{0}^{a}\frac{1}{\frac{\partial ...
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30 views

How and why an integral Transform is created?

I don't know if what I'm going to ask will make any sense, but I was just wondering about integral transforms. I am talking about, for example, Mellin Transform, or Laplace Transform or Hilbert ...
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19 views

Integral inversion

Say I know this function $$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$$ where $m(x)$ is a Fourier transform of an infinitely differentiable real function, whose maximal ...
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40 views

Existence of solutions in time and Laplace domains

I have not made use of Laplace transforms for many years since my education and I am a bit rusty on applying the various theorems associated with the transform. I have an equation $f(t)=0$ and I am ...
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1answer
32 views

How do you show $F$ has a well-defined inverse function $G : (0,1) \to \mathbb{R}$, which is strictly increasing [closed]

If $X$ be a random variable with cdf $F$, and $F$ is continuous and strictly increasing on $\mathbb{R}$ how do you show $F$ has a well-defined inverse function $G : (0,1) \to \mathbb{R}$, which is ...
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47 views

Integral calculation by using Mellin Transform

I want to use the Mellin Transform (MT) to calculate the integral: $\int_0^{1 } \exp(-2\rho^2) J_0(\pi \rho r)\rho \, d\rho$ in which $r>=0$ and real. I have calculated it by numerical methods. ...
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0answers
34 views

Fourier transforms of Gaussians with rational functions

Consider functions of the form $$ f(x)= e^{- x^2/\sigma^2} \frac{a_0 + a_1 x + \cdots +a_n x^n }{b_0+ b_1 x + \cdots + b_m x^m } $$ Under which circumstances do these functions have closed forms for ...
3
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1answer
2k views

Calculating an integral derived from the convolution of two Fourier transforms

Let $\sigma>0$ , $1<\alpha\leq 2$, and $-1\leq \beta \leq 1$. I am looking for a closed-form solution (or something near) for the following integral. $$\frac{1}{2 \pi } \text{PV}\int_{-\infty ...
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1answer
71 views

How to integrate this looking simple ODE?

I meet an ODE about $V(\theta)$ $$\frac{d^2V}{d\theta^2}+\frac{1}{2V}=0.$$ But I can not figure out how to integrate it to yield $$\left(\frac{dV}{d\theta}\right)^2+logV=C_1$$ or ...
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1answer
21 views

Does a symetric complex function $k(t,s)$ verify $\overline{k(t,s)}=k(t,s)$?

I am trying to figure out why an integral operator is self-adjoint. The operator is: $$K(f)=\int_{0}^{1} k(t,s)f(s)ds$$ From $L^2([0,1])$ to $L^2([0,1])$ and $0, \leq t,s \leq 1$ So I did a bit of ...
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27 views

Integral transforms involving logarithms?

We have the fourier-transform: $$F\{f\}(w) = \int_{-\infty}^\infty f(x)\exp(iwx)dx$$ Which has extremely many applications and interpretations throughout science and engineering. For instance since ...
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19 views

Conditions under which a convolution transformation is injective in the 1-d Torus

Let $X=[0,1)$ the 1-d torus. Given a bounded positive function $w\colon X\to\mathbb{R}$ with unit integral (I mean $w\geq 0$, $w\in L^\infty(X)$ and $\int_X w\; dx=1$), define \begin{align*} T_{w} ...
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9 views

What are the different type of Daubechies Wavelet transform?

Like Daub4 are there others named as Daub2, daub3 or we only have daub4 , daub8, daub16? What is the order of a transform(represented usually by N)? Does this order have any resemblance with the ...
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0answers
26 views

Stieltjes Transform is injective?

Let $\mu$ a probability measure on $\mathbb{R}$, we define the Stieltjes transform by : $$ S[\mu](\lambda)=\int_\mathbb{R} \frac{d\mu(t)}{t-\lambda} $$ For all ...
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0answers
14 views

Hankel transformation

How to solve the Hankel transform of $x^2e^{-ax}$ with kernel $x(J_2(px))$ where $J_n(x)$ denotes bessel function of order $n$ of first kind?
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1answer
78 views

What is the advantage of the Fourier Transform over the Hartley Transform?

