Holomorphic maps ${\mathbb{C}}^m\to {\mathbb{C}}^n$ and the complex Jacobian matrix

Fix $m,n\ge 1$, read ${\mathbb{C}}^m$ and ${\mathbb{C}}^n$ through Complex $m$-space and its real coordinate dictionary, and let $U\subseteq {\mathbb{C}}^m$ be open with $a\in U$. A map $L:{\mathbb{C}}^m\to {\mathbb{C}}^n$ 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 operations being those of the $\mathbb{C}$-vector spaces ${\mathbb{C}}^m$ and ${\mathbb{C}}^n$ (Vector space over a field, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ 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 weaker notion of an $\mathbb{R}$-linear map (A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates).

A map $F:U\to {\mathbb{C}}^n$ is holomorphic at $a$ when there is a $\mathbb{C}$-linear $L:{\mathbb{C}}^m\to {\mathbb{C}}^n$ with

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

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

Such an $L$ is unique. If $L_1,L_2$ both work, then $T=L_1-L_2$ is $\mathbb{C}$-linear with $\parallel T(h)\parallel \mathrm{/}\parallel h\parallel \to 0$; fixing $h\mathrm{\ne }0$ and replacing $h$ by $th$ for small real $t>0$ gives $\parallel T(h)\parallel \mathrm{/}\parallel h\parallel =\parallel T(th)\parallel \mathrm{/}\parallel th\parallel \to 0$, so $T(h)=0$. Write $DF(a):=L$.

The complex Jacobian $J_{\mathbb{C}}F(a)$ is the matrix of $DF(a)$ relative to the standard ordered bases of ${\mathbb{C}}^m$ and ${\mathbb{C}}^n$ (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 $0$, Coordinate columns $[v{]}_{\mathcal{B}}$ and matrices $[T{]}_{\mathcal{B}}^{\mathcal{C}}$ of linear maps relative to ordered bases), an $n\times m$ matrix over the field $\mathbb{C}$ (Finite rectangular matrices over a commutative ring, their entries, rows and columns); its $(j,k)$ entry is the $j$th coordinate of $DF(a)e_k$.

$n=1$ recovers the scalar definition. For $n=1$ the norm on ${\mathbb{C}}^1$ is the modulus and the displayed condition is that of Holomorphic functions on an open subset of ${\mathbb{C}}^m$; the Jacobian is then the single row of the coordinates of $DF(a)$.

The entries are the Wirtinger derivatives of the components. A map into ${\mathbb{C}}^n$ is holomorphic exactly when each of its components is ↗ shows that $F$ is holomorphic exactly when each component $F_j$ is, and then $(J_{\mathbb{C}}F(a){)}_{jk}={\mathrm{\partial }}_{z_k}{F}_j(a)$ with the operators of Wirtinger operators in ${\mathbb{C}}^m$. That identification is proved there and is not assumed here: this definition fixes the Jacobian as the matrix of the differential and nothing more.

The target dimension $n=0$. ${\mathbb{C}}^0$ has exactly one element, so every map into it is holomorphic with zero differential and empty Jacobian; nothing below needs that case and it is recorded only so that the convention is not left open.