today Ahlfors 0h5 53 85

83 Jensen formula

Say f is hold on IDF and assume fco 70and Arias qE Dplo are the zerosof f repp.esfjeyd

log Ifiosl log1 1 So bg fipeiosldo

last time pal loglal logtalo

Hadamard's 7hmrelates the number of zeros with the growth rate

of a function

e iz e iz

Ex fez sincz Ti

fca has roots at ne k

lfczsleleizltle.it eelZI2

FI na

Ucr of roots of fc inside Her

then for f since Va

ite un a a

linear inn

Mcr sup If I sap IfeelHer 121 r

thenfor

fat since µcn e ezhewing

concepts a genus horder of growth

Let f Q E be an entire functioningthe genus h of f is the smallest integersuch that we have h terms

fax e ftp.t.fany.ettaEIt tfaIh

and gtfo is a polynomial of degree e h

Ee fCE eZZ h _2

f z sincz h L

1 related to Hw 3 IT I E e1is convergent

Det order X of a function f

a inf a l Mor e er realnumber

Ee f Z e 11 2

true Hadamard If f is an entire function then

h E X E htt

Fremark if X is not an integer then h is

L Uniquely determined

55 Normal FamilyA family of hot's fuentions over an open

set I with some common properties and we

are looking for limits for a segue of functionsuQopen

subset

Let's consider functions from SL valued

in a metric space S

Recall a meteor space 5 is a set witha distance function dlx.DE Rzo X YES

di x g o x y

yo dcx.ly day 23 3 dex 27

d x g dly X

i Euclidean metric on

geRhdcx.yCxiyi5t CXnyn5

Sphere S E Rs unit spheregeodesic

find a tight line between

yi I

and y on the sphere

n measure its length

here the geodesic connecting andywill be a segment of a greatcircle passing through x y

Cl has 2 metresE t R2 Euclid

Q E E E S2 use the spheremetric metric

IRIE space generalO metric space

A function f I IS d continuous

Map SL S this is a setcan we put a metro on this

set

several different ways lepers

LP distance f.SE MapCA S

p f g dffcz.i.gczDJP.dk f

measurehere

T distance supnorm

Pcf g sap d fat gas2 Er

if we use E distance as metric on Maplethen fa f converges it means

fu f uniformly on R

If we want to describea

fu fon every compact subset F CSL howtoconstruct a metre

Construction of such metric in 3 stepsD Find an exhausting Seg of compact subsets

sE C Ez C Es C Eic R R U Ei

1

For example construct Fok sitEk 7 Erl 2 d a

ex a if D Q F 9121 Ek

t.isa.is

radius and alsoapproach 252

27 Modify the metric on S so thatS has bounded diameter d

He

X'YESgcx.ggd

a II 1 day

one can cheek S S also satisfiesthe distance function conditions

37 For any f g E MapCS4S7potdtafEeimDiPEsCf.gD ftp.gcfca.gczD.el

s

Dt an

Map Ris P is a metric space

chats P is a distance functioncm Pcf g o f g r

I pcf.gg so Pee f g k k

f g on Ek FkD 4 D tfEe e f g on R

2 p f g peg f131 pCf g tpcg.bz pcf h PE ai

satisfies theseproperties

check i If Tfn is a seq of fan in MapleDthen for f w r t the P distance

S fa f uniformly for everycompactsubset F Cr

PI since F Cr D U EkE If NED

i E is compact E EMEK for Klarge

i e E C T for some KEI why enoughK

largeenough K

want show PE f n f 0

but E C Ek PE fu f c PekCfn f

Pffn f zPe Ifn f Iz Pez fn f t

Peffniff t feet

PeaCfn fn E 2K P Cfn fbut K is fixed and PC fu f 0 byassaptia

PECfa f Pe Cf f E 2K fcfn.FI 20as n p

we have uniform convergence for enemyEk We need to show f e I N St t a N

Pff f EFind Ro large enough sit 2

ko s

then

Pffn f E EEE 2h Pedfn f I iEk

E'iii Peifnfs 17

isreno

torn w E 2k 2K E Ez

E Ez Lt Ez E