Complex Analysis Lecture Notes - Mathematics Home

Complex Analysis Lecture NotesDan created these Notes for the courseMath 205A: Complex AnalysisItaught at UC Davis in 2016 and 2018. With a few exceptions, the expositionfollows the textbookComplex Analysisby E. M. Stein and R. Shakarchi (Prince-ton University Press, 2003). The Notes were not heavily vetted for accuracy andmay contain minor typos or errors. You can help me continue to improve themby emailing me with any comments or corrections you am grateful to Jianping Pan, Anthony Nguyen, Christo-pher Alexander, Brynn Caddel, Jennifer Brown, and Brad Velasquez for com-ments that helped me improved the Notes . Figure 5 on page 23 was created byJennifer Analysis Lecture NotesDocument version: April 20, 2018 Copyrightc 2018 by Dan RomikEmail comments and feedback figure: a heat map plot of the entire functionz7 z(z 1) z/2 (z/2) (z).Created withMathematica 10using code by Simon Woods, available Introduction42 The fundamental theorem of algebra53 Analyticity, conformality and the Cauchy-Riemann equations84 Power series125 Contour integrals156 Cauchy s theorem187 Consequences of Cauchy s theorem228 Zeros, poles, and the residue theorem299 Meromorphic functions and the Riemann sphere3210 The argument principle3411 Applications of Rouch e s theorem3812 Simply-connect

Note that “complex analysis” is part of “analysis” — you need to develop facility with such estimates until they become second nature. Second proof: topological proof. Let w 0 = p(0). If w 0 = 0, we are done. Otherwise consider the image under pof the circle jzj= r. Specifically:

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