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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0answers
17 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 ...
1
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0answers
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 ...
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0answers
28 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 ...
2
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0answers
34 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. ...
1
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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
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 ...
1
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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 ...
1
vote
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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0answers
25 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 ...
2
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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} ...
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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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0answers
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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0answers
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?
5
votes
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 ...
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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
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 ...
1
vote
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 ...
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votes
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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0answers
10 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 ...
0
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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 ...
1
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0answers
21 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
votes
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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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 ...
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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: $$X[z]=\sum_{n=0}^2x[n]z^{-n}$$ ...
1
vote
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
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 ...
2
votes
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
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 ...
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0answers
32 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
votes
1answer
113 views

Fourier transforms and Dirac delta function

What is the Dirac delta function $\delta(t_1-t_2)$ in Fourier (frequency) space?
0
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1answer
25 views

How do you differentiate a Laplace transform?

Consider the Laplace transform of $\color{green}{t\cfrac{\mathrm{d^2}f}{\mathrm{d}t^2}}$: ...
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0answers
18 views

Fourier transform involving completion of square

Show that the Fourier transform of $$f(x) = k e^{\frac{-x^{2}}{2\sigma^{2}}}$$ for some $k \in \mathbb{R}$ is $$F( \lambda) = \frac{k}{\sigma}e^{\frac{-\sigma^{2}}{2\sigma^{2}}}$$ By ...
0
votes
1answer
53 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 ...
0
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0answers
15 views

Abel transform of $R_z(\Delta)$

The spectrum of the Laplace operator $\Delta$, as self-adjoint unbounded operator on $L^2$ is equal $]-\infty ,0]$. The resolvent $R_z$ is defined for $z \in \mathbb C \setminus ]-\infty ,0] $. Using ...
4
votes
1answer
67 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
8 views

Limitations on Linear Canonical Transforms

Preface: LCTs are used in physics and in signal processing, but this question is about the mathematics behind them moreso than any particular application. As such, I think it belongs here, but if I ...
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0answers
17 views

For a good understanding of Gauss' theorem

Why does any term that can be written as a divergence vanish when integrated over the volume V? And what is the physical meaning of this property? For example, $$\int_V\frac{\partial}{\partial ...
2
votes
0answers
25 views

A specific partial differential equation using Fourier Transform

I have the following PDE problem which I think sounds like a job for the Fourier transform: $ u_t + 2u_x = u_{xx} \space \space \space -\infty < x < \infty \space \space \space t>0 $ ...
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0answers
22 views

Hilbert transform of $\cos(\phi(t))$.

I am attempting to derive the Hilbert transform of $\cos{\phi(t)}$. I understand that the transform is given by \begin{align*} H[\cos(\phi(t)] = \frac{1}{\pi} \ p.v. \ \int_{-\infty}^{\infty} ...
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0answers
30 views

Nonstandard analysis and integral transforms

Can integral transforms be evaluated without limits(i.e Laplace transform) such as in non standard analysis? Can the improper integral be bounded by a hyperreal number? I am not very familliar with ...
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0answers
56 views

Finding the inverse of an Integral Transform

I'm working with the following integral transform - $\tilde{f}(y) = \int_{-\infty}^\infty dx\,\frac{f(x)}{x + i y}$ What would be the inverse of a such an integral transformation?
2
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3answers
92 views

Integrate $\int\sqrt{x+\sqrt{x^{2}+2}}$ $dx$ .

Q) $\int\sqrt{x+\sqrt{x^{2}+2}}$ $dx$ . Tried rationalising the numerator twice to get Numerator =-2 but not able to simplify denominator The question reduces to (as per my rationalising)
-1
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1answer
37 views

Integrate $\int\dfrac{x \cos x+1}{\sqrt{2x^{3}e^{\sin x}+x^{2}}}dx$ [closed]

$$\int\dfrac{x \cos x+1}{\sqrt{2x^{3}e^{\sin x}+x^{2}}}dx$$
6
votes
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 ...
1
vote
1answer
42 views

Integrate $\int\frac{\left(1+x\right)\sin x}{\left(x^{2}+2x\right)\cos^{2}x-\left(1+x\right)\sin2x}dx$

(Q) $\displaystyle \int\frac{\left(1+x\right)\sin x}{\left(x^{2}+2x\right)\cos^{2}x-\left(1+x\right)\sin2x}dx$ Tried a lot to expand denominator and reduce it to bring its derivative on top , but all ...
2
votes
3answers
52 views

Integrate $\int\frac{x+1}{x^{4}+6x^{2}+4x}dx$

This gives a cubic polynomial in denominator and nonfactorisable It looks so simple but i just am not able to solve it