1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

272 CHAPTER 7 • TAYLOR AND LAURENT SERIES

series in Equation (7-22) involving the positive powers of (zo -a) is actually the
Taylor series for f. The Cauchy-Goursat theorem shows that the coefficients for
the negative powers of (z 0 - a) equal zero. In this case, therefore, there are no
negative powers involved, and the Laurent series reduces to the Taylor series.
Theorem 7.9 delineates two important aspects of the Laurent series.

The uniqueness of the Laurent series is an important property because the

coefficients in the Laurent expansion of a function are seldom found by using