Application of uniform asymptotics to the second Painlevé transcendent
Connection problems of Painlevé differential equations of the form \(d^2\phi/d\eta^2= -\xi^2F(\eta,\xi)\phi\) are studied. These problems involve finding uniform approximations to solutions to this equation when the independent variable passes towards infinity along different directions in the complex plane. By the method used the need to match solutions is avoided. The treatment depends on the locations of the zeros of the function \(F\) in the limit. If they are isolated a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. If two of the zeros of \(F\) coalesce as \(|\xi|\to \infty\) then an approximation can be derived in terms of parabolic cylinder functions.
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