Complex Analysis
نویسندگان
چکیده
I. The Complex Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 II. Elementary Properties and Examples of Analytic Fns. . . . . . . . . . . . . . . . 3 Differentiability and analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Conformality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Cauchy–Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Möbius transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 III. Complex Integration and Applications to Analytic Fns. . . . . . . . . . . . . 11 Local results and consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Homotopy of paths and Cauchy’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Winding numbers and Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . .15 Zero counting; Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Morera’s Theorem and Goursat’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 IV. Singularities of Analytic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 Residue integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 V. Further results on analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 The theorems of Weierstrass, Hurwitz, and Montel . . . . . . . . . . . . . . . . . . . . . . . 26 Schwarz’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 The Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Complements on Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 VI. Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 The Poisson kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 Subharmonic functions and the solution of the Dirichlet Problem . . . . . . . . . 36 The Schwarz Reflection Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39
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