Complex Analysis/Complex Functions/Complex Derivatives

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Complex differentiability[edit]

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

, a contradiction.

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The Cauchy-Riemann equations[edit]

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[edit]

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[edit]

  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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