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DLMF: §32.5 Integral Equations ‣ Properties ‣ Chapter 32 Painlevé Transcendents

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32 Painlevé Transcendents Properties 32.4 Isomonodromy Problems 32.6 Hamiltonian Structure

§32.5 Integral Equations

ⓘ

Keywords:: Painlevé transcendents, applications, integral equations
Notes:: See Ablowitz and Segur (1977, 1981); also Ablowitz and Clarkson (1991).
Permalink:: http://dlmf.nist.gov/32.5
See also:: Annotations for Ch.32

Let $K(z,\zeta)$ be the solution of

32.5.1		$K(z,\zeta)=k\operatorname{Ai}\left(\frac{z+\zeta}{2}\right)+\frac{k^{2}}{4}\*% \int_{z}^{\infty}\!\!\!\int_{z}^{\infty}K(z,s)\operatorname{Ai}\left(\frac{s+t% }{2}\right)\operatorname{Ai}\left(\frac{t+\zeta}{2}\right)\,\mathrm{d}s\,% \mathrm{d}t,$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : real, $k$ : real and $K(z,\zeta)$ : solution Permalink: http://dlmf.nist.gov/32.5.E1 Encodings: TeX, pMML, png See also: Annotations for §32.5 and Ch.32

where $k$ is a real constant, and $\operatorname{Ai}\left(z\right)$ is defined in §9.2. Then

32.5.2		$w(z)=K(z,z),$
ⓘ Symbols: $z$ : real and $K(z,\zeta)$ : solution Permalink: http://dlmf.nist.gov/32.5.E2 Encodings: TeX, pMML, png See also: Annotations for §32.5 and Ch.32

satisfies $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$ and the boundary condition

32.5.3		$w(z)\sim k\operatorname{Ai}\left(z\right),$
$z\to+\infty$ .
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : asymptotic equality, $z$ : real and $k$ : real Permalink: http://dlmf.nist.gov/32.5.E3 Encodings: TeX, pMML, png See also: Annotations for §32.5 and Ch.32

32.4 Isomonodromy Problems 32.6 Hamiltonian Structure

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