Exponential Asymptotics for Integrals with Degenerate and Non-Isolated Critical Points

In this thesis we consider the exponentially improved asymptotic solutions to unbounded multidimensional steepest descent type integrals in the case where the phase function has sets of non-isolated critical points. These sets are connected components of the critical set of the phase function and we consider the case where these sets have both general order of degeneracy and general dimension. We consider first the case of isolated critical points of general order of degeneracy as a lead-in to the general problem.

In the isolated case, we justify the reduction of the study of the asymptotic behaviour of the general integral to the study of the asymptotic contribution to each individual critical point by appealing to the Morse lemma and results from the literature regarding the homology group of allowable integration surfaces. In the non-isolated case no such results exist, but we give a first step by appealing to the Morse-Bott lemma to suitably parameterise the integration surface. The analogous homological result does not yet exist and such a derivation is beyond the scope of this thesis, but we proceed to study individual contributions regardless, inspired by how readily the Morse-Bott lemma affords an analogous parameterisation of the integration surface in the non-isolated case.

Once this justification is established we focus on individual critical connected components of the phase function. A full hyperasymptotic expansion representing the repeatedly exponentially improved asymptotic contribution to the integral for critical points of this type is derived for the first time, with examples provided to demonstrate this new theory. The case of a general bounded integration region is briefly considered, but we demonstrate that work still needs to be done to extend the theory to this case.