An analog to the heat equation in complex space variables
| dc.contributor.advisor | Jones, Frank | |
| dc.creator | Tomlinson, Kathy Adiene | |
| dc.date.accessioned | 2009-06-04T00:39:54Z | |
| dc.date.available | 2009-06-04T00:39:54Z | |
| dc.date.issued | 1991 | |
| dc.description.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. | |
| dc.format.extent | 86 p. | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.callno | Thesis Math. 1991 Tomlinson | |
| dc.identifier.citation | Tomlinson, Kathy Adiene. "An analog to the heat equation in complex space variables." (1991) Diss., Rice University. <a href="https://hdl.handle.net/1911/16489">https://hdl.handle.net/1911/16489</a>. | |
| dc.identifier.uri | https://hdl.handle.net/1911/16489 | |
| dc.language.iso | eng | |
| 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. | |
| dc.subject | Mathematics | |
| dc.title | An analog to the heat equation in complex space variables | |
| dc.type | Thesis | |
| dc.type.material | Text | |
| thesis.degree.department | Mathematics | |
| thesis.degree.discipline | Natural Sciences | |
| thesis.degree.grantor | Rice University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy |
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