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Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations

Statement

Let $U \subseteq C$ be open, let $a \in U$ , and write $f = u + i v : U \to C$ . The following are equivalent:

$f$ is complex differentiable at $a$ .
Under $C ≅ R^{2}$ , the map $f$ is real totally differentiable at $a$ and $D f (a)$ is multiplication by a complex number.
The map $f$ is real totally differentiable at $a$ and $\partial_{\overset{ˉ}{z}} f (a) = 0$ .
The map $f$ is real totally differentiable at $a$ and satisfies the Cauchy–Riemann equations

$u_{x} (a) = v_{y} (a), u_{y} (a) = - v_{x} (a) .$

When these conditions hold,

$f^{'} (a) = u_{x} (a) + i v_{x} (a) = v_{y} (a) - i u_{y} (a) = \partial_{z} f (a) .$

Facts & Assumptions

Given: An open set $U \subseteq C$ , a point $a \in U$ , and a map $f = u + i v : U \to C$ .

[F1]

Complex differentiability at $a$ is existence of the limit $(f (a + h) - f (a)) / h$ as $h \to 0$ through nonzero increments with $a + h \in U$ (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions).

[F2]

By Real total derivatives transport through the complex coordinate isometry, real total differentiability at $a$ means that for some real-linear $L$ , $f (a + h) = f (a) + L h + r (h)$ with $∣ r (h) ∣ / ∣ h ∣ \to 0$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, A linear map $L : R^{m} \to R^{n}$ in Euclidean coordinates).

[L1]

If a map is totally differentiable at $a$ , then its directional derivatives exist and equal $D f (a) v$ ; its partial derivatives are the columns of its Jacobian matrix (A total derivative computes every directional derivative, and its matrix is the Jacobian, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

[L2]

Every real-linear map between Euclidean spaces has a unique matrix and is bounded by a constant times the Euclidean norm (Every Euclidean linear map has a unique matrix and satisfies $∥ L h ∥_{2} \leq K ∥ h ∥_{2}$ for some $K \geq 0$ ).

[L3]

Under $Φ (a + b i) = (a, b)$ , complex multiplication satisfies $(a + b i) (x + i y) = (a x - b y) + i (b x + a y)$ ( $C$ is the real coordinate plane, with coordinate arithmetic).

[F3]

For a real-differentiable $f$ , $D f (h) = (\partial_{z} f) h + (\partial_{\overset{ˉ}{z}} f) \overset{ˉ}{h}$ , with $\partial_{\overset{ˉ}{z}} f = \frac{1}{2} (u_{x} - v_{y}) + \frac{i}{2} (v_{x} + u_{y})$ (The Wirtinger derivatives $\partial_{z} f$ and $\partial_{\overset{ˉ}{z}} f$ , and antiholomorphic functions).

[L4]

For complex numbers, $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ and $∣ z ∣ = 0$ if and only if $z = 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Proof

technique · direct

1.1F1L4givenalgebra

Assume condition 1 and write $L = f^{'} (a)$ . For $h \neq 0$ put $r (h) = f (a + h) - f (a) - L h$ ; then $∣ r (h) ∣ / ∣ h ∣ = ∣ (f (a + h) - f (a)) / h - L ∣ \to 0$ .

1.2F1F2L4givenalgebra

Conversely assume condition 2, so $f (a + h) - f (a) = L h + r (h)$ with $∣ r (h) ∣ / ∣ h ∣ \to 0$ . For nonzero $h$ , division by $h$ gives $(f (a + h) - f (a)) / h = L + r (h) / h$ , and $∣ r (h) / h ∣ = ∣ r (h) ∣ / ∣ h ∣ \to 0$ ; hence condition 1 holds with $f^{'} (a) = L$ .

1.3L1L2L3

Write the matrix of $D f (a)$ as $(\begin{array}{cc} u_{x} & u_{y} \\ v_{x} & v_{y} \end{array})$ by [L1]. By [L3], it is multiplication by $α + i β$ exactly when it is $(\begin{array}{cc} α & - β \\ β & α \end{array})$ .

1.4F3algebra

By [F3], $\partial_{\overset{ˉ}{z}} f (a) = 0$ exactly when both $u_{x} - v_{y} = 0$ and $v_{x} + u_{y} = 0$ . Hence condition 3 is equivalent to condition 4.

2.1step 1.1F2L2L3

The map $h \mapsto L h$ is real-linear: its coordinate matrix is real linear by [L3], and The complex coordinate map preserves real scalar multiplication transports real linearity back to $C$ . so step 1.1 is the remainder condition [F2]. Thus condition 2 holds.

2.2step 1.3algebra

Therefore condition 2 is equivalent to condition 4: equality with the multiplication matrix is exactly $u_{x} = v_{y}$ and $u_{y} = - v_{x}$ . In that case $α = u_{x} = v_{y}$ , $β = v_{x} = - u_{y}$ , so the multiplier is $u_{x} + i v_{x} = v_{y} - i u_{y}$ .

3.1step 1.2step 2.2step 1.4F3∎

Under the equivalent conditions, [F3] and the Cauchy–Riemann equations give $\partial_{z} f (a) = u_{x} + i v_{x}$ , while steps 1.2 and 2.2 identify the same number with $f^{'} (a)$ . Thus all four conditions are equivalent and the displayed derivative formulas hold.

Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂zˉf=0∂zˉ​f=0, or with the Cauchy–Riemann equations

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations