The book comprises six chapters, each meticulously structured to build a comprehensive understanding of complex analysis. Chapter 1 covers the most fundamental concepts, providing the essential groundwork that permeates the entire text. Chapter 2 delves into normal families, the Riemann mapping theorem, and conformal mapping. Chapter 3 focuses on the zeros of analytic functions, while Chapter 4 explores the essential properties of harmonic and subharmonic functions. Chapter 5 introduces H^p spaces and the Fourier transform, highlighting their interconnections. The final chapter discusses uniform approximation by rational functions.Emphasizing foundational theories, modern methods, and key ideas in complex analysis, this book also presents some cutting-edge research problems and recent advancements. The topics are thoughtfully selected, and the exposition is clear and rigorous. This book is an excellent resource for graduate students and independent learners alike. It features many new and concise proofs of classical theorems, and offers numerous challenging exercises to deepen understanding.