Oct 7 , 2020
II f : u - G is holomorphic provided + 2- C- U
F'Cz) = him fcz → b) - 7- cz )
I → o '
Hag __→÷ ( Cauchy Riemann )
Def T : R2 - R2 R - linear
, invertible
.
v→ c- 1122
¢ Cri , E) = KITE,TÑ)
preserving ( unless a = b = o )
.
, b) =/ co,o)
a = a +bi -b a
⇒ Tu - Tr =
U . V
cos# 4,0) = -
11211 . 11011
=> 4 (u.ro) = & Ctu .tv)
(2-7=0 or else - -
⇒
"
I
= =
a- I
$ Power series & Analytic functions e e e. , an c-
is
n =D
Def / Theorem ( Ab- 1) F R o ER so such that
if I -7 - c / < R => (*) converges .
If 0 I r < R ⇒ (* ) converges absolutely & uniformly
in ( e , r )
radius of convergence .
or → v0 .
Definition f : u → so is analytic if * 2- c- U FR >oo
such that is
C- U .
M = 0
, else work 2-
new = 2- - C
.
⇒ Kant < Ig if NIN.
=> Ian 2- "
I < (g ) "
if nzxl .
is is
lfn 1 I ran , I ran < is => [ fn converges absolutely & un.am/y .
n =D
n = 0
n
.
⇒ lanz "
/ > (1¥) "
I => an 2-
→ f ' ri
Theorem (Rudin 8. 1) .
If [ an (2- - c) "
,
.
n -1 => f
.
n 2k
2- = C.
be => as =
, R ) .
Examples : exp , cos, Siri
1- - - - , R = v0 .
= is .1.1
__ g. Yeo )=e '
- 't
⇒ -
'
sins2- → cost2- = 1 .
Beware ! sin2- , cos 2- are not bounded functions as 2- C- E.
Cos init = e- " 'T
2- "
.
log 1 = 0 , 1=211-1
'
,
these are related to the logarithm .
.ir2- a- Z "
continuous function such that
t 2- C- U .
.
.
÷ .
2- c- U
2- = r e #
I
in
Indeed the two logarithms leas and dog 2- differ by
2 a- in => Log 2- - l (Z ) = 21T in .
Set 2- = 1
= =
-