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# Questions tagged [ca.classical-analysis-and-odes]

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

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Good afternoon, everyone. Does anyone know what the Riemann $P$-symbols mean when they contain more than three columns (i.e., to what ordinary differential equations they correspond)? Examples: $P\... 1answer 80 views ### When$0<\alpha<\pi$, what is$\iint_{0}^{\infty}e^{-(x^2+y^2+2xy\cos\alpha)}dxdy$[on hold] For$0<\alpha<\pi$, what is $$\iint_{0}^{\infty}e^{-(x^2+y^2+2xy\cos\alpha)}dxdy\, =?$$ 1answer 72 views ### Decay of Laplace (or Mellin) transform beyond region of convergence? Let$f:[0,\infty)\to \mathbb{R}$be a piecewise differentiable function with$f(0)=0$and$f'(t)$of bounded variation. Its Laplace transform$\mathcal{L}f$converges for$\Re s > 0$. Assume it can ... 1answer 329 views ### turn$\pi/n$, move$1/n$forward start at the origin, first step number is 1. turn$\pi/n$move$1/n$units forward Angles are cumulative, so this procedure is equivalent (finitely) to $$u(k):=\sum_{n=1}^{k} \frac{\exp(\pi i H_{n}... 1answer 1k views ### An equality about sin function? Empirical evidence suggests that, for each positive integer n, the following equality holds: \begin{equation*} \prod_{s=1}^{2n}\sum_{k=1}^{2n}(-i)^k\sin\frac{sk\pi}{2n+1}=(-1)^n\frac{2n+1}{2^n}, \... 1answer 343 views ### Computing spectra without solving eigenvalue problems There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ... 1answer 128 views ### Functions orthogonal to powers of 1/{\left(1+x^2\right)} Let f,g:\mathbb{R}\to\mathbb{R} be continuous functions with the following properties: f(x) and {g(x)}/x are bounded; {g(x)}/{\left(1+x^2\right)}\in L^1\left(\mathbb{R}\right); \lim_{x\to0}f(... 0answers 28 views ### Asymptotic dispersion of a periodic Sturm–Liouville problem From a physical problem of waves in periodic waveguides, I obtain the following Sturm–Liouville equation:$$ \left(\frac{d^2}{dx^2}-k^2+\omega^2\ V(x)\right)\psi(x)=0 $$where V(x+L)=V(x)>0 is a ... 0answers 167 views ### Packing number of a subset of Stiefel manifold Let \lambda_1\geq...\geq\lambda_r>0 be a sequence of real numbers such that \sum_{i=1}^r\lambda_i^2=1. Define a subset of the Stiefel manifold O(n,r) as$$S_t=\{U\in O(n,r): 1-\sum_{i=1}^{r}\... 4answers 2k views ### Prove that$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$Let$x>0$and$n$be a natural number. Prove that: $$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1.$$ This question is very similar to many contests problems, but ... 0answers 57 views ### How to tanslate the problem of computing a Kakutani fixed point to the problem of computing a Brouwer fixed point? I read from some paper that the problem of computing a Kakutani fixed point can be reduced to the problem of computing a Brouwer fixed point? How to make this translation? Can this translation be ... 0answers 123 views ### Lax Milgram for non coercive problem? I obtained the variational form of my problem. and the bilinear form is below. Bilinear Form Let$\Omega\subset\mathbb{R}$be an open set. For$u,v\in H^1_0(\Omega)$. I have $$a(u,v)=\int_\Omega u(t)... 0answers 42 views ### Prove or find correct 4-order power series expansion of a multivariate function [on hold] a(x) is an n-order positive definite matrix,$$F(x)=(1/2)?(?_{x}^{T}F(x))?a(x)?(?_{x}F(x)),$$F(x)=0 if and only if x=x_?, x is an n-dimensional vector, F(x) is a function of vector x,... 0answers 126 views ### A combinatorial / geometric interpretation of compositional inversion via matrix inversion There are several ways of finding the power or Taylor series for the compositional inverse of a function f(x) with f(0)=0\; given its series expansion, e.g., by using the classic Lagrange ... 2answers 543 views ### Function orthogonal to powers of 1/\left(1+x^2\right) Does there exist any continuous function f:\mathbb{R}\to\mathbb{R}, f(x)/(1+x^2)\in L^1(\mathbb R), such that f(0)=1 and$$\int_{-\infty}^{\infty}\frac{f(x)}{\left(1+x^2\right)^p}dx=0$\$ for ...

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