Complex Number Algebra as a Simple Case of Heaviside Operational Calculus

Abstract

In [1] the author offered the following postulational basis for an algebraic structure called a heaviside operational calculus. Let G be an additive Abelian group of operands. Let E be a set of endomorphisms on G. If a nonnull element, U, of G has the property that every element of G is the map of U under some endomorphism in E, we call U a unit element of G relative to E. Suppose G and E satisfy the following postulates: 1. The endomnorphisms in E are permutable. 2. G contains at least one unit element, U, relative to E. 3. Any endomorphisin in E that sends a unit element U into a nonnull operand sends every nonnull operand in G into a nonnull operand. It follows, as shown in [1], that E is an integral domain under the usual definitions of equality, sums, and products for endomorphisms. For some choices of G and E, E will also be a field. G and E (or G and a class of operators associated with E) determine a heaviside operational calculus. For example, let G be the class of entering, sectionally continuous functions of t. The unit step function, {u(t) }, will serve as a unit operand for postulate 2. Given a member, f, of G we find the following endomorphism on G that sends u intof: