Holomorphic functions on an open subset of ${\mathbb{C}}^m$

Fix $m\ge 1$, read ${\mathbb{C}}^m$ through Complex $m$-space and its real coordinate dictionary, and let $U\subseteq {\mathbb{C}}^m$ be open and $a\in U$.

A map $L:{\mathbb{C}}^m\to \mathbb{C}$ is $\mathbb{C}$-linear when $L(u+v)=L(u)+L(v)$ and $L(\lambda u)=\lambda L(u)$ for all $u,v\in {\mathbb{C}}^m$ and all $\lambda \in \mathbb{C}$, the vector-space operations being those of Vector space over a field over the field $\mathbb{C}$ ($\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$). Requiring the second clause only for real $\lambda$ gives the strictly weaker notion of an $\mathbb{R}$-linear map (A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates) read through the dictionary.

A function $f:U\to \mathbb{C}$ is complex differentiable at $a$ when there is a $\mathbb{C}$-linear $L:{\mathbb{C}}^m\to \mathbb{C}$ with

$f(a+h)=f(a)+L(h)+r(h),\frac{\mathrm{\mid }r(h)\mathrm{\mid }}{\parallel h\parallel }⟶0\text{ as }h\to 0,$

the quotient being considered for $h\mathrm{\ne }0$ with $a+h\in U$ and the norm being that of the dictionary (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms). The map $f$ is holomorphic on $U$ when it is complex differentiable at every point of $U$.

Such an $L$ is unique, so the notation $Df(a):=L$ is well posed. If $L_1$ and $L_2$ both satisfy the condition, then $T=L_1-L_2$ is $\mathbb{C}$-linear and $\mathrm{\mid }T(h)\mathrm{\mid }\mathrm{/}\parallel h\parallel \to 0$; fixing $h\mathrm{\ne }0$ and taking $h$ replaced by $th$ for real $t\in (0,1)$ small enough that $a+th\in U$, $\mathbb{C}$-linearity gives $\mathrm{\mid }T(h)\mathrm{\mid }\mathrm{/}\parallel h\parallel =\mathrm{\mid }T(th)\mathrm{\mid }\mathrm{/}\parallel th\parallel$, whose limit as $t\to 0$ is $0$; so $T(h)=0$ for every $h$.

No continuity and no local boundedness are built in. The definition asks for the linear approximation and nothing else. That a holomorphic function is continuous is proved on this page rather than assumed, and the two theorems that recover holomorphy from separate holomorphy — under continuity, and under local boundedness — are theorems precisely because those properties are not part of the definition. Defining holomorphy by local power-series representability or by the $C^1$ Cauchy–Riemann system, as some treatments do, would make one or other of them a tautology.

At $m=1$ this is the published one-variable notion. A $\mathbb{C}$-linear $L:\mathbb{C}\to \mathbb{C}$ satisfies $L(h)=hL(1)$, so with $c=L(1)$ the condition reads $f(a+h)=f(a)+ch+r(h)$ with $\mathrm{\mid }r(h)\mathrm{\mid }\mathrm{/}\mathrm{\mid }h\mathrm{\mid }\to 0$, which is exactly complex differentiability at $a$ with $f^{\mathrm{\prime }}(a)=c$ in the sense of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions; conversely that condition produces the $\mathbb{C}$-linear map $h\mapsto f^{\mathrm{\prime }}(a)h$.

Relation to the real total derivative. Reading $f$ as a map ${\mathbb{R}}^{2m}\to {\mathbb{R}}^2$ through the dictionary, the displayed condition is the total-differentiability condition of The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder with the extra requirement that the approximating linear map be $\mathbb{C}$-linear and not merely $\mathbb{R}$-linear. So a complex differentiable $f$ is real totally differentiable with $Df(a)$ as its real total derivative, and The total derivative at a point is unique says the two uses of the notation cannot disagree. The standard basis vectors $e_k$ of The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$ are the ones used to read off coordinates of $L$.