19 The Taylor seriesг Uniform converges of series es

Let an и 31 be complex numbers We recallthat я series

a

204по

is convergent it the sequence of itsparti trial sums

и

Snr Е АкKIO

has a finite limit S This limit is

called the sum of the series

А functional series

ЁКАwith the functions fu defined on a setМ ЕЙ converges uniformly on И ifit converges at all 2 c И and moreover

Его 3 c I s.t.tn N

II t.it I lfH Etxlt1IsE V zc M

wherefez к ЕЁ Ault

Example 18.1 Consider the series

Ё Е 41 1 19.1

InWe remark that series

rs.zw.lt L 19.1 converges forх

every 2 c М Indeedне It 171 El б

и

E г 2 Ее ZKIO

E Sn 2 1 2 2 1 2

111 E I Еи 11

Sn 1 1 1 ч сорин tisincn.ly1 E

I 21 where 2 rfosetis.ie1 E

But series 19.1 converges uniformly onlyon Me for any б о

Indeed

III ELIE ЕЕ

1Е 12 1 бE о

п го I

V X C МбSo 118.1 converges uniformly on Мб

Assume that 19.1 converges uniformlyon М then for any Е 20

3 c Ж s.t.tn 2 N

Ён.tk II7ae eI

1TukeZX x iO хго Thenи 11

IEI.EE ESo inequality 119.2 does not hold forх closed to 1 Consequently 119.1does not converges uniformly on

И LZ 12 121

Exercise 18.1 Show that the series

2 t.lt 19.3по

converges uniformly on М it the series

2 КАКи о

converges where kfnlt suplt.CZ2ЕМ

Solution Let 514 211121 и ЁЛКАwe first estimate for man

Knelt snltll IEI.CZ E

EI.tt ка't а ЁЛКИ и 2 c М

We fix Его

Since the series ЁНfull converges thereexists Nt I S t.tn т з И нам

т

2 ИДИ с EК не 2

by the Cauchy criterion see Th 19.3 Math I

So1517 Salt а А 2 ЕМ

V n.mr.NO

Making т we have

1517 Salt c Е 2 ЕМи из NoThis implies the uniform converges of48

2 The Taylor series

Theorem 19.1 Let t be holomorphic in Vand t.tv Then the function f пикуbe represented as я sum

114 cult to

inside anydisk В Z

t.ICRICVO21 li tIz Z I erik and denote by13 13 to г

The integral Cauchy formulaimplies that

На till азWe write

ну 1544 t.DE.EE

we multiply both sides by 113and integrate the series term wise alongК The series 29.4 converges uniformlySince

ЕЁ.tt gcIand

ЁршConsequently the term wise integrationis legitimate and we obtain

114 17 t.EE lt toT

2Cn z Zo

where113 d

a tilк Д 2 уж

I 1.2

DEDef 19.1 The power series

7 Cult to 19.5

where c f1 19.6I f и

is the Taylor series of the functionat the point to or centered at to

The Cauchy inequality Let the function fbe holomorphic in a closed disk

Бч L It tolerand let its absolute value on the circle ктоbe bounded by a constant М Then

161 E I и 0,1Ч I

Proof From 19.6 we have

ЛЕ II t.o I.LIМчп

Theorem 19.2 Liouville If the function Iis holomorphic in the whole complex planeand bounded then it is equal identicallyto a constantProof According to Th 19.1 the function fmay be represented by а Taylor series

tltl E.at

in anyclosed disk Бк 112 1 ER Кст

with the coefficients that do not dependon R Since f is bounded on в we

have 111211 EM.tt C в

Ву Cauchy inequalitiestu 0,32

Cult о R го

Therefore С L О and

f 7 СоВ