Follow-up to: Mathematical reason for the validity of the equation: $S=1+x^2S$ and General question on relation between infinite series and complex numbers
(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)?