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Additional resources for Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory
1 a v)); hence, canonical bases of C' and Cm. 2 Chain Rule. If g eHoI(Y, Z), then For n = m, ôz = • . j (3z the determinant J1(a):=det (a) exists; it is called the complex functional determinant (or Jacobian determinant) of at a. With that notation, we have the following fact: f (a) is an isomorphism (a) 0. 3 Inverse Mapping Theorem. Suppose that fE Hol (X, C") and a E X C"; then f is a biholomorphic mapping from an open neighborhood of a onto an open neighif (a) 0. borhood of f(a) i1 (a) = JId(a) = id.
Assume the contrary; then there exists a polydisk P(x0; r0) X I I with x0 X on which the Taylor series T of f at X0 converges. In particular, I and T coincide on r0). We can construct recursively a sequence PX(xk; r&) 1)cx P(xk; rk). Then in ¶IR with Xk€ Px(xo; r0) and Tk < 1/k such that P(xk+ 1; r0) form a subsequence of (a1) in the corresponding points aJ(k) E PX(Xk; that converges to a point y P(X0; r0). Although T converges on P(x0; r0), we have that IT(y)I = limf(aJ(k))I = iv) The implications iii) v) are trivial, since each involves only exchanging an existential quantifier with a universal quantifier.
L2Aa. 4 would not hold for X = P2(1) if the stipulation is non- constant" were dropped from ii). 5 Example. For X C2\rR2, the restriction-mapping q: isan isornorphism of topological algebras. 6 iii), to show that x = (x1,x2)eC2\X = a + 81 ( we have that dist(bdD, aX) = = 1, is nonconstant (ä5(x1 + i,X2 + 2) = and I <2 + i,x2 + 2i)). Since tx—al = 12 is surjective. For 1 = dist(bdD,ÔX), A. 4 yields that f is holomorphically extendible to a polydisk containing x. The assertion follows from E. 12a.