z.IE l eRecED.Cefz e3Pt3iO henz3e3Pt3i0.3Pt3i03PRele e cosczo feel e costacosta

ISo for 30E f Iz t 21h cos 8 o

O E n t t E CBOCso OIT E6 It

Iii niShaded region has cosho O

7T and the unshaded regionI6 has cosczo LO

fG I grows as 171 70 with cosc30 OIft l decays as 121 30 with CBC303 0

y I 1 h CBCIfcz oscillates if cos 03 0

The asymptotic behavior willnot change if we consider Pcz7esince there is an R 0 and C o sit for all El RI.tt ns1PCz3l E C 121 where n degCPFor any fixed 0if cos3070

limeIPA est 7 figg f EP eeoso lime

e'Pasospz eeti0 p pp o P

if cos30 so

fing IPA e l E Liff c een.eeosoc.ligyee3Poso nIo

as Iipm Paso np D since ouzo sotoo

if cos30 0 then le I L and lim le't Past limpest p12120 12136So the grow or decaybehavior is roughly unchanged except in the

interface where oscillation occurs

2 2 D or as E w o is an essential

singularity since as 171 70 Hal is neither boundedor simply go to hence 2 P is neither a removable

singularity nor a pole hence 2 0 is an essential singularity

2 fez On E faz has

Zeros 2 1 2 3poles Z Z 2 D

each of them has order 1

Near 2 L we have Taylor expansion

let u Z l then

flatus taffeta u 1 atu 2 u l UU 2 a Hutu'tU 212112W J y city inu 2 1 ut

2ut U2

Near 2 P use w Yz and expand around w othose fits i E IFw

To I w l 3W l12W t DIwc I 2W1th 2 t

FI Find the winding number around 0 for the followingcurve

a f f C C unit circle fc z3 EZus r fez HII7 45

Solin the winding number is

a ner or if two dw dzwEfcc

numberof zeros of food inside Cof poles of f inside c

3 O 3since the roots are 7 0 2 IFL

2 her o zero of polding of f in C2 2 0

Ht For any point 2 ED away from the poles we havea small disk DECZ that is freefrompoles ZiZz 3

2a

polesZeo

El Zz Zz f23 3

Dada i

Then for any Z E DIZ such that12 Znl E A Enlant

Then Ifczyl E En TEY E T En tant kAgAnd If't't'll E EE Tain e IT A CD

Hence f is holomorphic in a neighborhood Dscz of Lfor any E ID I Zi Zz iZn

gleio joiI y Let Flo Z be a functionon 0,212 X D then by Thur5.2 in Ch2 we know

fat f dei z f et.fi ei0idoOoJo HO Z do

is a holomorphic function on D

Similarly view FcQZ as a function on0,21T x IC DT is hot'c in Z continuous in

both variables hence fit is also holk on 14213

Let C Eff 1gal then for any121 1 we haveg

f glfczsltfswyp.IT nteuIIuI e o as so

2 and CD The story goes as follows

we first consider 8th Z for ZEC ItuNEIL

then forn 30 fez

HIEIO 121 I

mo fcHeo

zn 12171

In general for ZOE G we have the difference ofthe two boundary values from inside and outside being ga7 fcr.to Liff fcr.to gtfo

This is clear if gas is a finite linear combinationof Z got Az En't Ar Zn't AN 2nThis is also true more generally for any gotsmooth function on C or even just continuous function