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<8 Denoted as lim am= a or an→a(m→>) 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|<|a.-a+b.-b bn-b≤{xn-a|≤n-al+1-b → TH4.1.1 Complex series am=a, +ibn)(n=1, 2,converges toa=+iban→ a and b→b(n→∞) That is lima =a=atib iff lim a =a and lim b=b n→ n→>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 = = = =