The Hartley_transform is defined as $$ H(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) \, \mbox{cas}(\omega t) \mathrm{d}t, $$ with $\mbox{cas}(\omega t) = \cos(\omega t) + \sin(\omega ...
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36 views

Find the Fourier transform for this function

Find the Fourier transform for this function $$f(x)=e^{x-e^x}$$ My Solution:- $T[f(x)]=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx} f(x)dx$ $=\frac{1}{\sqrt{2\pi}} ...
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0answers
8 views

Is it possible to define a hankel transform for a function depending of a complex variable

Hankel transform is defined by $F_{\nu}(k) = \int_0^{\infty}f(r)J_{\nu}(kr)rdr$, and the inverse transform by $f(r) = \int_0^{\infty}F_{\nu}(k)J_{\nu}(kr)kdk$, In my problem, r is a complex ...
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1answer
18 views

How do I calculate the Fourier-transform of $f(t) = \max(t-1,0) (t \in \mathbb{R})$?

I get $$\hat f(w) = \int_{-\infty}^{+\infty}\max(t-1,0)e^{-i\omega t}dt$$ $$= -\int_{-\infty}^{1}(t-1)e^{-i\omega t}dt$$ $$ = \lim_{p\to\infty}\left(\int_{-p}^1e^{-i\omega t} - \int_{-p}^1 t\cdot ...
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1answer
31 views

Gaussian quadrature with a to $[0,1]$ reference domain instead of a $[-1,1]$ reference domain?

For 1-d Gaussian quadrature with two points per element we have the following formula to transform an integral from an arbitrary domain $[a, b]$ to the reference domain $[-1,1]$ on which various ...
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14 views

What reparametrization of vector parameters makes the Jeffreys prior correspond to the uniform prior?

What reparametrization of vector of parameters $\theta$ makes the Jeffreys prior $$\sqrt{\det I(\theta)}$$ correspond to the uniform prior? A change of parametrization from $\theta$ to $\eta$ changes ...
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1answer
40 views

Inverse $Z$ transform of $\frac{1}{z-a}$

I don't really get what's happening here and I haven't been able to find a single example on how to get the inverse $Z$-transform of $\frac{1}{z-a}$. Can anyone show the way?
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15 views

How would you express this integral in cylindrical polar coordinates?

How would you express the integral \begin{gather*} \int_{0}^{1}\int _{0}^{\sqrt{1-x^{2}}}\int_{0}^{1-x^{2}-y^{2}} e^{z} \ dz \ dy \ dx \end{gather*} In cylindrical polar coordinates, would it be ...
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15 views

Is this the correct domain of integration for this double integral, under the following coordinate transformation?

Suppose you had the double integral $\iint \limits_{A} \frac{y^{2}}{x^{4}}e^{xy} \ dx \ dy$, where $A$ is the region defined by $x>0, \ y>0$ satisfying $x^{2} \leq y \leq 2x^{2}, \ \frac{1}{x} ...
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0answers
23 views

What would the limits of integration be for this double integral?

Suppose you had the double integral $\iint \limits_{A} \frac{y^{2}}{x^{4}}e^{xy}dx \ dy$, where $A$ is the area defined by $x>0, \ y>0, \ x^{2} \leq y \leq 2x^{2}, \ \frac{1}{x^{2}} \leq y \leq ...
2
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0answers
27 views

Fourier transform problem

I have to show that the Fourier transform of the function $f(x) = \ln(x)$ is: $$\mathfrak{F}[\ln(x)](k) = \frac{1}{k}\sqrt{\frac{\pi}{2}} - \frac{1}{|k|}\sqrt{\frac{\pi}{2}} + i\ ...