Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology?

Solution 1:

A book that makes a real effort to separate the analytic and topological aspects of complex analysis is Steven Krantz and Robert Greene's Function Theory of One Complex Variable. Like most of Krantz' other textbooks,it's extremely well written and takes the student deep into a graduate course in complex analysis.The prerequisites are minimal,too-just a good careful background in calculus of one and several variables.Indeed,one of the original touches of the book is the authors emphasize the connections between multivariable calculus in the plane and complex analysis in one variable.

I think you may find it exactly what you're looking for. Also, I'd like to second the recommendation of Jones and Singerman-it's a wonderful book focusing on a modern presentation of the relationship between geometry and complex analysis. Krantz/Greene generally focuses on the purely analytic aspects of the subject-as a result, the 2 books complement each other beautifully.I think that combination will work very well for a graduate course.

Solution 2:

Strictly speaking the answer is "no", there is no such book. But this is perhaps a good approximation to what you are asking:

MR0890746 Jones, Gareth A.; Singerman, David Complex functions. An algebraic and geometric viewpoint. Cambridge University Press, Cambridge, 1987.