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Series of functions of a complex variable

Series of functions of a complex variable.

Definition 7.2.1 Let be given a sequence $(f_n(z))_{n \geq 0}$ of functions of a complex variable

. Suppose that all the functions

are defined on the same domain

. The expression

$\displaystyle \underset{k=0}{\overset{+\infty }{\sum}} f_k(z) =f_0(z)+f_1(z)+f_2(z)+ \dots + f_n(z) + \dots$

is called the infinite series with as its general term.

In fact for every $z \in D$ , we definehere a series of complex numbers.

Definition 7.2.2 The sequence $(S_n)_{n \geq 0}$ whose general term is $S_n=\underset{k=0}{\overset{n}{\sum}} f_k = f_0+f_1 + \dots + f_n$ is called the sequence of partial sums of the given series.

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