Advanced Engineering Mathematics, 7th Edition

18.4 Cauchy’s Integral Formulas

INTRODUCTION

In the last two sections we saw the importance of the Cauchy–Goursat theorem in the evaluation of contour integrals. In this section we are going to examine several more consequences of the Cauchy–Goursat theorem. Unquestionably, the most significant of these is the following result:

The value of an analytic function f at any point z₀ in a simply connected domain can be represented by a contour integral.

After establishing this proposition we shall use it to further show that

An analytic function f in a simply connected domain possesses derivatives of all orders.

The ramifications of these two results alone will keep us busy not only for the remainder of this section but in the next chapter as ...