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Results tagged with ca.classical-analysis-and-odes
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user 48831
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
24
votes
Binomial again, and again
Here is a proof which doesn't use the identity $\int_{-\infty}^\infty {n \choose x}\,dx= 2^n$:
Using the representation ${ n \choose x}=\frac{1}{2\pi}\int_{-\pi}^\pi e^{-ixt}\left(1+e^{it}\right)^n\, …
- 3,255
21
votes
Accepted
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
I sketch the arguments for $C(x)$, the arguments for $L(x)$ are essentially the same.
The specific form of the sum suggests probabilistic arguments.
Let $X_x$ be a $\mathrm{Poiss}(x^2)$-distributed …
- 3,255
14
votes
Accepted
Integral $\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}...
Here is another approach, which also gives the rational term.
(I) To see how it works let $n\geq 2$ and consider first
the simpler case
\begin{align*}
\mathbb{E}\bigg(\frac{1}{X_1+\ldots+X_n}\bigg)= …
- 3,255
5
votes
(Sharp) inequality for Beta function
One can also use Jensen's inequality. Let (for $\sigma>0$) $G_\sigma$ denote a random variable with $\Gamma(1,\sigma)$-distribution, i.e. having Lebesgue density
$$f_\sigma(t)=\frac{t^{\sigma-1}}{\G …
- 3,255