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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

2 votes
1 answer
207 views

infinite integral defining Bessel functions

Landau in the first equation of Über die Gitterpunkte in einem Kreise uses the following formula for the Bessel function of the first kind: $$\frac{1}{2\pi i } \int_{1-\infty}^{1+i\infty} \frac{\mathr …
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2 votes
0 answers
111 views

Inequality about exponential integrals

I am reading about Dirichlet polynomials in the book Analytic Number Theory by Iwaniec-Kowalski. During the proof of Theorem 9.1 for any positive real numbers $T, N$ they define a piecewise linear and …
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0 votes
0 answers
286 views

What is the "best" good kernel?

A family of functions $k_n(x):[-\pi,\pi]\to \mathbb R$ for $n\in \mathbb N$ is said to be a good kernel if all the following are satisfied: $\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$, …
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2 votes
0 answers
114 views

Least positive value of a random polynomial

Fix a positive even integer $d$ and consider the polynomial $f(x)=c_d x^d+\ldots+c_1x+c_0$, where the $c_i$ are independent random variables that follow the uniform distribution in the interval $[-1,1 …
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