Proof of the Inverse Function Theorem in Complex Analysis 📂Complex Anaylsis

Proof of the Inverse Function Theorem in Complex Analysis

Theorem ¹

A function $f : \mathbb{C} \to \mathbb{C}$ that is analytic at $\alpha$ and satisfies $f ' (\alpha) \ne 0$ exists in the region $\mathcal{N} \left( f(\alpha) \right)$, where $f^{-1}$ exists.

Explanation

Consider the condition given by $f ' (\alpha) \ne 0$.

When thought of as a real function, it implies the function is either increasing or decreasing, which is a condition for the existence of an inverse function. Geometrically, this refers to a Smooth function, indicating there are no abrupt changes in direction or bends. What must be noted in the inverse function theorem is that even if such conditions are met, it doesn’t mean the inverse function itself exists universally but rather within a local limit.

Proof

It suffices to show that equation $w = f(z)$ has a unique solution for $w \in \mathcal{N} (f(\alpha))$.

If we set $$ \beta := f(\alpha) \\ g(z) := f(z) - \beta $$, then $g(\alpha) = 0$ and $g ' (\alpha) \ne 0$. This means that $\alpha$ is a simple zero of $g$, and there exists $\rho >0$ that satisfies $g(z) \ne 0$ in $|z - \alpha | \le \rho$.

For circle $\mathscr{C}: |z - \alpha| = \rho$, define $h(z) := -\gamma$ that satisfies $$ m := \min_{\mathscr{C}} |g(z)| \\ |\gamma| < m $$. Then, the following holds in $\mathscr{C}$: $$ g(z) \ne 0 \\ |h(z) | = | - \gamma | = |\gamma| < m \le |g(z)|$$

Rouché’s Theorem: If $g$ and $h$ are analytic on the simple closed path $\mathscr{C}$ and satisfy $|h(z)| \le |g(z)|$ on $\mathscr{C}$, then $g$ and $g + h$ have the same number of zeros inside $\mathscr{C}$.

By Rouché’s Theorem, $g$ and $g + h = g - \gamma$ have the same number of zeros inside $\mathscr{C}$.

However, as seen before, since $g$ has only a single simple zero $\alpha$, the zero satisfying $g(z) - \gamma = 0$ is also unique inside $\mathscr{C}$. Therefore, it can be concluded that equation $g(z) = \gamma$ has a unique solution inside $\mathscr{C}$.

If we now let $w = \beta + \gamma$, $$ f(z) - \beta = w - \beta $$ it means that $w = f(z)$ has a unique solution inside $\mathscr{C}$ at $\mathcal{N}(\alpha): |z - \alpha| < \rho$.