CHAPTER 1 Basic properties of holomorphic functions Preview of differences between one and several variables For any n 1, the holomorphy or complex differentiability of a function on a domain in Cn implies its ≥analyticity: a holomorphic function has local representations by convergent power series. This amazing fact was discovered by Cauchy in the years 1830-1840 and it helps to explain the nice properties of holomorphic functions. On the other hand, when it comes to integral representations of holomorphic functions, the situation for n 2 is much more complicated than for n = 1: simple integral formulas in terms of boundary≥ values exist only for Cn domains that are products of C1 domains. It turns out that function theory for a ball in Cn is different from function theory for a \polydisc", a product of discs. The foregoing illustrates a constant theme: there are similarities between complex analysis in several variables and in one variable, but also differences and some of the differences are very striking. Thus the subject of analytic continuation presents entirely new phenomena for n 2. Whereas every C1 domain carries noncontinuable holomorphic functions, there are C≥n domains for which all holomorphic functions can be continued analytically across a certain part of the boundary (Section 1.9). The problems in Cn require a variety of new techniques which yield a rich theory. Sections 1.1 { 1.8 deal with simple basic facts, while Sections 1.9 and 1.10 contain previews of things to come. NOTATION. The points or vectors of Cn are denoted by z = (z1; : : : ; zn) = x + iy = (x1 + iy1; : : : ; xn + iyn): For vectors z and w in Cn we use the standard `Euclidean' norm or length and inner product, 1 2 z = z = z 2 + : : : + z 2 ; j j k k j 1j j nj (z; w) = z; w = z w = z w + : :: + z w : h i · 1 1 n n Subsets of Cn may be considered as subsets of R2n through the correspondence (x + iy ; : : : ; x + iy ) (x ; y ; : : : ; x ; y ) : 1 1 n n $ 1 1 n n Ω will always denote a (nonempty) open subset of the basic underlying space, here Cn. We also speak of a domain Ω in Cn, whether it is connected or not. A connected domain will often be denoted by D if that letter is not required for a derivative. 1.1 Holomorphic functions. Later on we will use the terms `analytic' and `holomorphic' interchangeably, but for the moment we will distinguish between them. According to 1 Weierstrass's definition (about 1870), analytic functions on domains Ω in Cn are locally equal to sum functions of (multiple) power series [cf. Definition 1.51]. Here we will discuss holomorphy. In order to establish notation, we first review the case of one complex variable. Let Ω be a domain in C R2. For Riemann (about 1850), as earlier for Cauchy, a complex- valued function ∼ f(x; y) = u(x; y) + iv(x; y) on Ω provided a convenient way to combine two real-valued functions u and v that occur together in applications. [For example, a flow potential and a stream function.] Geometrically, f = u + iv defines a map from one planar domain, Ω, to another. Let us think of a differentiable map (see below) or of a smooth map (u and v at least of class C1). We fix a Ω and write 2 z = x + iy; z = x iy; − z a = ∆z = ∆x + i∆y; z a = ∆z = ∆x i∆y: − − − Then the differential or linear part of f at a is given by @f @f df = df(a) = (a)∆x + (a)∆y @x @y 1 @f 1 @f 1 @f 1 @f = + ∆z + ∆z: 2 @x i @y 2 @x − i @y In particular dz = ∆z; dz = ∆z: It is now natural to introduce the following symbolic notation: 1 @f 1 @f @f 1 @f 1 @f @f + = ; = 2 @x i @y @z 2 @x − i @y @z since it leads to the nice formula @f @f @f @f df(a) = (a)∆z + (a)∆z = dz + dz: @z @z @z @z [Observe that @f=@z and @f=@z are not partial derivatives in the ordinary sense { here one does not differentiate with respect to one variable, while keeping the other variable(s) fixed. However, in calculations, @f=@z and @f=@z do behave like partial derivatives. Their definition is in accordance with the chain rule if one formally replaces the independent variables x and y by z and z. For a historical remark on the notation, see [Remmert].] We switch now to complex notation for the independent variables, writing f((z + z)=2; (z z)=2i) simply as f(z): By definition, the differentiability of the map f at a (in the real sense)− means that for all small complex numbers ∆z = z a = ρeiθ we have − def (1a) ∆f(a) = f(a + ∆z) f(a) = df(a) + o( ∆z ) as ∆z 0: − j j ! 