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Power Series, Analyticity and Elementary Functions MATH 566 LECTURE NOTES 2: POWER SERIES, ANALYTICITY AND ELEMENTARY FUNCTIONS TSOGTGEREL GANTUMUR 1. Uniform and normal convergences For a set Ω ⊆ C and a function f :Ω ! C, we define the uniform norm kfkΩ = sup jfj: Ω We say that a sequence ffkg of functions fk :Ω ! C converges uniformly in Ω to f, if kfk − fkΩ ! 0 as k ! 1. Recall that this is equivalent to the sequence ffkg being a Cauchy sequence, i.e., kfk − f`kΩ ! 0 as k; ` ! 1. P P Theorem 1 (Weierstrass M-test). If gn :Ω ! C and n kgnkΩ < 1, then the series n gn converges uniformly in Ω. P Proof. With fk = n≤k gn, we have for ` < k X X X kfk − f`kΩ = k gnkΩ ≤ kgnkΩ ≤ kgnkΩ; `<n≤k `<n≤k n>` which tends to 0 when ` ! 1. So ffkg is a Cauchy sequence, hence converges. P n ¯ Example 2. The geometric series n z converges uniformly in Dr (and also in Dr) as long n as r < 1. To be pedantic, with r < 1 and with functions gn : Dr ! C given by gn(z) = z , P the series n gn converges uniformly in Dr. However, the convergence is not uniform in the P n open unit disk D1, and n z does not converge if jzj ≥ 1. Convergence behaviour of frequently occurring sequences in complex analysis can be cap- tured conveniently by the notion of locally uniform convergence. Recall that U ⊆ Ω is a neighbourhood of z 2 Ω in Ω if there is an open set V ⊆ C such that z 2 V \ Ω ⊆ U. Definition 3. A sequence ffkg of functions fk :Ω ! C is called locally uniformly convergent in Ω if for each z 2 Ω there is a neighbourhood U 3 z in Ω such that ffkg converges uniformly in U. P n Example 4. The series n z converges locally uniformly in D1. Continuity is preserved under locally uniform convergence. In this regard uniform continu- ity is a very strong type of convergence; it naturally preserves uniform continuity. Theorem 5. If ffkg ⊂ C(Ω) converges to f :Ω ! C locally uniformly, then f 2 C(Ω). Proof. Let z 2 Ω, and let U 3 z be an open neighbourhood such that kf − fkkU ! 0. Let " > 0. Then choose % > 0 such that B%(z) ⊂ U, and k such that kf − fkkB%(z) ≤ ". Moreover, let δ > 0 be such that jfk(z) − fk(w)j ≤ " whenever w 2 Bδ(z). Then it is clear that jf(z) − f(w)j ≤ 3" whenever w 2 Bδ(z) \ B%(z). Date: September 28, 2010. 1 2 TSOGTGEREL GANTUMUR One drawback of (locally) uniform convergence is that a rearrangement of a uniformly convergent series can be divergent or converge to a limit different than the original series. So uniform convergence can be compared to conditional convergence of series of numbers. An improvement of uniform convergence that can be compared to absolute convergence of series of numbers is the notion of normal convergence. P Definition 6. A series gn of functions gn :Ω ! C is called normally convergent in Ω if P n P kgnkΩ < 1. Moreover, we say gn converges locally normally in Ω if for each z 2 Ω n n P there is an open neighbourhood U 3 z such that n kgnkU < 1. The Weierstrass M-test immediately implies that every (locally) normally convergent series is (locally) uniformly convergent. It is also obvious from the definition that any subseries of a (locally) normally convergent series is (locally) normally convergent. 2 Theorem 7. Let fk;` :Ω ! C for k; l 2 N, and let σ : N ! N be a bijection. Define the sequence fgng by gσ(k;`) = fk;`. Then the followings are equivalent: P (a) The series gn is locally normally convergent. n P P (b) For each z 2 Ω there is an open neighbourhood U 3 z such that the series ( kfk;`kU ) P k ` converges. In particular, for all k 2 N, the series fk;` is locally normally convergent. ` P P (c) For each z 2 Ω there is an open neighbourhood U 3 z such that the series ( kfk;`kU ) P ` k converges. In particular, for all ` 2 N, the series k fk;` is locally normally convergent. If any (so all) of the above conditions is satisfied, then there holds that X X X X X ( fk;`) = ( fk;`) = gn: ` k k ` n P Proof. First we prove the implication (a) ) (b). Let U ⊂ Ω such that N = kgnkU < 1. P n This obviously implies that for all k 2 N, Mk = ` kfk;`kU < 1. Let " > 0 and let mk be such that P kf k ≤ 2−k". So for any m we have `>mk k;` U X X X X ( kfk;`kU ) ≤ ( kfk;`kU ) + 2" ≤ N + 2": k≤m ` k≤m `≤mk P P P Now we shall prove that g = n gn is equal