Intro Complex Analysis, Lec 16, Taylor Polynomials, Complex Exponential, Trig & Hyperbolic Functions

Lecture 16. (0:00) Goals and a demonstration of what happens near where the derivative of an analytic function is zero. (5:42) Visualizing the amplitwist based on the modulus of the derivative and the argument of the derivative (based on Tristan Needham's "Visual Complex Analysis"). (9:35) Find a (finite) Taylor expansion of a polynomial function about z = 2 in two ways: 1) substitution, 2) Taylor's formula. (20:07) Comments about functions versus multi-valued functions. (21:20) The complex exponential function: formula, periodicity, mapping properties. (28:05) The complex cosine and sine functions: derivatives, identities, modulus of cosine along the imaginary axis. (39:34) The complex hyperbolic functions: identities, relationships to trigonometric functions, geometric/numerical demonstration of sin(iz) = i*sinh(z). (46:15) Introduction to the complex logarithm

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