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