TERMWISE DERIVATIVES OF COMPLEX FUNCTIONS

Transcription

1 TERMWISE DERIVATIVES OF COMPLEX FUNCTIONS This writeup proves a result that has as oe cosequece that ay complex power series ca be differetiated term-by-term withi its disk of covergece The result has other cosequeces as well Recall some results that are already established Oe is that the uiform limit of cotiuous fuctios is cotiuous: Let S be a subset of C Cosider a sequece of cotiuous fuctios o S, {ϕ 0, ϕ, ϕ 2, } : S C Suppose that the sequece coverges uiformly o S to a limit fuctio The ϕ is also cotiuous ϕ : S C This was show i the previous writeup o compactess ad uiformity The other result to recall is that we ca pass uiform limits through itegrals Specifically, let Ω be a regio i C ad let a be ay poit of Ω Some closed ball B cetered at a lies i Ω Let = B be the boudary circle of B, traversed oce couterclockwise Suppose that a sequece of cotiuous fuctios {φ 0, φ, φ 2, } : C coverges uiformly o to a limit φ : C We kow that φ must also be cotiuous The as also was show i a previous writeup, the limit of the itegrals is the itegral of the limit, lim φ (ζ) dζ = φ(ζ) dζ Now we ca state ad prove our mai result The followig theorem is due to Weierstrass Let Ω be a regio i C Cosider a sequece of differetiable fuctios o Ω, {ϕ 0, ϕ, ϕ 2, } : Ω C Suppose that the sequece coverges o Ω to a limit fuctio ϕ : Ω C ad that the covergece is uiform o compact subsets of Ω The () The limit fuctio ϕ is differetiable (2) The sequece {ϕ } of derivatives coverges o Ω to the derivative ϕ of the limit fuctio (3) This covergece is also uiform o compact subsets of Ω Proof First, to show that the limit fuctio ϕ is cotiuous, let z be ay poit of Ω Some closed ball B cetered at z lies i Ω, ad the covergece of {ϕ } to ϕ is uiform o the compact set B The restrictio of the limit fuctio ϕ to B is therefore cotiuous, ad so ϕ itself is cotiuous at the iterior poit z Sice z is arbitrary, ϕ is cotiuous o Ω

2 2 TERMWISE DERIVATIVES OF COMPLEX FUNCTIONS Next, to show that ϕ is differetiable, let = B be the boudary circle of the closed ball B from the previous paragraph, traversed oce couterclockwise, ad let z be ay poit iside Cosider a auxiliary sequece of fuctios o, with limit φ (ζ) = ϕ (ζ), = 0,, 2,, ζ z φ(ζ) = ϕ(ζ) ζ z Sice {φ } coverges uiformly o to φ, we may exchage a itegral ad a limit, ϕ(z) = lim ϕ (z) = lim ϕ (ζ) ζ z dζ = lim = φ (ζ) dζ φ(ζ) dζ = ϕ(ζ) ζ z dζ This shows that the cotiuous fuctio ϕ has a Cauchy itegral represetatio, makig it differetiable Third, use the Cauchy itegral represetatio of derivatives to argue similarly (with a modified auxiliary sequece {φ }) that the sequece {ϕ } of derivatives coverges to the derivative ϕ of the limit fuctio, lim ϕ (z) = lim ϕ (ζ) (ζ z) 2 dζ = ϕ(ζ) (ζ z) 2 dζ = ϕ (z) Fially, we eed to argue that this covergece is uiform o compact subsets of Ω I the special case that the compact set is the closed ball B havig half the radius of the ope ball B, let c > 0 deote the half-radius, so that ζ z c for all ζ ad z B It follows by Cauchy s formula for the derivative ad the usual estimatio techiques that for all z B, ϕ (z) ϕ (z) = ϕ(ζ) ϕ (ζ) (ζ z) 2 dζ ϕ(ζ) ϕ (ζ) 2π c 2 dζ = C sup{ ϕ(ζ) ϕ (ζ) : ζ } But {ϕ } coverges to ϕ uiformly o the compact subset of Ω So, give ε > 0, there exists a startig idex 0 such that 0 = ϕ (z) ϕ (z) < ε for all z B To complete the argumet, let K be ay compact set of the whole regio Ω About each poit a of K there is a ball B = B a as i the previous discussio Let B a be the ball about a of half the radius of B a These balls give a ope cover of K, K = {a} B a a K By compactess, K has a fiite subcover, a K K B a B a k

