What is a good complex-analysis textbook for someone without too much real analysis background?

Question

I'm a sophomore in HS and I've been self-studying pure maths for about 6 months now. I really enjoy complex analysis and have had a lot of fun tackling dogbones, keyholes, and even evaluating the basel problem and fresnel integrals, as well as understanding how important results such as Jordan's Lemma, Cauchy's Integral Theorem, and residues at infinity are derived. I only have basic Real Analysis background, but I want to have a solid textbook for self-study. It should have a decent amount of exercises (and solutions), stay semi-rigorous, and not leave anything important out. I don't mind a challenge as long as it doesn't assume I have crazy prior knowledge about Real Analysis and Number Theory. I know most of univariate and a good chunk of multivariate Calc. Thanks!!

Answer 1

I would say that two standard references are Lars Ahlfors Complex analysis, an introduction to the theory of analytic functions of one complex variable and Serge Lang Complex Analysis. They are for a first (but rigorous) introduction to complex analysis. Note that since they are both well-known books, you can find nice exercises and also some of them solved here or online. Ahlfors' book also has a chapter that helps you bridge the possible gap you may have from introductory real analysis to analysis in metric spaces and some multivariable analysis (implicit function theorem...), with a bit of topology of metric spaces that you may need. Another book may be also John Conway Functions of one complex variable, it is introductory, it starts from the very basics of complex numbers and has the needed material for topology of $\mathbb{C}$ and metric spaces. These 3 are basically equivalent, it's a matter of tastes, you may try to have a quick view at some libraries and then pick, also based on the solutions of some exercises that you may want to solve and then check the (suggested) solutions. Sometimes Gamelin's Complex Analysis is suggested since very intuitive and with many applications, but I honestly have never seen it.

Another suggestion may be to follow the syllabi and the websites of some professors that on their website publish the notes of their classes. Sometimes they are well-written and may be useful. You can basically type the name of the university, the name "complex analysis" and see if you find some stuff.