# When - during the study and development of- - and how were complex numbers introduced in the study of [real-valued] power series (expansions)?

(This question seems broad at this stage, but I will (re-)edit as answers and responses come by.)

It is said that complex numbers play a central role in the study of power series.

My question: When - during the study and development of- - and how were complex numbers introduced in the study of [real-valued] power series (expansions)? Specifically, what was the first motivation to introduce complex numbers to study power series (expansions)?

• I don't know much about the historical development, but one clearly central role is that the radius of convergence of a power series is determined by all singularities in the complex plane -- and the radius of convergence again determines which real interval the series converges for (modulo inclusion/exclusion of the endpoints). Oct 5, 2011 at 19:38

It is my personal inference from cursory reading that $${1 \over (1-x)} = \sum_{n=0}^{\infty}x^n \quad \forall \|x\| \lt 1$$ combined with Bernoulli's observation (circ. 1702) that $${1 \over {1 + x^2}} = {1 \over 2} \left({1 \over {1 - ix}} + { 1 \over { 1 + ix }}\right)$$ and that $$\int \! {1 \over {1+ax}} \mathrm{d}x = {1 \over a} ln(1+ax) + C,$$