2 Complex differentiability of such a function f at a requires the existence of ∆f @f @f ∆z (1b) lim = lim + + o(1) : ∆z!0 ∆z @z @z ∆z Note that ∆z=∆z = e−2iθ: Thus for a differentiable map, one has complex differentiability at a precisely when the Cauchy-Riemann condition holds at a : @f @f 1 @f (a) = 0 or = : @z @x i @y [If @f=@z = 0, the limit (1b) as ∆z 0 can not exist.] The representation f = u + iv 6 ! gives the familiar real Cauchy-Riemann conditions ux = vy; uy = vx. For the complex derivative one now obtains the formulas − 0 ∆f @f @f 1 @f (1c) f (a) = lim = = = = ux + ivx = ux iuy: ∆z!0 ∆z @z @x i @y − Observe that complex differentiability implies differentiability in the real sense. Functions f which possess a complex derivative at every point of a planar domain Ω are called holomorphic. In particular, analytic functions in C are holomorphic since sum functions of power series in z a are differentiable in the complex sense. On the other hand, by Cauchy's integral form− ula for a disc and series expansion, holomorphy implies analyticity, cf. also Section 1.6. HOLOMORPHY IN THE CASE OF Cn: Let Ω be a domain in Cn R2n and let ∼ f = f(z) = f(z1; : : : ; zn) be a complex-valued function on Ω: (1d) f = u + iv : Ω C: ! Suppose for a moment that f is analytic in each complex variable zj separately, so that f has a complex derivative with respect to zj when the other variables are kept fixed. Then f will satisfy the following Cauchy-Riemann conditions on Ω : @f def 1 @f 1 @f (1e) = = 0; j = 1; : : : ; n: @z 2 @x − i @y j j j Moreover, the complex partial derivatives @f=@zj will be equal to the corresponding formal derivatives, given by @f def 1 @f 1 @f (1f) = + ; @z 2 @x i @y j j j cf. (1c). Suppose now that the map f = u + iv of (1d) is just differentiable in the real sense. [This is certainly the case if f is of class C1:] Then the increment ∆f(a) can be written in the form (1a), but this time ∆z = (∆z1; : : : ; ∆zn) and the differential of f at a is given by 3 n @f @f n @f @f df(a) = (a)∆x + (a)∆y = dz + dz : @x j @y j @z j @z j 1 j j 1 j j X X Thus df = @f + @f [del f and del-bar or d-bar f], where n n def @f def @f @f = dz ; @f = dz : @z j @z j 1 j 1 j X X With this notation, the Cauchy-Riemann conditions (1e) may be summarized by the single equation @f = 0: Definition 1.11. A function f on Ω Cn to C is called holomorphic if the map f is “differentiable in the complex sense": ⊂ ∆f(a) = @f(a) + o( ∆z ) as ∆z 0 j j ! at every point a Ω. In particular a function f C1(Ω) is holomorphic precisely when it satisfies the Cauchy-R2 iemann conditions. 2 More generally, a function f defined on an arbitrary nonempty set E Cn is called holomorphic, notation ⊂ (1g) f (E); (also for open E = Ω!) 2 O if f has a holomorphic extension to some open set containing E . The notation (E) for the class or ring of holomorphic functions on E goes back to a standard notationO for rings, cf. [Van der Waerden] Section 16. The letter is also appropriate as a tribute to the Japanese mathematician Oka, who has made fundamenO tal contributions to complex analysis in several variables, beginning about 1935, cf. [Oka]. A function f (Ω) will have a complex derivative with respect to each variable zj at every point of Ω,2henceO by Cauchy's theory for a disc, f will be analytic in each complex variable zj separately. A corresponding Cauchy theory for so-called polydiscs will show that every holomorphic function is analytic in the sense of Weierstrass, see Sections 1.3 and 1.6. Thus in the end, holomorphy and analyticity will come to the same thing. REMARK. The expressions for df, @f and @f (with variable a) have the appearance of differential forms. First order differential forms p1dx1 + q1dy1 + : : : + qndyn or u1dz1 + v1dz1 + : : : + vndzn; with dx1; dy1; : : : ; dyn or dz1; dz1; : : : ; dzn as basis forms (!), will frequently be used as a notational device. Later on we will also need higher order differential forms, cf. Chapter 10 for a systematic discussion. 4 1.2 Complex affine subspaces.