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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2answers
942 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 ...
2
votes
2answers
169 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 ...
4
votes
2answers
180 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 $u=0$ ...
6
votes
1answer
340 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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0answers
165 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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vote
1answer
163 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 ...
2
votes
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 $t$.
5
votes
2answers
494 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: ...
1
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1answer
169 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 ...
2
votes
3answers
678 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 ...
2
votes
1answer
325 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!
3
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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 ...
3
votes
0answers
224 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 ...
3
votes
2answers
393 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 ...
0
votes
1answer
292 views

Inverse Laplace Transform

What's the inverse laplace transform of s/((s-.5)^2+1)?
0
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1answer
728 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 ...
4
votes
2answers
187 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 $$f''(x)$$?
7
votes
1answer
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 ...
2
votes
1answer
249 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 ...
0
votes
1answer
116 views

When to use the Functional Determinant in Polar Coordinate Transformation

I am currently learning about polar coordinate transformation, especially for integrating over certain regions. Let's say we have to calculate $\int_{n}{xy \; dx dy}$ Then I think the correct ...
15
votes
1answer
2k views

Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?

the Fourier transformation of a scalar function with respect to one variable might be defined as $\mathcal{F}\left[w\right](\omega )\equiv ...
38
votes
4answers
6k 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 ...
20
votes
7answers
11k views

Laplace transformations for dummies

Is there a simple explanation of what the Laplace transformation do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm ...
0
votes
1answer
838 views

Inverse of Laplace transform

There is a very simple expression for the inverse of Fourier transform. What is the easiest known expression for the inverse Laplace transform? Moreover, what is the easiest way to prove it?