Intro Complex Analysis, Lec 12, Cauchy-Riemann Eqs (Rectangular & Polar), Intro Harmonic Functions

Lecture 12.
(0:00) Announcements.
(0:22) Lecture topics.
(0:59) Review differentiability and analyticity.
(3:19) Derive the Cauchy-Riemann equations symbolically (their necessity when differentiability is assumed). Mention the sufficiency of the Cauchy-Riemann equations when the partial derivatives are continuous.
(16:28) Check the Cauchy-Riemann equations on Mathematica for complicated examples.
(24:53) Geometric meaning of the Cauchy-Riemann equations with contour maps (of level curves) on Mathematica, both in terms of local linearity and partial derivatives as rate of change, and in terms of orthogonal trajectories.
(33:26) The polar form of the Cauchy-Riemann equations. Derive with the Chain Rule from Multivariable Calculus and the conversion equations for polar coordinates.
(47:11) Attempt to confirm this using level curves and linear approximations.
(55:25) Analytic functions have real and imaginary parts that satisfy Laplace's equation. They are called harmonic functions.

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thank you for sharing this series of lectures, really helpful