Intro Complex Analysis, Lec 25, Cauchy Integral Formula, Applications, Liouville's Theorem

Lecture 25. The lecture also includes a proof of a theorem that can be used to prove the generalized Cauchy integral formula (once you assume the truth of the Cauchy integral formula). (0:00) Mathematical beauty and Philippians 4:8. (0:54) Animation of the flow of a linear vector field (you could relate these ideas to vector fields generated by analytic functions). (3:23) Cauchy integral formula. (5:58) Generalized Cauchy integral formula. (7:06) Application to evaluation of integrals and check the answers with parameterizations on Mathematica. (19:20) Theorem statement about differentiating a special kind of integral (which can be used to prove the generalized Cauchy integral formula from the ordinary Cauchy integral formula) and an example on Mathematica. (30:47) Verbally describe Liouville's Theorem and its proof. (33:56) Liouville's Theorem can be used to prove the Fundamental Theorem of Algebra (and describe basic idea of proof). (36:12) Verbal description of (two forms of) the Maximum Modulus Principle. (37:44) Go back to some details of the proof of theorem about differentiating a special kind of integral.

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[parthvishnu]

"Liouville's Theorem", explained very well, Sir.