Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket

Holomorphic maps. Let $m,p\ge 1$ and let $D\subseteq {\mathbb{C}}^m$ be open. Write each coordinate $z_k=x_k+i{y}_k$ and identify $D$ with the corresponding open subset of ${\mathbb{R}}^{2m}$ by $(z_1,\dots ,z_m)\mapsto (x_1,y_1,\dots ,x_m,y_m)$. A map $F:D\to {\mathbb{C}}^p$ is complex differentiable at $a\in D$ when there is a map $L:{\mathbb{C}}^m\to {\mathbb{C}}^p$ that is $\mathbb{C}$-linear — $L(u+v)=L(u)+L(v)$ and $L(\lambda u)=\lambda L(u)$ for $\lambda \in \mathbb{C}$ (Linear map between vector spaces over the same field, A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$) — with

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

the norm being the Euclidean norm of ${\mathbb{R}}^{2m}$ and ${\mathbb{R}}^{2p}$ (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder). Such an $L$ is unique by the argument of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions, applied componentwise; it is denoted $DF(a)$. The map $F$ is holomorphic on $D$ when it is complex differentiable at every point of $D$; for $m=p=1$ this is the notion of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions. For holomorphic $F$ and $k\le m$ put ${\mathrm{\partial }}_{z_k}F(a):=DF(a)(e_k)$, the $k$-th coordinate derivative; for $p=1$ this is the partial derivative of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ in the complex coordinate direction.

Real analytic functions. Let $N\ge 1$ and let $V\subseteq {\mathbb{R}}^N$ be open. A function $\phi :V\to \mathbb{R}$ is real analytic at $x\in V$ when there are an open set $W\subseteq {\mathbb{C}}^N$ with $x\in W$ and a holomorphic $F:W\to \mathbb{C}$ with $F(\xi )=\phi (\xi )$ for every $\xi \in W\cap V$; $\phi$ is real analytic on $V$ when it is real analytic at every point of $V$. Thus real analyticity is defined here exactly as admitting a local holomorphic extension.

Polydiscs. For $a\in {\mathbb{C}}^m$ and a polyradius $\rho =({\rho }_1,\dots ,{\rho }_m)$ with $0<{\rho }_k\le \mathrm{\infty }$ put

${\mathrm{\Delta }}_{\rho }(a):=\{z\in {\mathbb{C}}^m:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }<{\rho }_k\text{ for all }k\},$

the open polydisc, which is open and convex (Open ball, closed ball and sphere in a metric space, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). If every ${\rho }_k$ is finite, its distinguished boundary is

${\mathrm{\Gamma }}_{\rho }(a):=\{z\in {\mathbb{C}}^m:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }={\rho }_k\text{ for all }k\}\mathrm{.}$

When any radius is infinite, no distinguished boundary is defined.

Complex phase domains. Let $d\ge 1$. A complex phase domain is a nonempty open $D\subseteq {\mathbb{C}}^{2d}$ whose points are written $Z=(z,w)$ with $z=(z^1,\dots ,z^d)$, $w=(w_1,\dots ,w_d)\in {\mathbb{C}}^d$. In these coordinates, ${\mathrm{\partial }}_{z^i}$ and ${\mathrm{\partial }}_{w_i}$ denote the coordinate derivatives defined above for the corresponding entries of $Z$. On $D$ define the complexified canonical two-form and the complexified Poisson bracket of holomorphic functions $f,g:D\to \mathbb{C}$ by

${\omega }_{\mathbb{C}}:={\sum }_{i=1}^{d}d{z}^i\wedge d{w}_i,\{f,g\}:={\sum }_{i=1}^d({\mathrm{\partial }}_{z^i}f{\mathrm{\partial }}_{w_i}g-{\mathrm{\partial }}_{w_i}f{\mathrm{\partial }}_{z^i}g),$

and for a holomorphic $f:D\to \mathbb{C}$ its complex Hamiltonian field $X_f:=J\mathrm{\nabla }f$, where the same entries of $J=\left(\begin{array}{cc}{0}_d& I_d\\ -I_d& 0_d\end{array}\right)$ from A phase domain, its canonical one-form and its standard symplectic two-form are regarded as a matrix in $M_{2d}(\mathbb{C})$, and $\mathrm{\nabla }f=({\mathrm{\partial }}_{z^1}f,\dots ,{\mathrm{\partial }}_{z^d}f,{\mathrm{\partial }}_{w_1}f,\dots ,{\mathrm{\partial }}_{w_d}f)$ is the gradient column of coordinate derivatives. If $D_{\mathbb{R}}:=D\cap {\mathbb{R}}^{2d}$ is nonempty, it is open in ${\mathbb{R}}^{2d}$, and the restrictions of ${\omega }_{\mathbb{C}}$ and $J$ to $D_{\mathbb{R}}$ are exactly the standard two-form and matrix of A phase domain, its canonical one-form and its standard symplectic two-form.

Distance domains. For a nonempty $U\subseteq {\mathbb{R}}^{2d}$ and $r>0$ put

$D(r):=\{\zeta \in {\mathbb{C}}^{2d}:\mathrm{dist}(\zeta ,\bar{U})<r\},\mathrm{dist}(\zeta ,\bar{U}):=\inf \{\parallel \zeta -x\parallel :x\in \bar{U}\},$

where $\bar{U}$ is the closure of $U$ in ${\mathbb{R}}^{2d}$ and the norm is the Euclidean norm of ${\mathbb{C}}^{2d}\cong {\mathbb{R}}^{4d}$. Each $D(r)$ is open, contains $U$, and $D(r)\subseteq D(r^{\mathrm{\prime }})$ for $r\le r^{\mathrm{\prime }}$; if $U$ is star-shaped with respect to $0$ (Star-shaped open subsets of Euclidean space), then every $D(r)$ is star-shaped with respect to $0$ as well.

• The definition of holomorphy used here is the $\mathbb{C}$-linear total-derivative definition, matching the one-variable published definition. The equivalence with power-series representability and with continuity plus separate holomorphy is not assumed: it is proved in Iterated Cauchy formula, power-series expansion and Cauchy estimates for functions continuous and separately holomorphic on a polydisc and Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits on this page, together with the differentiability of the coordinate derivatives.

• No choice principle is used. All objects are the standard complexifications of the real canonical objects; no physical interpretation is attached to $z,w$ or to ${\omega }_{\mathbb{C}}$ in this definition.

• To verify the distance-domain claim, first note that $\bar{U}$ is star-shaped about $0$: if $x_j\in U$ converges to $x\in \bar{U}$, then $t{x}_j\in U$ and $t{x}_j\to tx$, so $tx\in \bar{U}$ for every $0\le t\le 1$. If $\zeta \in D(r)$, the inequality $\mathrm{dist}(\zeta ,\bar{U})<r$ supplies $x\in \bar{U}$ with $\parallel \zeta -x\parallel <r$. Thus $tx\in \bar{U}$ and $\mathrm{dist}(t\zeta ,\bar{U})\le \parallel t\zeta -tx\parallel =t\parallel \zeta -x\parallel <r$ for $0\le t\le 1$. Hence $t\zeta \in D(r)$, proving that $D(r)$ is star-shaped with respect to $0$.