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Chapter 1 Holomorphic and Real Analytic Calculus

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Chapter 1 Holomorphic and Real Analytic Calculus This version: 28/02/2014 Chapter 1 Holomorphic and real analytic calculus In this chapter we develop basic analysis in the holomorphic and real analytic settings. We do this, for the most part, simultaneously, and as a result some of the ways we do things are a little unconventional, especially when compared to the standard holomorphic treatments in, for example, texts like [Fritzsche and Grauert 2002, Gunning and Rossi 1965,H ormander¨ 1973, Krantz 1992, Laurent-Thiebaut´ 2011, Range 1986, Taylor 2002]. We assume, of course, a thorough acquaintance with real analysis such as one might find in [Abraham, Marsden, and Ratiu 1988, Chapter 2] and single variable complex analysis such as one might find in [Conway 1978]. 1.1 Holomorphic and real analytic functions Holomorphic and real analytic functions are defined as being locally prescribed by a convergent power series. We, therefore, begin by describing formal (i.e., not depending on any sort of convergence) power series. We then indicate how the usual notion of a Taylor series gives rise to a formal power series, and we prove Borel’s Theorem which says that, in the real case, all formal power series arise as Taylor series. This leads us to consider convergence of power series, and then finally to consider holomorphic and real analytic functions. Much of what we say here is a fleshing out of some material from Chapter 2 of [Krantz and Parks 2002], adapted to cover both the holomorphic and real analytic cases simultaneously. 1.1.1 Fn Because much of what we say in this chapter applies simultaneously to the real and complex case, we shall adopt the convention of using the symbol F when we wish to refer to one of R or C. We shall use x to mean the absolute value (when F = R) or j j the complex modulus (when F = C). In like manner, we shall denote by x¯ the complex conjugate of x if F = C and x¯ = x if F = R. We use Xn x; y = xj y¯ j h i j=1 2 1 Holomorphic and real analytic calculus 28/02/2014 to denote the standard inner product and n X 1=2 2 x = xj k k j j j=1 to denote the standard norm on Fn in both cases. We shall need notions of balls and disks in Fn. Balls are defined in the usual way. n Given x0 F and r R>0, the open ball with radius r and centre x0 is 2 2 n n B (r; x0) = x F x x0 < r ; f 2 j k − k g with n n B (r; x0) = x F x x0 r f 2 j k − k ≤ g n n similarly denoting the closed ball. For x0 F and for r R , denote 2 2 >0 n n D (r; x0) = x F xj x0j < rj; j 1;:::; n ; f 2 j j − j 2 f gg which we call the open polydisk of radius r and centre x0. The closure of the open polydisk, interestingly called the closed polydisk, is denoted by n n D (r; x0) = x F xj x0j rj; j 1;:::; n : f 2 j j − j ≤ 2 f gg n n n If r R>0 then we denote rˆ = (r;:::; r) R>0 so that D (rˆ; x0) and D (rˆ; x0) denote polydisks2 whose radii in all components are2 equal. If U Fn is open, if A U, and if f : U Fm is continuous, we denote ⊆ ⊆ ! f A = sup f(x) x A : (1.1) k k fk k j 2 g A domain in Fn is a nonempty connected open set. m n m r A map f : U F from an open subset U F to F will be of class C , r Z 0 , !r ⊆ 2 ≥ [f1g if it is of class C in the real variable sense, noting that Cn R2n. ' 1.1.2 Multi-index and partial derivative notation n A multi-index is an element of Z 0. For a multi-index I we shall write I = (i1;:::; in). We introduce the following notation:≥ 1. I = i + + in; j j 1 ··· 2. I! = i ! in!; 1 ··· I i1 in n 3. x = x1 xn for x = (x1;:::; xn) F . ··· 2 n n Note that the elements e1;:::; en of the standard basis for F are in Z 0, so we shall n ≥ think of these vectors as elements of Z 0 when it is convenient to do so. The following property of the set of≥ multi-indices will often be useful. 28/02/2014 1.1 Holomorphic and real analytic functions 3 1.1.1 Lemma (Cardinality of sets of multi-indices) For n Z>0 and m Z 0, 2 2 ≥ ! n n + m 1 card I Z 0 I = m = − : f 2 ≥ j j j g n 1 − Proof We begin with an elementary lemma. m ! ! X n + j 1 n + m 1 Sublemma For n Z>0 and m Z 0, − = . 