The asymptotics of the solution of the Cauchy problem for the two-dimensional homogeneous wave equation (Q2703925)

Let \(K(x,y,t)=\{(\xi,\eta)\mid(x-\xi)^2+(y-\eta)^2\leq t^2 \}\). Two asymptotic expansions of the Poisson integral NEWLINE\[NEWLINE J(x,y,t)=\int_K\frac{f(\xi,\eta) d\xi d\eta}{\sqrt{t^2-(x-\xi)^2-(y-\eta)^2}}, NEWLINE\]NEWLINE where \(f(\xi,\eta)=O((\xi^2+\eta^2)^{-n})\) for \(|\xi|+|\eta|\to\infty\) (\(n>0\)) are obtained. Let \(r,\psi\) be polar coordinates. It is stated that inside the cone \(r<t-t^\delta\) (\(0<\delta<1/2\)) the asymptotic expansion \(J(x,y,t)=\sum_{m=0}^\infty t^{-m-1}A_m(r/t,\psi) (t\to\infty)\), where \(A_m(z,\psi)=(1-z^2)^{-m-1/2} P_m(z,\psi)\) and \(P_m\) are polynomials of degree \(m\) in \(z\), is uniformly valid. Moreover, for \(r\approx t\), i.e, near the cone boundary, another expansion \(J(x,y,t)=\sum_{m=0}^\infty t^{-m-1/2}B_m(r-t,\psi) (t\to\infty)\) is uniformly valid for \(|r-t|\leq t^\delta\), where \(B_n\) can be expressed via the integrals over a half-plane.