§32.5 Integral Equations

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

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

32.5.1 $K(z,\zeta)=k\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{z+\zeta}{2}\right)+\frac{k^{2}}{4}\*\int _{z}^{\infty}\!\!\!\int _{z}^{\infty}K(z,s)\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{s+t}{2}\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{t+\zeta}{2}\right)dsdt,$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $dx$ : differential of $x$ , $\int$ : integral, $z$ : real, $k$ : real and $K(z,\zeta)$ : solution
Permalink:: http://dlmf.nist.gov/32.5.E1
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where $k$ is a real constant, and $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ 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
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satisfies $\mbox{P}_{{\mbox{\scriptsize II}}}$ with $\alpha=0$ and the boundary condition

32.5.3 $w(z)\sim k\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right),$ $z\to+\infty$ .

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\sim$ : asymptotic equality, $z$ : real and $k$ : real
Permalink:: http://dlmf.nist.gov/32.5.E3
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