This mathematical study was initially written as a set of notes in complex analysis over the spring and summer of 2013. Its purpose is to clarify the techniques and results of complex analysis by using a dialogue between master and student. Its inspiration is the belief that this form of open conversation can show the reader both the tension and structure of the material. For this reason, dialogue can prove to be a very powerful and eloquent form of mathematical instruction.The book begins with an introduction to sequences and topology before tackling the topics of complex differentiability, Taylor series, contour integration, and the calculus of residues. After all this, complex functions are studied as conformal mappings, leading naturally to the Riemann mapping theorem.Before going on to the proof of this theorem, the focus shifts to the study of heat and Fourier analysis. The techniques of complex analysis are used to aid in studying the Fourier series and transform. The study then returns to conformal mappings in the context of solving Laplace's equation on various regions. After this, the topic becomes the study of special functions, culminating in Riemann's proof of the prime number theorem, and then shifting focus once again to the doubly-periodic elliptic functions. The final chapter is not standard in an introductory complex analysis book, and concerns the study of Theta functions and modular forms. There is a separate appendix of color plots to accompany this book.This work was mostly modeled after Elias Stein's lecture series at Princeton.

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