##### Abstract

Consider the operator$${\bf P} = {\partial\over\partial t} + \alpha{\partial m\over \partial z m},\qquad \alpha \in {\bf C} - \{0\}$$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.

##### Citation

Tomlinson, Kathy Adiene. "An analog to the heat equation in complex space variables." (1991) PhD diss., Rice University. http://hdl.handle.net/1911/16489.