Properties of Entire Functions

Abstract

We now show that if f is entire and if$g(z) = \left\{ {\begin{array}{*{20}{c}} {f(z) - f(a)} & {z \ne a} \\{f'(a)} & {z = a} \\ \end{array} } \right.$then the Integral Theorem (4.15) and Closed Curve Theorem (4.16) apply to g as well as to f. (Note that since f is entire, g is continuous; however, it is not obvious that g is entire.)We begin by showing that the Rectangle Theorem applies to g.