Intro Complex Analysis, Lec 13, Preimages, Laplace's Equation, Harmonic and Analytic Functions

Lecture 13. (0:00) Topics for the lecture. (0:32) Level curves of the real and imaginary parts of a complex-valued function are preimages of horizontal and vertical lines under the mapping. (3:26) Symbolically confirm that hyperbolic functions parameterize hyperbolas. (6:33) Animate hyperbolas being mapped under f(z) = z^2. (12:13) Review Cauchy-Riemann equations in polar coordinates and check them for f(z) = z^3 in both rectangular and polar coordinates. (18:21) Laplace's equation (a partial differential equation), notation, overview of applications, harmonic functions, and connections to analytic functions. (27:49) Generate a harmonic function by taking the real part of an analytic function. (30:56) Find the harmonic conjugate by integration and the Cauchy-Riemann equations. (37:34) Example to illustrate a solution to Laplace's equation along the closed unit disk, using polar coordinates to describe the boundary conditions, and then imagine it to be a temperature distribution. (49:41) Infinitesimals, complex derivatives, and the amplitwist concept (due to Tristan Needham).

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