Advanced Implicit Function Theorems

We will now consider implicit function theorems in both the real analytic and the complex analytic (holomorphic) categories. These are obviously closely related, as the problem in the real analytic category can be complexified (by replacing every xj with a zj) and thereby turned into a holomorphic problem. Conversely, any complex analytic implicit function theorem situation is a fortiori real analytic and can therefore be treated with real analytic techniques. And both categories are subcategories of the C∞ category.