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A Short Introduction to Several Complex Variables

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A Short Introduction to Several Complex Variables A short introduction to several complex variables Oliver Knill Caltech, 1996 This text is a short two hour introduction into the theory of several complex variables. These lectures were given in May 1996 at Caltech to the class Ma 108 substituting for someone else. As prerequisite, the topic requires some familiarity with complex analysis in one dimension. 1 Motivation and Plan The higher dimensional generalization of complex analysis in one variables is an important part of mathematics. Our short visit of this theory will allow us to repeat some facts of the theory in one complex variable. A motivating example, where multi-dimensional complex analysis can occur in mathematical research, is the iteration of multi-dimensional analytic maps like for example the H´enonmap family C2 ! C2 z z2 + c − aw T : ! ; w z where a; c are complex parameters. If a = 0, one obtains, by restriction to the first coordinate, the iteration of a one dimensional map z 7! z2 + c. A theorem in dynamical system theory assures that the fixed point (zc; wc) near the center is stable: if we start iterating with an initial condition near that point, we stay near that point. This stability does no more hold in C2. The theory to understand the iteration of the H´enonmap in C2 heavily needs the theory of complex variables in several dimensions. The aim of this two hour introduction is 1. to show that part of complex analysis in several variables can be obtained from the one- dimensional theory essentially by replacing indices with multi-indices. Examples of results which extend are Cauchy's theorem, the Taylor expansion, the open mapping theorem or the maximum theorem. 2. to show in two examples that there are new features in several dimensions. For example, a multi-dimensional Riemann mapping theorem or a multi-dimensional Piccard theorem does no 1 more hold. The theory becomes richer in higher dimensions. 2 Holomorphic functions Notation. Cn denotes the n-dimensional complex vector space. A point in Cn is written as z = (z1; z2; : : : ; zn−1; zn), where zj = xj + iyj are complex numbers. We write @f 1 @ @ = ( − i )f @zj 2 @xj @yj for the complex partial derivatives of a function f : Cn ! C. We also use the notation for n 2 Nn; z 2 Cn @mf @m1 @m1 m = m ··· m f(z) : (@z) (@z1) 1 (@zn) n Definition. Let D ⊂ Cn be an open set. A continuous function f : D ! C is called holomorphic in D or analytic in D, if for all z 2 D and 1 ≤ j ≤ n, the complex partial derivatives @f exist @zj and are finite. Remark. In other words, a function D ! C is holomorphic if it is continuous and holomorphic in each of its variables. Proposition 2.1 (Basic properties of holomorphic maps) If f; g are holomorphic in D, then f + g; f − g; f · g are holomorphic in D. If g(z) 6= 0 for all z 2 D, then also f=g is holomorphic in D. If fn is a sequence of holomorphic maps in D which converges uniformly on compact subsets of D to f, then f is holomorphic in D. Proof. These facts follow directly from the one-dimensional theory: if f is holomorphic in D, then it is holomorphic in each of the variables separately. That is, if we fix z1; : : : ; zj−1; zj+1; : : : ; zn, then the map zj 7! f(z1; : : : ; zj; : : : ; zn) is holomorphic as a map from Dj ! C, where Dj = fzj j (z1; : : : ; zj; : : : ; zn) 2 D g. Examples. 3 5 2 1) The function f :(z1; z2) 7! 