Editing Inverse function theorem (section)

== Holomorphic inverse function theorem ==
There is a version of the inverse function theorem for [[holomorphic map]]s.

{{math_theorem|name=Theorem|math_statement=<ref>{{harvnb|Griffiths|Harris|1978|loc=p. 18.}}</ref><ref>{{cite book |first1=K. |last1=Fritzsche |first2=H. |last2=Grauert |title=From Holomorphic Functions to Complex Manifolds |publisher=Springer |year=2002 |pages=33–36 |isbn=978-0-387-95395-3 |url=https://books.google.com/books?id=jSeRz36zXIMC&pg=PA33 }}</ref> Let <math>U, V \subset \mathbb{C}^n</math> be open subsets such that <math>0 \in U</math> and <math>f : U \to V</math> a holomorphic map whose Jacobian matrix in variables <math>z_i, \overline{z}_i</math> is invertible (the determinant is nonzero) at <math>0</math>. Then <math>f</math> is injective in some neighborhood <math>W</math> of <math>0</math> and the inverse <math>f^{-1} : f(W) \to W</math> is holomorphic.}}

The theorem follows from the usual inverse function theorem. Indeed, let <math>J_{\mathbb{R}}(f)</math> denote the Jacobian matrix of <math>f</math> in variables <math>x_i, y_i</math> and <math>J(f)</math> for that in <math>z_j, \overline{z}_j</math>. Then we have <math>\det J_{\mathbb{R}}(f) = |\det J(f)|^2</math>, which is nonzero by assumption. Hence, by the usual inverse function theorem, <math>f</math> is injective near <math>0</math> with continuously differentiable inverse. By chain rule, with <math>w = f(z)</math>,
:<math>\frac{\partial}{\partial \overline{z}_j} (f_j^{-1} \circ f)(z) = \sum_k \frac{\partial f_j^{-1}}{\partial w_k}(w) \frac{\partial f_k}{\partial \overline{z}_j}(z) + \sum_k \frac{\partial f_j^{-1}}{\partial \overline{w}_k}(w) \frac{\partial \overline{f}_k}{\partial \overline{z}_j}(z)</math>
where the left-hand side and the first term on the right vanish since <math>f_j^{-1} \circ f</math> and <math>f_k</math> are holomorphic. Thus, <math>\frac{\partial f_j^{-1}}{\partial \overline{w}_k}(w) = 0</math> for each <math>k</math>. <math>\square</math>

Similarly, there is the implicit function theorem for holomorphic functions.<ref name="holomorphic implicit">{{harvnb|Griffiths|Harris|1978|loc=p. 19.}}</ref>

As already noted earlier, it can happen that an injective smooth function has the inverse that is not smooth (e.g., <math>f(x) = x^3</math> in a real variable). This is not the case for holomorphic functions because of:
{{math_theorem|name=Proposition|math_statement=<ref name="holomorphic implicit" /> If <math>f : U \to V</math> is an injective holomorphic map between open subsets of <math>\mathbb{C}^n</math>, then <math>f^{-1} : f(U) \to U</math> is holomorphic.}}