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32 Painlevé Transcendents Properties 32.4 Isomonodromy Problems 32.6 Hamiltonian Structure

§32.5 Integral Equations

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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 $K(z,\zeta)=k\operatorname{Ai}\left(\frac{z+\zeta}{2}\right)+\frac{k^{2}}{4}\*% \int_{z}^{\infty}\!\!\!\int_{z}^{\infty}K(z,s)\operatorname{Ai}\left(\frac{s+t% }{2}\right)\operatorname{Ai}\left(\frac{t+\zeta}{2}\right)\,\mathrm{d}s\,% \mathrm{d}t,$

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real, : 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 is a real constant, and $\operatorname{Ai}\left(z\right)$ is defined in §9.2. Then

32.5.2

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Symbols:: : 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 $w(z)\sim k\operatorname{Ai}\left(z\right),$ $z\to+\infty$ .

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : asymptotic equality, : real and : 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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