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Results tagged with ca.classical-analysis-and-odes
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user 113397
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
1
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Does a minimiser exist for this Gaussian-like functional?
Let $f(x)$ be a strictly increasing function such that $\lim\limits_{x\to\pm\infty}f(x)=\pm\infty$ and $\lim\limits_{x\to+\infty}f'(x)e^{f(x)-x}=+\infty$. If $f(a)=0$ for some prescribed $a\in\Bbb R$, …
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43
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Is $\int_0^\infty{dx\over x^{x^{x^x}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^x}}}}}<\int_0^\in...
On MSE I asked whether each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ was less than $2$ and received answers on bounding t …
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18
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Are these continued fractions of integrals known?
Simplified repost of Are these continued fractions of integrals known? on MSE
EDIT: The period of the oscillations of $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ appea …
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3
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Curious asymptotics of real part of ratio between Lambert W branches
This question was inspired by the inactive thread How to find this value of $A$? but the focus there was on the divergence of the imaginary part.
It seems that for a given nonzero real $x$, $$\opera …
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4
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$\int_L^\infty \exp(- t - y/t) \, dt = \text{?}$
References to the study of these functions which are frequently used in hydrological models. No precise bounds in the following papers but at least they give a starting point.
Harris (2008) "Incomple …
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7
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Signs of Kravchuk matrix asymptotically produce a large circular region with hyperbolic sink...
The Kravchuk matrix of dimension $n+1$ is such that its entries satisfy $$K_{i,j}^{(n+1)}=[x^i](1+x)^{n-j}(1-x)^j\quad\forall0\le i,j\le n.$$ It enjoys properties such as involution and has various ex …
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