Questions tagged [integral-transforms]
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 transforms.
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Can the gamma function be generalized to quaternions and how? [duplicate]
The gamma function is a generalization of the operator !n. The question is: Can the concept of the gamma function be generalized to quaternion analysis and the use of quaternions, and how?
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How many bits would be required to store 2D FDCT result of 8-bit sample data?
The 2D FDCT can be expressed as a multiplication between an 8x8 matrix of 8-bit integers (lets call it A) and another 8x8 matrix containing fraction values (lets call it B) that are all less than 1. ...
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Laplace transform of exponential functions with derivatives.
I have been trying to calculate the Laplace transform of these troublesome exponential functions:
Having $\alpha \in \mathbb{R^+}$
1.$\mathcal{L}\left\{e^{n \alpha t}\frac{f(t)}{t^2} \right\},n \in \...
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Write triple integral as cylindrical coordinate of given region, confused in determining lower and upper bound.
Given $E$ is a region as follows:
$$E=\left\{0\leq x\leq 1, 0\leq y\leq \sqrt{1-x^2}, \sqrt{x^2+y^2}\leq z\leq \sqrt{2-x^2-y^2}\right\}.$$
Write triple integral
$$\iiint_\limits{E}xydzdydx$$
as triple ...
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How to derive the 1/s rule for Laplace transform of an integral?
How do we derive the rule $\mathcal{L}\{ \int_0^t f(\tau) d\tau\}=F(s)/s$ where $F(s)=\mathcal{L}\{f(t)\}$ and the symbol $\mathcal{L}$ represents the Laplace transform operator ?
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Inverse Fourier in two dimensions
I want to compute the Fourier inverse in 2D of the following integral
$\displaystyle\int_{\mathbb{R}^2}\Big(\frac{\zeta_1}{\zeta_2}\Big)^k F(\zeta_1,\zeta_2)e^{i(\zeta_1 x_1 + \zeta_2 x_2)}d\zeta_1d\...
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How to solve this integral equation $ \int_{-\infty}^\infty f(z)x^z dz = F(x)$ for f(x)?
My question is: solving $f(x)$ with known $F(x)$ and equation
$$ \int_{-\infty}^\infty f(z)x^z dz = F(x).$$
I met this problem when I tried to extend the idea of generating functions for discrete ...
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How can I solve $\sum_{i=1}^{M-1} (M+i)^{M+i+1/2}/i^{i+1/2}$?
I am trying to solve an equation in Mathematica:
$$
\sum_{i=1}^{M-1} \frac{(M+i)^{M+i+\frac{1}{2}}}{i^{i+\frac{1}{2}}}
$$
Does a general solution exist for this expression?
And if $M \to \infty$, can ...
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Wavelet admissibility and orthogonality
I have been studying the continuous wavelet transform and came across the following result on the Wikipedia:
A wavelet $\psi(t) \in L^1(\mathbb{R})\cap L^2(\mathbb{R})$ is admissible if it has a ...
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Relation between the inverse Laplace and inverse Mellin transforms
If we have the answer to the inverse Melin transformation of an expression, can we arrive at the inverse Laplace transform of that expression?
$$M^{-1}\left (\frac{1}{\Gamma (c+s)\Gamma (d-s)} \right ...
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Radon Transform of Gaussian function
I am trying to find the radon transform of the gaussian function
$$f(x,y) = e^{-(x^2 + y^2)}$$
Now, I am using the formula for radon transform as
$$
[\mathcal{R}f]{(\rho, \theta)} = \int_{-\infty}^{\...
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What is the inverse of an integral transformation that turns second order ODEs into first order ones?
Let $\mathcal{S}_f:\mathcal{C^\infty\rightarrow C^\infty}$ be an integral transform such that, for any $\{f,\psi\}\subseteq\mathcal{C}^\infty$, $\int_0^\infty f(x)dx=\infty$, we have that:
$$\mathcal{...
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How can i find the kernel of this integral transform?
I'm trying to define a class of integral transforms $\mathfrak{S}:\mathcal{C}^\infty\rightarrow\mathcal{C}^\infty$ with the following property:
$$\mathfrak{S}_{\psi}\{f(x)\psi(x)\}(t)=\alpha_f(t)\...
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Definition of Convolution of functions of two variables
We know that, If $f$ and $g$ are functions then their convolution is defined as,
$(f*g)(x) =\int_{-∞}^{∞} f(t)g(x-t) dt$
(This is the convolution structure for Fourier transform)
But, what if $f$ and $...
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Laplace Transform of the Product of a function and the Step Function
I need to find the Laplace transform of the product of a function $f(t)$ with the Unit Heaviside Step Function $H(t-c)$, i.e., $\text{L}[H(t-c)f(t)]$.
Given that
$$\text{L}[H(t)f(t-c)] = e^{-sc}F(s)\...
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