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COMPLEX ANALYSIS1 Douglas N. Arnold2
 

Summary: COMPLEX ANALYSIS1
Douglas N. Arnold2
References:
John B. Conway, Functions of One Complex Variable, Springer-Verlag, 1978.
Lars V. Ahlfors, Complex Analysis, McGraw-Hill, 1966.
Raghavan Narasimhan, Complex Analysis in One Variable, Birkh¨auser, 1985.
CONTENTS
I. The Complex Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
II. Elementary Properties and Examples of Analytic Fns. . . . . . . . . . . . . . . . 3
Differentiability and analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
The Logarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
Conformality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
Cauchy­Riemann Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
M¨obius 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

  

Source: Arnold, Douglas N. - School of Mathematics, University of Minnesota

 

Collections: Mathematics