3 Let K j = K B a j TERMWISE DERIVATIVES OF COMPLEX FUNCTIONS 3 for j =,, k The K = K K, ad by the previous paragraph, the covergece of {ϕ } to ϕ is uiform o each of the fiitely-may sets K j Cosequetly the covergece is uiform o K: Give ε > 0, the correspodig global startig idex 0 is the maximum of fiitely may local oes This completes the proof As metioed, the applicatio of the Weierstrass Theorem that we have i mid here is that the fuctios ϕ are the partial sums of a power series, ϕ (z) = a j (z c) j, = 0,, 2, while ϕ is the full power series, j=0 ϕ(z) = a j (z c) j j=0 I this case, the result is that ay power series ca be differetiated term by term withi its disk of covergece, ad the resultig power series has the same disk of covergece For aother applicatio of the Weierstrass Theorem, cosider the Euler Riema zeta fuctio, ζ(s) = s = Sice / s = / Re(s), the sum coverges absolutely o the right half plae Ω = {Re(s) > }, ad the covergece is uiform o compacta Thus ζ(s) is aalytic o Ω For a third applicatio of the Weierstrass Theorem, let Ω = C Z, a regio i C Defie for each N, ϕ : Ω C, ϕ (z) = z + ( z j + ), z + j ad defie the correspodig limit fuctio ϕ : Ω C, j= ϕ(z) = z + j= ( z j + ) z + j It ca be show that the sequece {ϕ } coverges to ϕ uiformly o compact subsets of Ω, ad so by the Weierstrass Theorem, all derivatives of ϕ exist o Ω The series ϕ ca be writte as a sum over Z rather tha as a sum of paired terms, but at the cost of itroducig modificatios to the idividual terms to force covergece, ϕ(z) = z + 0 j Z ( z j + j Its derivative is a similar sum, though o loger with modified terms, ϕ : Ω C, ϕ (z) = (z j) 2 j Z )

4 4 TERMWISE DERIVATIVES OF COMPLEX FUNCTIONS I fact, ϕ(z) = π cot πz ad ϕ (z) = π 2 csc πz, but showig this takes a little work Eve the periodicity of ϕ is ot obvious, because of the modified terms i its defiig sum For the derivatives of ϕ, periodicity it is more clear, although eve with the modificatios goe, the argumet relies o the fact that a absolutely coverget sum ca be rearraged with o effect o its value We will pursue all of this i aother writeup For a fourth applicatio of the Weierstrass Theorem, let ω ad ω 2 be ozero complex umbers such that ω /ω 2 is ot real These two umbers spa a lattice i C, Λ = ω Z ω 2 Z For each N let ad let Λ = {ω Λ : ω 0, ω }, Λ = {ω Λ : ω 0} Let Ω = C Λ, a regio i C Defie for each N, ( : Ω C, (z) = z 2 + Λ (z ω) 2 ω 2 This is the sequece of partial sums of the Weierstrass -fuctio, : Ω C, (z) = z 2 + ( (z ω) 2 ) ω 2 Λ As with the series for the cotaget, the modificatios /ω 2 to the summads for are required to make the sum absolutely coverget Ulike the series for cotaget, this series does ot represet a fuctio that is already familiar It is a geuiely ew fuctio arisig from complex aalysis It ca be show that the sequece { } coverges to uiformly o compact subsets of Ω, ad so by the Weierstrass Theorem, all derivatives of exist o Ω as similar sums, though o loger with modified terms I particular we have : Ω C, (z) = 2 Λ (z ω) 3 The Weierstrass -fuctio ad its derivatives are doubly periodic with respect to the lattice Λ, meaig for example that (z + ω) = (z) for all ω Λ For the derivatives of this is fairly clear, usig absolute covergece The double periodicity of itself is ot obvious, because of the modified factors i the series, but it follows quickly from the facts that is eve ad is doubly periodic We may discuss doubly periodic fuctios later i the course Aother applicatio of the Weierstrass Theorem i this course will be more abstract, i the proof of a result called the Riema Mappig Theorem The proof will costruct a sequece of fuctios that come ever closer to havig desired properties, ad the the Weierstrass theorem will the guaratee the existece of a limit fuctio, for which the desired properties will hold )

5 TERMWISE DERIVATIVES OF COMPLEX FUNCTIONS 5 A odd cosequece of the academic semester system is that sometimes a first complex aalysis course will use the Weierstrass Theorem to prove that power series are aalytic ad termwise differetiable (eve though we have already see that this result is readily proved without the theorem), but the ot get to ay other applicatios of the Weierstrass Theorem, such as the properties of the Weierstrass -fuctio This creates a doubly false impressio that the power series result is difficult ad that the theorem serves oly oe purpose