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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)?

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    $\begingroup$ 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). $\endgroup$
    – hmakholm left over Monica
    Oct 5, 2011 at 19:38

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There is a nice brief history of complex numbers at University of Alberta MATH department. I also found a very nice historical resource compiled here, and of course Mathematics and Its History by John Stillwell is a very good source.

From what I have been able to gather by doing some basic perusal is that while imaginary numbers are attributed to Cardano, Bombelli did formal systematic analysis in 1572.

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 $$\int \! {1 \over {1+ax}} \mathrm{d}x = {1 \over a} ln(1+ax) + C,$$

we get an inkling of the timeline when a relationship between the two can be seen in Western mathematics. Euler knew about these relationships as well. Wikipedia article has better explanation and references regarding this.

Let me know if you find something profound in there somewhere.

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