# §32.5 Integral Equations

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

 32.5.1 $K(z,\zeta)=k\mathrm{Ai}\left(\frac{z+\zeta}{2}\right)+\frac{k^{2}}{4}\*\int_{z% }^{\infty}\!\!\!\int_{z}^{\infty}K(z,s)\mathrm{Ai}\left(\frac{s+t}{2}\right)% \mathrm{Ai}\left(\frac{t+\zeta}{2}\right)\mathrm{d}s\mathrm{d}t,$

where $k$ is a real constant, and $\mathrm{Ai}\left(z\right)$ 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 $w(z)\sim k\mathrm{Ai}\left(z\right),$ $z\to+\infty$. ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: 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