2 2 ≥ n 1 n j=0 − Proof Recall that for j; k Z 0 with j k we have 2 ≥ ≤ ! k k! = : j j!(k j)! − We claim that if j; k Z>0 satisfy j k then 2 ≤ ! ! ! k k k + 1 + = : j j 1 j − This is a direct computation: k! k! (k j + 1)k! jk! + = − + j!(k j)! (j 1)!(k j + 1)! (k j + 1)j!(k j)! j(j 1)!(k j + 1)! − − − − − − − (k j + 1)k! + jk! = − j!(k j + 1)! − ! (k + 1)k! (k + 1)! k + 1 = = = : j!((k + 1) j)! j!((k + 1) j)! j − − Now we have m ! m ! m ! m ! X n + j 1 X n + j 1 X n + j X n + j 1 − = 1 + − = 1 + − n 1 n 1 n − n j=0 − j=1 − j=1 j=1 ! ! ! n + m n n + m = 1 + = ; n − n n as desired. H We now prove the lemma by induction on n. For n = 1 we have ! m card j Z 0 j = m = 1 = ; f 2 ≥ j g 0 which gives the conclusions of the lemma in this case. Now suppose that the lemma k+1 holds for n 1;:::; k . If I Z 0 satisfies I = m, then write I = (i1;:::; ik; ik+1) and take 2 f g 2 ≥ j j 4 1 Holomorphic and real analytic calculus 28/02/2014 I = (i ;:::; i ). If i = j 0; 1;:::; m then I = m j. Thus 0 1 k k+1 2 f g j 0j − Xm k+1 n card I Z 0 I = m = card I0 Z 0 I0 = m j f 2 ≥ j j j g f 2 ≥ j j j − g j=0 m ! m ! X k + m j + 1 X k + j 1 = − = 0 − k 1 k 1 j=0 − j0=0 − ! ! k + m (k + 1) + m 1 = = − ; k (k + 1) 1 − using the sublemma in the penultimate step. This proves the lemma by induction. Multi-index notation is also convenient for representing partial derivatives of multi- variable functions. Let us start from the ground up. Let (e1;:::; en) be the standard n n n basis for F and denote by (α1;:::; αn) the dual basis for (F )∗. Let U F and let ⊆ n f : U F. The F-derivative of f at x0 U is the unique F-linear map D f (x0) L(F ; F) for which! 2 2 f (x) f (x0) D f (x0)(x x0) lim − − − = 0; x x0 x x ! − 0 provided that such a linear map indeed exists. Higher F-derivatives are defined k k recursively. The kth F-derivative of f at x0 we denote by D f (x0), noting that D f (x0) k n 2 Lsym(F ; F) is a symmetric k-multilinear map [see Abraham, Marsden, and Ratiu 1988, Proposition 2.4.14]. If Dk f (x) exists for every x U and if the map x Dk f (x) is k 2 7! continuous, f is of F-class C or k-times continuously F-differentiable. This slightly awkward and nonstandard notation will be short-lived, and is a consequence of our trying to simultaneously develop the real and complex theories. k We denote by L (Fn; F) the set of k-multilinear maps with its usual basis (αj αj ) j ;:::; jk 1;:::; n : f 1 ⊗ · · · ⊗ k j 1 2 f gg Thinking of D f (x0) as a multilinear map, forgetting about its being symmetric, we write Xn k k @ f D f (x0) = (x0)αj1 αjk : @xj1 @xjk ⊗ · · · ⊗ j1;:::;jk=1 ··· Let us introduce another way of writing the F-derivative. For j1;:::; jk 1;:::; n , n 2 f g define I Z 0 by letting im Z 0 be the number of times m 1;:::; n appears in 2 ≥ 2 ≥ 2 f g the list of numbers j1;:::; jk. We recall from Section F.2.3 the product between k l A L (Fn; F) and B L (Fn; F) given by 2 sym 2 sym (k + l)! A B = Sym (A B); k!l! k+l ⊗ m where, for C L (Fn; F), 2 sym 1 X Sym (C)(v ;:::; v ) = C(v ;:::; v ): m 1 m m! σ(1) σ(m) σ Sm 2 28/02/2014 1.1 Holomorphic and real analytic functions 5 Then, as discussed in Section F.2.4, we can also write X 1 @k f Dk f x x i1 in ( 0) = i i ( 0)α1 αn i1! in! @x 1 @x n · · · i1;:::;in Z 0 1 n 2 ≥ ··· ··· i + +in=k 1 ··· n For I = (i1;:::; in) Z 0, we write 2 ≥ I I @j j f D f (x0) = (x0): @xi1 @xin 1 ··· n We may also write this in a different way: I I @j j f D f (x0) = (x0): @x1 @x1 @xn @xn | {z··· } ··· | {z··· } i1 times in times Indeed, because of symmetry of the F-derivative, for any collection of numbers j1;:::; j I 1;:::; n for which k occurs ik times for each k 1;:::; n , we have j j 2 f g 2 f g I I @j j f D f (x0) = (x0): @xj1 @xj I ··· j j In any case, we can also write X 1 Dk f (x ) = DI f (x )αi1 αin 0 I! 0 1 · · · n I Zn 2 0 I =≥k j j We shall freely interchange the various partial F-derivative notations discussed above, depending on what we are doing.
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