Complex_Analysis-Deitmar.pdf

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COMPLEX ANALYSIS
ANTON DEITMAR
CONTENTS
1 THE COMPLEX NUMBERS
3
2 HOLOMORPHY
7
3 POWER SERIES
9
4 PATH INTEGRALS
14
5 CAUCHY'S THEOREM
17
6 HOMOTOPY
19
7 CAUCHY'S INTEGRAL FORMULA
25
8 SINGULARITIES
31
9 THE RESIDUE THEOREM
34
10 CONSTRUCTION OF FUNCTIONS
38
11 GAMMA & ZETA
45
1
COMPLEX ANALYSIS
2
12 THE UPPER HALF PLANE
47
13 CONFORMAL MAPPINGS
50
14 SIMPLE CONNECTEDNESS
53
COMPLEX ANALYSIS
3
1 THE COMPLEX NUMBERS
PROPOSITION 1.1 THE COMPLEX CONJUGATION HAS THE
FOLLOWING PROPERTIES:
(A) Z + W = Z + W,
(B) ZW = Z W,
(C) Z 1 = Z 1 , OR
Z
W
=
W ,
(D) Z = Z,
(E) Z + Z = 2RE(Z), AND ZZ = 2IIM(Z).
Z
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COMPLEX ANALYSIS
4
PROPOSITION 1.2 THE ABSOLUTE VALUE SATISES:
(A) JZJ= 0,Z = 0,
(B) JZWJ=JZJJWJ,
(C) JZJ=JZJ,
(D) JZ 1 J=JZJ 1 ,
(E) JZ + WJJZJ+JWJ, (TRIANGLE INEQUALITY).
PROPOSITION 1.3 A SUBSET AC IS CLOSED I FOR EVERY
SEQUENCE (A N ) IN A THAT CONVERGES IN C THE LIMIT
A = LIM N!1 A N ALSO BELONGS TO A.
WE SAY THAT A CONTAINS ALL ITS LIMIT POINTS.
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COMPLEX ANALYSIS
5
PROPOSITION 1.4 LETODENOTE THE SYSTEM OF ALL OPEN SETS
IN C. THEN
(A);2O, C2O,
(B) A; B2O)A\B2O,
(C) A I 2OFOR EVERY I2I IMPLIES
S
I2I A I 2O.
PROPOSITION 1.5 FOR A SUBSET KC THE FOLLOWING ARE
EQUIVALENT:
(A) K IS COMPACT.
(B) EVERY SEQUENCE (Z N ) IN K HAS A CONVERGENT SUBSEQUENCE
WITH LIMIT IN K.

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