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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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
228 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
426 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)?
0
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1answer
863 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
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 $$f''(x)$$?
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
253 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
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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 ...
19
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 ...
48
votes
5answers
8k 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
14k 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 ...
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1answer
1k 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?