fez - courses.wikinana.org
Transcript of fez - courses.wikinana.org
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