An alternative method for the asymptotic expansion for a double integral (Q1910849)

A method is proposed for obtaining an asymptotic expansion for the double integral \[ \int^1_0 \int^1_0 x^\alpha y^\beta (\log x)^\ell (\log y)^j f(x,y) g(\lambda x^\alpha y^\beta) dxdy \] as \(\lambda \to \infty\). It is assumed that \(f\) should be smooth and that \(g\) is bounded near the origin and is of zero order at infinity. The coefficients in the expansion are Hadamard finite part integrals.