All Questions
8 questions
1
vote
0
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144
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integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 501
0
votes
1
answer
127
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Characterizing the integral as a function of $n$
Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...
- 25
6
votes
1
answer
408
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On an asymptotic integral decay
Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that
$$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$
for all $\lambda > 0$. Does it follow that $...
- 4,115
8
votes
3
answers
429
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A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
- 4,115
1
vote
0
answers
144
views
Zeroes of Mellin transform
There exist a "standard" or canonical way to construct a real valued function whose Mellin transform has a prescribed set of zeroes? Clearly for some set of zeroes this could be impossible ...
- 131
1
vote
0
answers
38
views
Solving an equation containing Laplace transform
Consider the equation
\begin{equation}
\frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}%
(y)(s_{2})=\mathcal{L(}y)\mathbf{(}p),
\end{equation}
where $\mathcal{L}$ is the ...
- 47
2
votes
0
answers
65
views
On a question relating integral equation:
I don't know if the following question qualifies as research level. If it isn't, sorry.
Set the following terminology:
$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$
$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
- 41
1
vote
0
answers
308
views
Inverse Laplace transform of a non-negative function
Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform,
$$
f(s)=\int_0^\infty e^...
- 1,488