Intro Complex Analysis, Lec 26, Sequences and Series of Functions, Maximum Modulus on Mathematica

Lecture 26. (0:00) Plan for the lecture. (0:44) Initial example for series: expand f(z) = 5/(3+7z) as a Taylor series about z = 8 by using some algebra tricks and the formula for the sum of a geometric series. (12:32) Sequences of real-valued functions of a real variable and pointwise convergence (examples on Mathematica). (24:55) Definition of convergence of a (complex) series as the limit of a sequence of partial sums (when the limit exists). (30:40) Geometric series example (find the function and find the disk of convergence, (almost) confirming the disk of convergence with the Ratio Test). (38:20) Give an illustration of the Fundamental Theorem of Algebra and the Maximum Modulus Principle on Mathematica, optimizing the modulus over a closed disk by analyzing the behavior along the boundary of the disk. The Extreme Value Theorem guarantees there will be a global maximum value somewhere on the disk. The Maximum Modulus Principle guarantees it occurs along the boundary. Parametrize the circular boundary of the disk to find the global maximum value. There is also a Maximum Principle for harmonic functions.

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