Complex Analysis/Complex Functions/Complex Derivatives

Complex differentiability

Let us now define what complex differentiability is.

Definition 2.3.1:

Let $S \subseteq \mathbb C$, let $f: S \to \mathbb C$ be a function and let $z_0 \in S$. $f$ is called complex differentiable at $z_0$ if and only if there exists a $w \in \mathbb C$ such that:

$\lim_{z \to z_0 \atop z \in S} \frac{f(z) - f(z_0)}{z - z_0} = w$

Example 2.3.2

The function

$f: \mathbb C \to \mathbb C, f(z) = \overline z$

is nowhere complex differentiable.

Proof:

Let $z_0 \in \mathbb C$ be arbitrary. Assume that $f$ is complex differentiable at $z_0$, i. e. that

$\lim_{z \to z_0 \atop z \in \mathbb C} \frac{\overline z - \overline{z_0}}{z - z_0}$

exists.

We choose

$A := \{z \in \mathbb C | \Re z = \Re z_0 \}$

and

$B := \{z \in \mathbb C | \Im z = \Im z_0 \}$

. Due to lemma 2.2.3, which is applicable since of course $\mathbb C$ is open, we have:

$\lim_{z \to z_0 \atop z \in A} \frac{\overline z - \overline{z_0}}{z - z_0} = \lim_{z \to z_0 \atop z \in \mathbb C} \frac{\overline z - \overline{z_0}}{z - z_0} = \lim_{z \to z_0 \atop z \in B} \frac{\overline z - \overline{z_0}}{z - z_0}$

But

$\lim_{z \to z_0 \atop z \in A} \frac{\overline z - \overline{z_0}}{z - z_0} = \lim_{z \to z_0 \atop z \in A} \frac{\Re (z - z_0) - i\Im (z - z_0)}{\Re (z - z_0) + i\Im (z - z_0)} = -1$

and

$\lim_{z \to z_0 \atop z \in B} \frac{\overline z - \overline{z_0}}{z - z_0} = \lim_{z \to z_0 \atop z \in B} \frac{\Re (z - z_0) - i\Im (z - z_0)}{\Re (z - z_0) + i\Im (z - z_0)} = 1$

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The Cauchy-Riemann equations

We can define a natural bijective function from $\mathbb C$ to $\mathbb R^2$ as follows:

$\Phi(x + iy) := (x, y)$

In fact, $\Phi$ is a vector space isomorphism between $\mathbb C^1$ and $\mathbb R^2$.

The inverse of $\Phi$ is given by

$\Phi^{-1}: \mathbb R^2 \to \mathbb C, \Phi^{-1}(x, y) = x + iy$

Theorem and definitions 2.3.3:

Let $O \subseteq \mathbb C$ be open, let $f: O \to \mathbb C$ be a function and let $z_0 = x_0 + i y_0 \in O$. If $f$ is complex differentiable at $z_0$, then the functions

$u: \Phi(O) \to \mathbb R, u(x, y) = \Re f(x + iy)$

and

$v: \Phi(O) \to \mathbb R, v(x, y) = \Im f(x + iy)$

are well-defined, differentiable at $(x_0, y_0)$ and satisfy the equations

$\partial_x u (x_0, y_0) = \partial_y v (x_0, y_0)$

and

$\partial_y u (x_0, y_0) = - \partial_x v (x_0, y_0)$

. These equations are called the Cauchy-Riemann equations.

Proof:

1. We prove well-definedness of $u$ and $v$.

Let $(x, y) \in \Phi(O)$. We apply the inverse function on both sides to obtain:

$x + iy \in \Phi^{-1}(\Phi(O)) = O$

, where the last equality holds since $\Phi$ is bijective (for any bijective $f: S_1 \to S_2$ we have $f^{-1}(f(S_3)) = f(f^{-1}(S_3)) = S_3$ if $S_3 \subseteq S_1$; see exercise 1).

3. We prove differentiability of $u$ and $v$ and the Cauchy-Riemann equations.

We define

$S_1 := \{z \in \mathbb C | \Re z = \Re z_0 \} \cap O$

and

$S_2 := \{z \in \mathbb C | \Im z = \Im z_0 \} \cap O$

Then we have:

\begin{align} \partial_x u (x_0, y_0) & = \lim_{x \to x_0} \frac{u(x, y_0) - u(x_0, y_0)}{x - x_0} & \\ & = \lim_{x \to x_0} \frac{\Re f(x + iy_0) - \Re f(x_0 + iy_0)}{x - x_0} & \\ & = \Re \left( \lim_{x \to x_0} \frac{f(x + iy_0) - f(x_0 + iy_0)}{x - x_0} \right) & \text{continuity of } \Re \\ & = \Re \left( \lim_{z \to z_0 \atop z \in S_2} \frac{f(z) - f(z_0)}{z - z_0} \right) & \\ & = \Re \left( \lim_{z \to z_0 \atop z \in S_1} \frac{f(z) - f(z_0)}{z - z_0} \right) & \text{lemma 2.2.3} \\ & = \Re \left( \lim_{y \to y_0} \frac{f(x_0 + iy) - f(x_0 + i y_0)}{iy - iy_0} \right) & \\ & = \Re \left( (-i) \lim_{y \to y_0} \frac{f(x_0 + iy) - f(x_0 + i y_0)}{y - y_0} \right) & i^{-1} = -i \\ & = \Im \left( \lim_{y \to y_0} \frac{f(x_0 + iy) - f(x_0 + i y_0)}{y - y_0} \right) & \\ & = \partial_y v (x_0, y_0) \end{align}

From these equations follows the existence of $\partial_x u (x_0, y_0)$ and $\partial_y v (x_0, y_0)$, since for example

$\lim_{z \to z_0 \atop z \in S_2} \frac{f(z) - f(z_0)}{z - z_0}$

exists due to lemma 2.2.3.

The proof for

$\partial_y u (x_0, y_0) = - \partial_x v (x_0, y_0)$

and the existence of $\partial_y u (x_0, y_0)$ and $\partial_x v (x_0, y_0)$ we leave for exercise 2.

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Holomorphic functions

Definitions 2.3.4:

Let $S \subseteq \mathbb C$ and let $f: S \to \mathbb C$ be a function. We call $f$ holomorphic if and only if for all $z_0 \in S$, $f$ is differentiable at $z_0$. In this case, the function

$f': S \to \mathbb C, f'(z_0) = \lim_{z \to z_0 \atop z \in S} \frac{f(z) - f(z_0)}{z - z_0}$

is called the complex derivative of $f$. We write $H(S)$ for the set of holomorphic functions defined on $S$.

Exercises

1. Let $S_1, S_2, S_3$ be sets such that $S_3 \subseteq S_1$, and let $f: S_1 \to S_2$ be a bijective function. Prove that $f^{-1}(f(S_3)) = f(f^{-1}(S_3)) = S_3$.
2. Let $O \subseteq \mathbb C$ be open, let $f: O \to \mathbb C$ be a function and let $z_0 = x_0 + i y_0 \in O$. Prove that if $f$ is complex differentiable at $z_0$, then $\partial_y u (x_0, y_0)$ and $\partial_x v (x_0, y_0)$ exist and satisfy the equation $\partial_y u (x_0, y_0) = - \partial_x v (x_0, y_0)$.

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