Complex Analysis/Function series, power series, Euler's formula, polar form, argument
Series of complex functionsEdit
Given a sequence of (possibly holomorphic functions) we can, in the case of convergence, form the series
which depends on and may be seen as a complex function, where domain of definition equals domain of convergence. We first note the obvious definitions (we count from zero since this will allow for the important special case of power series without any modifications).
Within the interior of the domain of convergence, the complex differentiability of all the implies the differentiability of the corresponding series, and moreover we may differentiate term-wise.
In this section, we specialize the considerations of the previous section down to the case , where is a sequence of constants.