University Mathematics Handbook (2015)

XII. Complex Functions

Chapter 6. Taylor and Laurent Series

6.1  Regular and singular points of complex function

a.   is a regular point of function if the function is analytic in that point.

b.   is a singular point of function if the function is not analytic in any neighborhood of a.

c.  If a function is analytic in domain than each point of is regular.

6.2  Expansion of Taylor series

a.  Let be analytic on . Let be any point on and be a circle with its center at and radius contained in . Then, there exists a unique power series, with a radius of convergence of at least converging to , which is

(*)

where , , is the maximum value of on circle .

(*) is a Taylor series of at a.

In the case of , (*) is Maclaurin series.

b.  The radius of convergence of Taylor series at a, converging to function , equals to the shortest distance between point and the singular point of closest to .

6.3  Zeroes of Analytical Function

a.  Point is called a zero of if .

b.   is a zero of order of function if

c.   is a zero of order of function if and only if can be represented the following way:

where is an analytic function on and .

d.   is called isolated zero of function if there exists a neighborhood of which does not contain further zeroes of .

e.  The zeroes of a non-zero analytic function are isolated.

6.4  Laurent Series

a.  A two sided series containing positive and negative powers of

is called a Laurent series.

The first series on the right side is a power series converging in circle . The second series in the same side, after substituting , turns into a power series converging in circle . Therefore, the second series converges at . If, in addition, , then the domain of convergence of a Laurent series is the common domain of convergence of the two series . Otherwise, there is no point where the series converges.

b.  If is analytic in ring , then there exists a unique Laurent series converging to in that ring, with the coefficients