Operation Calculus and the Finite Part of Divergent Integrals

Citation

Boehme, Thomas Kelman (1960) Operation Calculus and the Finite Part of Divergent Integrals. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/P33D-3W91. https://resolver.caltech.edu/CaltechETD:etd-02222006-154540

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis the operational calculus of J. Mikusinski is utilized to study the finite part of divergent convolution integrals. In Chapters 2 and 3 the idea of an analytic operator function is utilized. An operator function f(z) is said to be an analytic operator function on an open region S of the complex plane if there is an operator [...] such that af(z) = {af(z, t)} has a partial derivative with respect to z which is continuous on [...]. Let f(z) be an analytic operator function and suppose that {f(z, t)} is a continuous function on [...]. Suppose also that for each t > 0 f(z, t) is an analytic function of z on a larger region S* > S. Let f*(z) be an analytic operator function on S* which is such that f*(z) = f(z) on S. Then the operator function f*(z) is called [FP f (z, t)] on S*. The relationship between the operator product g[FP f(z,t)] and [...] is studied for the case when {f( z,t)} = [...], where m is function which possesses continuous derivatives of some order on [...]. In Chapter 4 the solutions to the singular integral equation [...] all t > 0 are found from considering the operators [...]. In Chapter 5 a type of generalized wave function is discussed.

Thesis Files