An analog to the heat equation in complex space variables

Consider the operator${\bf P} = {\partial\over\partial t} + α{\partial m\over \partial z m},\qquad α \in {\bf C} - \{0\}$where $$\partial\over{\partial z}$$ is the usual complex operator:${\partial\over\partial z} = {1\over 2} \left({\partial\over\partial x} - i{\partial\over\partial y}\right).$When m = 2 and $\alpha$ = $-$1, P bears a remarkable resemblance to the heat operator in one space variable. The "only" difference is that the space variable is now complex. In spite of this superficial similarity, P is quite different from the heat operator. It is neither hypoelliptic nor parabolic. The key result is a formula for a fundamental solution, E. It is obtained formally using Fourier transforms. The formula is a linear combination of Fresnel-like integrals, divided by z and a power of t. It is a $$C\infty$$ function except across t = 0. It has a homogeneity property which is similar to the one the standard fundamental solution for the heat operator possesses. It has a skew-reflection property in the time variable. The proof that E is a fundamental solution is done by applying PE to a test function. It is similar to the standard analogous proof for the heat equation. The main difference is that E is not integrable for fixed non-zero t. Thus we do our calculations with Fourier transforms. This requires making some of the formal arguments in the derivation of E into rigorous ones. The basic tools for this are approximating functions, Cauchy's integral theorem, and Lebesgue's dominated convergence theorem.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Natural Sciences

Grantor

Rice University

Year dc:date.issued

1991

Author dc:creator

• Tomlinson, Kathy Adiene

Advisor dc:contributor.advisor

• Jones, Frank

• Mathematics

Statement dc:rights

• Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1911/16489

OAI identifier oai:identifier

oai:repository.rice.edu:1911/16489