to f = k( ` fk;`). To this end, let m be such that P (P kf k ) ≤ ", and let f~ = P P f . Then we have k>m ` k;` U " k≤m `≤mk k;` ~ X X X X kf − f"kU ≤ ( kfk;`kU ) + ( kfk;`kU ) ≤ 3": k>m ` k≤m `>mk P Similarly, for sufficiently large p, the partial sumg ~p = n≤p gn satisfies ~ X X X X kg~p − f"kU ≤ ( kfk;`kU ) + ( kfk;`kU ) ≤ 3"; k>m ` k≤m `>mk and so we have kf − gkU ≤ kg − g~pkU + 6": Sinceg ~p ! g and " is arbitrary, we conclude f = g. For the other direction (b) ) (a), we start with the definition of Mk and the condition P M = k Mk < 1 for a suitable U. Then we have for any p X X X kgnkU ≤ kfk;`kU ≤ M; n≤p k≤m `≤m where m is such that f` ≤ mg2 ⊇ σ−1(fn ≤ pg). The equivalence of (a) and (c) can be proven analogously. POWER SERIES, ANALYTICITY AND ELEMENTARY FUNCTIONS 3 P Corollary 8. Suppose that the series n gn converges locally normally in Ω to g. Let N = [kMk be a disjoint decomposition of N, where k is from a countable set and each Mk = fmk;`g P is countable. Then for each k, the series g~k = ` gmk;` converges locally normally in Ω, and P moreover the series k g~k converges locally normally in Ω to g. Proof. Let us say k runs over K ⊆ N, and for each k 2 K, ` runs over Lk ⊆ N. Define the set 2 M ⊆ N by M = f(k; `): k 2 K and ` 2 Lkg, and let fk;` = gmk;` if (k; `) 2 M, and fk;` = 0 otherwise. Then we get the proof by applying the above theorem with, e.g., σ(k; `) = 2mk;` 2 2 for (k; `) 2 M, and σ(k; `) = 2τ(k; `) − 1 for (k; `) 2 N n M, where τ : N n M ! N is a bijection. The above corollary with each Mk having a single element gives the following rearrangement result. P Corollary 9. Suppose that the series gn converges locally normally in Ω to g. Then given n P any bijection τ : N ! N, the rearranged series n gτ(n) also converges locally normally in Ω to g. P P Corollary 10. If f = m fm and g = n gn are locally normally convergent series in Ω, 2 P then for every bijection σ : N ! N, the series p hp with elements hσ(m;n) = fmgn converges locally normally in Ω to fg. Proof. For any z 2 Ω we can certainly find a neighbourhood U 3 z such that X X kfkg`kU ≤ kfkkU kg`kU < 1; ` ` and that X X X X X X ( kfkg`kU ) ≤ kfkkU ( kg`kU ) = ( kfkkU )( kg`kU ) < 1: k ` k ` k ` Note that these imply X X X fkg` = fkg; and ( fkg`) = fg: ` k ` The proof is established upon employing Theorem 7 (b) with fk;` = fkg`. Remark 11. Apart from \local" convergences, one can talk about \compact" convergences, which require a convergence in all compact subsets of Ω. Examples include compactly uni- form convergence (which is often simply called compact convergence) and compactly normal convergence. For subsets of the complex plane, it can be shown by covering arguments that these compact convergences are equivalent to their local counterparts (In fact the equiva- lence holds in any locally compact topological space). Another frequently occurring term is \absolute convergence", which means that the real valued function jfj :Ω ! R enters the discussion. For example, \uniformly absolute convergence" designates the uniform conver- P gence of the function series jfnj. Note that this is in contrast to the normal convergence Pn where we consider the series n kfnkΩ of numbers. In particular, normal convergence implies uniformly absolute convergence, but the converse is not true as can be seen from a sequence of bumps of height 1=n going off to infinity. The local version of the above \uniformly absolute convergence" is \locally uniformly absolute convergence"; one now can guess what it should mean. It is instructive to try to prove Theorem 7 and its corollaries for locally uniformly absolutely converging series. Remark 12. There seems to be no universally used \unambiguous language" in literature con- cerning the terminology on various modes of convergence. For instance, depending on which book you read, \normal convergence" may mean \locally uniform convergence" or \locally 4 TSOGTGEREL GANTUMUR normal convergence", and \absolute convergence" may mean \uniform absolute convergence" or \pointwise absolute convergence". 2. Power series We understand by a power series an expression of the form 1 X n f(z) = an(z − c) ; (1) n=0 with the coefficients an 2 C, and the centre c 2 C.
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