By Reinhold Remmert
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Extra resources for Theory of complex functions
Sample text
As in R we have in C the fundamental Convergence criterion of CAUCHY. For any complex sequence (en) the following are equivalent: i) (en) is convergent. ii) (en) is a Cauchy sequence. Given E> 0, choose k k, where c :=limen. Then Proof. - cl + lc- en I < E for all m, n ~ k. Renl :5 lc... -en I , l9c... Ren) and (9en) are each Cauchy sequences. Because of the completeness of R, they converge to numbers a and bin R. l the sequence (en) in C converges to a + bi. 0 The notion of a Cauchy sequence can be defined in every metric space X: A sequence (en), en EX, is called a Cauchy sequence in X if for every E > 0 there is a k.
Because max(IRal, l~al) :5 lal :5 IRal + l~al, it further follows (from the majorant criterion) that The complex series :L av is absolutely convergent precisely when each of the real series :L Rav and :L ~av is absolutely convergent. 3. The rearrangement theorem. If Lv>o av is absolutely convergent, then every "reafTangement" of this series afso converges and to the same limit: for every bijection T of N. Proof. The proof that most readers have doubtlessly seen for R works as well in C. It runs as follows: Let s := Lv>O av.
J and infer that for an appropriate finite constant C, lanl :::; Cbn for all n. a:: Exercise 4. Let (lln)n~o and (bn}n~o be sequences of complex numbers and suppose that i} the sequence of partial sums Sm := E:':o a,. is bounded; ii) lim bn = 0; iii) the sum E:;"= 1 Ibn - bn-11 is finite. bn is convergent. Hint. b,. _I)b,. for n > o. Exercise 5. To each (m, n) E N2 associate a complex number am,n· Suppose the numbers am,n are "somehow" organized into a sequence c,.. Show that the following statements are equivalent: i} The series E Ck is absolutely convergent.
