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For $C^{1}$ functions, holomorphy, complex linearity of the real derivative, and the Cauchy–Riemann system agree

Statement

Let $m \geq 1$ , let $U \subseteq C^{m}$ be open and let $f : U \to C$ be of class $C^{1}$ in the real coordinates ( $C^{k}$ maps and multi-index derivative notation in Euclidean space, applied to the real and imaginary parts of $f$ on $U$ read as an open subset of $R^{2 m}$ ). Let $a \in U$ . The following are equivalent.

$f$ is complex differentiable at $a$ .
$f$ is real totally differentiable at $a$ and its real total derivative $D f (a)$ is $C$ -linear.
$f$ is real totally differentiable at $a$ and $\partial_{{\overset{ˉ}{z}}_{k}} f (a) = 0$ for every $k < m$ — the several-variable Cauchy–Riemann system.

The $C^{1}$ hypothesis is used only to pass from the Cauchy–Riemann system to real total differentiability; the implications from 1 to 2 and between 2 and 3 hold at any point with no regularity beyond what each condition states.

Facts & Assumptions

Given: An open $U \subseteq C^{m}$ , a $C^{1}$ function $f : U \to C$ and a point $a \in U$ ; $C^{m}$ is read through Complex $m$ -space and its real coordinate dictionary.

[L1]

$f$ is complex differentiable at $a$ when there is a $C$ -linear $L$ with $f (a + h) = f (a) + L (h) + r (h)$ and $∣ r (h) ∣ / ∥ h ∥ \to 0$ ; that condition is the total-differentiability condition with the extra requirement that $L$ be $C$ -linear (Holomorphic functions on an open subset of $C^{m}$ ).

[L2]

$f$ is totally differentiable at $a$ when there is an $R$ -linear $L$ with $∥ f (a + h) - f (a) - L h ∥ / ∥ h ∥ \to 0$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder), and such an $L$ is unique (The total derivative at a point is unique).

[L3]

At a point where all $2 m$ real partial derivatives exist, $\partial_{z_{k}} f = \frac{1}{2} (\partial_{x_{k}} f - i \partial_{y_{k}} f)$ and $\partial_{{\overset{ˉ}{z}}_{k}} f = \frac{1}{2} (\partial_{x_{k}} f + i \partial_{y_{k}} f)$ ; at a point of real total differentiability, $D f (a) h = \sum_{k < m} (\partial_{z_{k}} f (a)) h_{k} + \sum_{k < m} (\partial_{{\overset{ˉ}{z}}_{k}} f (a)) \overline{h_{k}}$ (Wirtinger operators in $C^{m}$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[L4]

An $R$ -linear $T$ has a unique representation $T (h) = \sum_{k} c_{k} h_{k} + \sum_{k} d_{k} \overline{h_{k}}$ , and $T$ is $C$ -linear exactly when every $d_{k} = 0$ (A real-linear functional on $C^{m}$ is complex linear exactly when its antiholomorphic part vanishes).

[L5]

If every partial derivative of $f$ exists near $a$ and is continuous at $a$ , then $f$ is totally differentiable at $a$ (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

[L6]

$f$ is of class $C^{1}$ when its first-order coordinate partial derivatives exist and are continuous ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

[L7]

For one complex variable, complex differentiability at $a$ , real total differentiability with $D f (a)$ multiplication by a complex number, and real total differentiability with $\partial_{\overset{ˉ}{z}} f (a) = 0$ are equivalent, and then $f^{'} (a) = \partial_{z} f (a)$ (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).

[L8]

A holomorphic function of several variables is continuous and separately holomorphic with $D f (a) h = \sum_{k < m} (\partial_{z_{k}} f (a)) h_{k}$ (A holomorphic function of several variables is continuous and separately holomorphic), and it is smooth in the real coordinates (Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic).

Proof

technique · direct

1.1

Condition 1 implies condition 2. If $f$ is complex differentiable at $a$ with $C$ -linear $L$ , then $L$ is in particular $R$ -linear and the same remainder condition is the one in [L2] read through the dictionary, so $f$ is real totally differentiable at $a$ and, by the uniqueness in [L2], $D f (a) = L$ is $C$ -linear.

givenL1L2

1.2

Condition 2 implies condition 1. If $f$ is real totally differentiable at $a$ with $C$ -linear $D f (a)$ , the remainder condition of [L2] is exactly that of [L1] for the $C$ -linear map $D f (a)$ , so $f$ is complex differentiable at $a$ .

givenL1L2

1.3

Conditions 2 and 3 are equivalent. Real total differentiability is common to both, and given it, [L3] represents $D f (a)$ in the form of [L4] with $c_{k} = \partial_{z_{k}} f (a)$ and $d_{k} = \partial_{{\overset{ˉ}{z}}_{k}} f (a)$ ; by the uniqueness in [L4] the map $D f (a)$ is $C$ -linear exactly when every $\partial_{{\overset{ˉ}{z}}_{k}} f (a)$ vanishes.

givenL3L4

1.4

The $C^{1}$ hypothesis enters only here: it makes the first-order real partial derivatives exist near $a$ and be continuous by [L6], so [L5] supplies the real total differentiability that conditions 2 and 3 name. Without it, the Cauchy–Riemann system alone constrains the partial derivatives and asserts nothing about the existence of $D f (a)$ .

givenL5L6

2.1

Steps 1.1, 1.2, 1.3 and 1.4 give the three-way equivalence for a $C^{1}$ function. At $m = 1$ the statement is [L7], with $D f (a)$ being $C$ -linear exactly when it is multiplication by a complex number, namely $\partial_{z} f (a)$ ; and by [L8] a holomorphic function of several variables is automatically $C^{1}$ , so the $C^{1}$ hypothesis restricts only the direction that starts from the Cauchy–Riemann system.

step 1.1step 1.2step 1.3step 1.4L7L8∎

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Sources

H. P. Boas, Lecture Notes on Multidimensional Complex Analysis, Ch. 1 (standard reference, not scraped)

For C1C1 functions, holomorphy, complex linearity of the real derivative, and the Cauchy–Riemann system agree