Embark on a comprehensive journey into complex analysis with this introductory lecture. Explore the foundations of complex numbers, including their representation as z=x+iy and their role in solving differential equations. Master basic arithmetic operations with complex numbers, such as addition, subtraction, multiplication, and division. Delve into Euler's formula and gain insights into complex numbers in polar coordinates. Understand the concepts of radius and phase angle, and get a glimpse of future topics in the course. Perfect for mathematics enthusiasts and students seeking a solid grounding in complex analysis fundamentals.