Chapter 4 Complex Series §4. I Series §4.2 Power series §4.3 Taylor's series §4.4 Laurent series
§4.1 Series §4.2 Power Series §4.3 Taylor’s Series §4.4 Laurent Series Chapter 4 Complex Series
§4. 1 Series 1. Complex Sequences Def. 4.1.1 Sequences a,=(a,+ib,),n=1,2,,there exists a complex numbera=a+ib.a> converges toa, or a is the limit of thean),if VE>0,3N(e)>0, such that n 2 N implies that a -a) The limit a is unique
§4.1 Series 1. Complex Sequences ( ) ( ) Def. 4.1.1 Sequences = , =1,2,...,there existsa complex number = , converges to , or is the limit of the ,if 0, 0, such that implies that . Denoted as lim or . n n n n n n n n n a ib n a ib N n N n → + + − = → → The limit is unique.
The following equations a|00 n→00
Thefollowingequations n n n n n n n n a a a a b b b b a a b b − − − + − − − − + − TH 4.1.1 ( ) ( ) Complex series = =1,2,..., converges to = and . That is lim iff lim and lim . n n n n n n n n n n n a ib n a ib a a b b n a ib a a b b → → → + + → → → = = + = =
2. Complex series Lan=(a,+ib(n=1, 2, >sequences The sum of the sequences ∑an=a1+a2+a3+…+an+…> series The nth partial sum of the series, usually denoted by s is the sum of the first n terms that is n=∑ C2=C1+a,+.+C
2.Complex Series ( ) 1 2 3 1` = 1,2,... sequences The sum of the sequences: ... ... series n n n n n n a ib n = + = → = + + + + + → 1 2 1 The nth partial sum of the series, usually denoted by , is the sum of the first terms, that is ... . n n n i n i S n S = = = + + +
Def: a)If the sequence of the partial sumsSn has a limit S that is lim S,=S, then the series is said to converge n→00 and s is called as the sum of the series and we write that am=Sn n b)a series that does not converge is said to diverge c) a series∑ m-1 Is said to absolutely converge ∑ a. converges
Def: a) If the sequence of the partial sums has a limit S, that is lim , then the series is said to converge, and is called as the sum of the series,and we write n n n S S S S → = 1 1 1 that . b) A series that does not converge is said to diverge. c) A series is said to absolutely converge if converges. n n n n n n n S = = = =
3. Criterion TH41.2 ∑an=∑(an+1b) converge s∑ a and∑ b, converge TH4.1.3 ∑ an converges→an,>0n→∞) C collar an→→0(n→>∞)∑ an diverge TH4,1,4 ∑oe2 a. converge
3.Criterion n 1 1 n=1 n=1 ( ) converge and converge. n n n n n n a ib a b = = = + 1 converges 0( ) n n n n = → → n TH 4.1.2 TH 4.1.3 Corollary 0 ( ) n → 1 disverge n n = TH 4.1.4 1 converge n n = 1 converge n n =
84.2 Power Series 1. Function series Def.4.2.1(,(=)(n=1, 2, is a complex function sequences, defined on complex domain d ∑m()=f(-)+/(-)+…+m()+ n=1 is called complex function series yZo∈D,20 is the converge point of∑()i∑f(= 0丿 converge. Convergence domain=zE DZf(=) converge S(a)=>/(), 2 convergence domain, is called sum function
§4.2 Power Series ( )( ) ( ) 1 2 ( ) ( ) ( ) 1 Def. 4.2.1 1,2,... is a complex function sequences,defined on complex domain D, ... ... is called complex function series. n n n n f z n f z f z f z f z = = = + + + + 1.Function Series ( ) ( ) ( ) ( ) 0 0 0 1 1 1 1 z D,z is the converge point of ,if converge. Convergence domain= | ( ) , ,is called sum function. n n n n n n n n f z f z z D f z converge S z f z z convergence domain = = = = =
2. Power series Def. a series of the form ∑c1(2-=)=c+c(2-0)+…+n(z-=)y+ n=0 1s called as a power series Specially, when zo=0, we get ∑ Cz=C+C2++cz+ The complex constants c, are the cofficients of the power series
0 0 1 0 0 0 0 0 1 0 Def. A series of the form ( ) ( ) ... ( ) ... is called as a power series. Specially,when 0, we get ... ... The complex constants n n n n n n n n n n c z z c c z z c z z z c z c c z c z = = − = + − + + − + = = + + + + are the cofficients of the power series. n c 2.Power Series
Ouestions: 1. For what values of z does the series converge? 2. Is the sum an analytic function? 3. Is the power series representation of a function unique?
Questions: 1.For what values of z does the series converge? 2.Is the sum an analytic function? 3.Is the power series representation of a function unique?
TH 4.2. 1(Abel theory) 1)∑cn= converge at=≠0 →∑cn" absolutely converge on n=
TH 4.2.1 (Abel theory) 0 0 0 0 1 0 1 0 1). converge at 0 absolutely converge on ; 2). diverge at diverge on . n n n n n n n n n n n n c z z c z z z c z z c z z z = = = = 1 z x y . O 0 z