2z1 + sin(z1) − z2 is holomorphic on D = C . z2+z2 2 2) The function f :(z1; z2) 7! e 1 2 is holomorphic in C . We will see in a moment that it can k1 k2 P1 z1 z2 be written as f(z1; z2) = . k1;k2=0 k1!k2! P1 k 3) The function f :(z1; z2) 7! k=0 z2z1 is holomorphic in D = fjz1j < 1g × C. z2 4) The function f :(z1; z2) 7! is holomorphic in Cnf1g×C. It is the analytic continuation 1−z1 of 3) from D to C n f1g × C. Short exercise. Find the region in C2, where the sum 1 X k k f(z1; z2) = z1 z2 k=0 2 is analytic. Find an explicit analytic continuation of f to a larger domain. Remark. It is actually not necessary to assume that f is continuous: a theorem of Hartogs assures that the continuity assumption is redundant in the definition of analyticity. Note the difference between the real and the complex multi-dimensional case, because as you well know, a real function f : Rn ! R for which all partial derivatives exist, needs not to be continuous: Definition. Let K ⊂ Cn be any set. A function f : K ! C is called holomorphic on K if each point a 2 K has an open neighborhood D such that D \ K is closed and such that there exists a function fD which is holomorphic on D such that fD = f on D \ K. Example. If f is holomorphic on D and K is a closed subset of D then f is holomorphic on K. Proof. Take for any point a 2 K the open set D. 3 Cauchy, Taylor and Co. Definition.A closed polydisc D(a; r) = Dn(a; r) with center a 2 Cn and radius r is the set n fz 2 C j jzj − ajj ≤ rj; j = 1;:::; n g. The interior D(a; r) is called the open polydisc or simply polydisc. The boundary of the polydisc is the topological boundary. It consists of the set of points in the closed polydisc which satisfy jz − ajj = rj for some r. n iθj The distinguished boundary T (a; r) of the polydisc is the set fzjzj = a + re ; 0 ≤ θj < 2πg. n n If a = 0; rj = 1, we use also the notation D or T . n n Pn 2 2 + We denote by B (a; r) = fz 2 C j j=1 jzjj = r g the ball with radius r 2 R around a. Figure 3a) The polydisc D ⊂ C2 Figure 3b) The ball B ⊂ C2 shown in the the coordinates jz1j; jz2j. shown in the the coordinates jz1j; jz2j. Remark. Note that the distinguished boundary of the polydisc is topologically a real n-dimensional torus Tn, whereas the boundary has real dimension 2n − 1. Theorem 3.1 (Cauchy Formula) n Let f be holomorphic on the closed polydisc D (a; r). Then, for all z 2 Dn(a; r) one has 1 Z f(w) f(z) = n dw : (2πi) Tn(a;r) w − z 3 1 Qn 1 We used the notation dw = dw1dw2 : : : dwn and = . w−z j=1 wj −zj Proof. The proof goes by induction on the dimensions. For n = 1, it is the usual Cauchy formula. Assume the formula has been proven in dimension n − 1. Define the analytic function g(z) = f(z; z2; : : : ; zn). By the one dimensional Cauchy formula applied to the function g on the disc fjz1 − a1j < r1g, one has Z 1 f(w1; z2; : : : ; zn) f(z1; z2; : : : ; zn) = dw1 : (1) (2πi) T(a1;r1) w1 − z1 If we fix w1, the function (z2; : : : ; zn) ! f(w1; z2; : : : ; zn) is a function of n − 1 variables. By the induction assumption, we have Z 1 f(w1; z2; : : : ; zn) f(w1; z2; : : : ; zn) = n−1 dw2 : : : dwn : (2) (2πi) Tn−1(a;r) w − z If we plug in Equation ?? into the right hand side of Equation ??, we obtain the Cauchy's formula in n dimensions. Corollary 3.2 (Holomorphic maps are smooth) If f is holomorphic, then (@=@zj)f is holomorphic. Moreover, all partial deriva- tives @nf=(@z)n exist. Proof. Differentiate on the right hand side of the Cauchy formula gives Z (k) 1 f(w) f (z) = n k+1 k! dw : (3) (2πi) Tn(a;r) (w − z) More notation. We use the notation k = (k1; k2; : : : ; kn) and say k ≥ 0 if all coordinates of the k Qk kj multi-index k satisfy kj ≥ 0. We also use the abbreviation (z − a) = j=1(zj − aj) and @kf f (k)(a) = (a) (@z)k as well as k! = k1!k2! ··· kn!. Corollary 3.3 (Cauchy estimates) Assume f is holomorphic in D(a; r) and that jf(z)j ≤ M for all z 2 D(a; r). Then, for a 2 D(a; r), we have Mk! jf (k)(a)j ≤ : rk Proof. This follows directly from the Cauchy formula by majorizing the integral: replace f inside the integral by M and do the estimate! Corollary 3.4 (Taylor Series Expansion) If f is holomorphic in the open polydisc D(a; r), then X 1 f(z) = f (k)(a)(z − a)k : k! k≥0 4 1 Proof. The proof is the same as in one variable: expand z ! w−z into a power series 1 1 X zk = : w − z wk+1 k=0 k Qn kj (Remind the notation z = j=1 zj also in the following). By Cauchy's formula and the Cauchy formula for the derivative, one has (k) Z k X f (a) k 1 X (z − a) (z − a) = n f(w) k+1 dw k! (2πi) n (w − a) k T (a;r) k 1 Z f(w) = n dw = f(z) : (2πi) Tn(a;r) w − z Corollary 3.5 (Principle of analytic continuation) Assume f is holomorphic in an open connected subset (=domain) of Cn.
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