As the title says, I'm looking for a modern, rigorous book on complex analysis to restudy the subject from scratch, hoping to study after Riemann surfaces and their connection with algebraic curves and cohomology.
I took a course long ago on the subject using a dense, elegant French book by Dolbeault, which use differential geometry objects like differential forms and Stokes' theorem.
My background in metric and point-set topology is good, but I lack a solid understanding of integration(both the multiple Riemann and Lebesgue!) and differential geometry, which forces me to admit or not think too deeply about some concepts, such as what a surface is, its orientation, or some regularity arguments of integral functions(why its continue,differentiable,it's has compact support which implies....)
So, I'm looking for a self-contained, modern book in complex analysis that introduces or recall the full statements of all the necessary concepts or theorems he needs from calculus,topology, measure,Lebesgue integration and differential geometry in a rigorous way. Thanks in advance!
(Someone suggest me in another forum a good book called Complex Made Simple ,but i still wish to find others books in the style of Dolbeault book or Berenstein Complex Variables: An Introduction book.)
Edit: There is two forums :What is a good complex analysis textbook, barring Ahlfors's? and Complex Analysis Book [duplicate] asking similar questions but what I'm looking for is some book that is not encyclopedic and talks about differential forms,orientation and their integration,green or stokes theorem ,simply connected sets and state or introduce all the definitions and theorems he needs from calculus and integration like what the type and definition of integration he will use,it's properties and the convergence theorems.....something between Henri Cartan book and Bernstein Carlos book with good appendices about calculus or measure theory he needs.
