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