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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15 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 ...
3
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1answer
426 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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0answers
21 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
29 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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1answer
72 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 ...
2
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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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0answers
33 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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3answers
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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0answers
30 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 ...
2
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1answer
981 views

Inversion formula for the Abel transform

I need an inversion formula for the Abel transform $$ F(y) = 2\int_y^\infty\frac{f(r)r\,dr}{\sqrt{r^2-y^2}}. $$ Hint: The inversion formula found on Wikipedia appears to be incorrect. The ...
21
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1answer
193 views

Exponential integral $ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt$

Now since the sum $$ \sum_{n=0}^\infty \frac{x^n}{n!},\quad x\in\Bbb R, $$ does have some relatively nice properties, is the same true for its analogues integral? If we take the gamma function to be a ...
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3answers
2k views

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer number of them. Is there a unified ...
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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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1answer
67 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
365 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
52 views

Laplace Transform

The question I had was Find the Laplace transform of $$f(t)=10e^{-200t}u(t).$$ Would it be correct to take out the 10 because it is a constant, find the Laplace transform of $e^{-200t}$ and then ...
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0answers
24 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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0answers
15 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} ...
4
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2answers
162 views

Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$

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 ...
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0answers
7 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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2answers
264 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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23 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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12 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
23 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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1answer
91 views

Bilateral Laplace transform

My knowledge of Bilateral Laplace transform is less. Here are the few questions I need answer. What is the condition for existence of bilateral Laplace transform? How is the condition for existence ...
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0answers
30 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
6 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
16 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 ...
6
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1answer
74 views

Integrate $\int\frac{x^{6}-2x^{3}}{\left(x^{3}+1\right)^{3}}dx$ .

$$\int\frac{x^{6}-2x^{3}}{\left(x^{3}+1\right)^{3}}dx$$ I tried adding and subtracting 1 to bring a square expression with numerator as $(x^3-1)^2 -1$ but always going to partial fraction which ...
3
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2answers
472 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\nolimits_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + ...
3
votes
1answer
420 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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0answers
9 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
28 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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0answers
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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0answers
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 ...
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0answers
20 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\ ...
0
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1answer
26 views

Solving 2nd order linear ODE with integral transformation

I have this differential equation $-u''(x)+\mu \cdot u(x)=f(x)$ where $x \in (0,\pi)$ with boundary conditions $u'(0)=u'(\pi)=0$ where $c$ is a constant. I checked the values of $\mu$ where I have ...
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6answers
604 views

What does it mean when two functions are “orthogonal”, why is it important?

I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means ...
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1answer
38 views

How would you solve this surface integral?

Suppose you had the surface integral $\iint \limits_{A} = x^{3}(1-x^{4}-y^{4})dx \ dy$ where $A$ is the region defined by $x \geq 0, \; y \geq 0, \; x^{4}+y^{4} \leq 1$. How would you solve this ...
4
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1answer
66 views

Mellin transform of Gumbel distribution

The probability density function (PDF) of Gumbel distribution is given as: $$f\left(x\right)=\frac{\exp \left(-\left(\exp \left(-\frac{x-\mu}{\beta }\right)+\frac{x-\mu}{\beta }\right)\right)}{\beta ...
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0answers
11 views

Why does count of Z -Transform of sequence change?

I was looking at the reference [1] below and noted the author defined the Z-transform for [1, 2, 3] as $$[6, \frac{11}{4}, 2]$$ I worked it out as follows: $$X[z]=\sum_{n=0}^2x[n]z^{-n}$$ ...
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0answers
21 views

Identify a probability distribution with coordinate transformation

I have a problem with this task: We have a random variable $X:\Omega \rightarrow \mathbb R^2$, which is uniformly distributed on $K:= \{(x_1,x_2) \in \mathbb R^2 : \sqrt{x_1^2+x_2^2} \le 1 \}$ Now I ...
3
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1answer
48 views

Proving that $(2 \pi i)^{-1} \int e(\pi v^2/y^2) x^v y^{-1} dv = e(-\pi (\log x)^2 y^2 /4)$

I've seen a particular integral transform (an inverse Mellin Transform) used a few times, but I don't know how it's proved. In particular, I'm trying to prove $$\frac{1}{2\pi i} \int_{(2)} e^{\pi ...
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0answers
45 views

The resolvent kernel

Good morning. How can we prove that $$ \Gamma (s,t, \Lambda ) - \Gamma (s,t, M ) = \Lambda - M \times \int { \Gamma(s,t, \Lambda ) \times \Gamma (s,t,M) \, dx }$$ I tried to use the formula of ...
2
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1answer
40 views

Does the Discrete Fourier Transform assume a periodic signal, or one that dies off?

I keep hearing that the DFT assumes a periodic signal. E.g. the first answer in this MATLAB Q&A site. This doesn't make any sense to me. According to the derivations I've seen of the DFT one ...
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0answers
30 views

Radial Green's function

I would like to solve an equation of the form $$ \bigg(\frac{d}{dr^2} + m^2 \bigg)f(r) = g(r), $$ for $f(r)$. Normally I would just find the Green's function $G(r,r')$, which is defined by $$ ...
2
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1answer
111 views

Fourier transforms and Dirac delta function

What is the Dirac delta function $\delta(t_1-t_2)$ in Fourier (frequency) space?
2
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1answer
546 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
25 views

How do you differentiate a Laplace transform?

Consider the Laplace transform of $\color{green}{t\cfrac{\mathrm{d^2}f}{\mathrm{d}t^2}}$: ...