All Questions
709 questions
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Integral with inequality
Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
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- 573
-1
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1
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519
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Poisson kernel is the Cauchy distribution, reference?
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
-1
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1
answer
208
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Does this function belong to $L^2(\mathbb{D})$?
Edit: After the answer of Prof. Eremenko to the previous version, I realized that a weaker assumption works for the main motivation of this post. so I revise the question.
The unit ...
- 356
-1
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1
answer
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Natural candidates for sub-half-exponential which limit to half-exponential function from below
There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth.
However sub-half-exponentials (functions whose composition grows ...
- 1,826
-1
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1
answer
204
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Cauchy reduction formula with measure (a variation)
The Cauchy reduction formula conveniently compresses $n$ integrations of a function $F(x)$ into a single integral. Here I am interested in reducing the following "curved-space" ...
- 141
-1
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1
answer
369
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Would this go to 0 [closed]
Let $t_{m}$ be the sup of the sum of the pairwise distances
between any $2m$ points in the unit disk. Does $t_{m}/m^{2}$ go to
$0$ as $m\rightarrow\infty$?
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-2
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1
answer
314
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Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$
(Question is short and straight-forward. )
What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ??
By "nice and non-trivial" I mean contains no ...
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2
answers
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Is there a transformation or a proof for these integrals?
Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
Question. Is this true? If so, is there an underlying transformation or just a ...
- 43.2k
-6
votes
1
answer
614
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Proof of formula for $\pi$ [closed]
The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...
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