In the 3rd complex analysis video I would like to introduce the polar form of a complex number. It may seem a little odd to bring this in so early in the series, but I think it will help me greatly when I cover multiplication and division. Multiplying functions is easier to comprehend geometrically if you think in polar coordinates. Secondly we take quick look at Euler's identity. You don't really need to understand how it is derived just yet, because we will cover the exponential function in a later video. However I feel that the exponential form needs a little explaining, otherwise it would seem to come out of nowhere. Lastly, we develop our visualization tools a little by looking making enhanced phase portraits. We can add contour lines for locations of equal magnitude and argument. Take the time to practice reading the cosine phase portrait with different constants added to it. 2D phase portraits are quite a useful way to visualise and understand complex functions. (The promised video of extra phase portraits is coming soon) In this video: 00:00 Introduction 00:46 Polar Coordinates 03:10 How to represent the polar form. 04:56 Radians (a recap) 06:39 Examples 08:00 Euler's Identity 10:47 Enhanced Phase Portraits 14:52 3D Phase Portraits. In this series: 1 - https://www.youtube.com/watch?v=jU7QW6AjUf4 Introduction to Complex Numbers. 2 - https://www.youtube.com/watch?v=nT3WYFxvPLk Adding and Subtracting Complex Numbers 3 - https://www.youtube.com/watch?v=O3aJCGbyfR8 Polar Coordinates of Complex Numbers 4 - [Coming Soon] Multiplication of Complex Numbers and Functions 5 - [Coming Soon] Division of Complex Numbers and Functions 6 - [Coming Soon] Complex Differentiation and Analytic Functions Extra Visuals (No Commentary): 1 - https://www.youtube.com/watch?v=3qEJeP6qQGA Trigonometric Functions Visualised (3D) 2 - [Coming Soon] Phase Portraits of Trigonometric Functions