Advanced Engineering Mathematics, 7th Edition

19.3 Laurent Series

INTRODUCTION

If a complex function f fails to be analytic at a point z = z₀, then this point is said to be a singularity or a singular point of the function. For example, the complex numbers z = 2i and z = −2i are singularities of the function f(z) = z/(z² + 4) because f is discontinuous at each of these points. Recall from Section 17.6 that the principal value of the logarithm, Ln z, is analytic at all points except those points on the branch cut consisting of the nonpositive x-axis; that is, the branch point z = 0 as well as all negative real numbers are singular points of Ln z. In this section we will be concerned with a new kind of “power series” expansion of f about an isolated singularity z₀. This new series will ...