Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic

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

A continuous separately holomorphic function on an open set is holomorphic, and for a continuous function holomorphy, separate holomorphy and local power-series representability agree (Osgood's lemma: continuous and separately holomorphic implies holomorphic).

For $f$ continuous and separately holomorphic on ${\mathrm{\Delta }}_{\rho }(a)$ and $r_k<{\rho }_k$, there are coefficients with $\mathrm{\mid }{c}_{\alpha }\mathrm{\mid }\le M{\prod }_{k<m}{r}_k^{-{\alpha }_k}$ and $f={\sum }_{\alpha }{c}_{\alpha }(z-a{)}^{\alpha }$ on ${\mathrm{\Delta }}_r(a)$ (A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc).

Under such a coefficient bound the sum is holomorphic on ${\mathrm{\Delta }}_r(a)$, differentiates termwise with ${\mathrm{\partial }}_{z_k}$, the derived series obeys a bound of the same shape on every smaller polyradius, every iterated ${\mathrm{\partial }}_z^{\beta }$ of the sum exists, and ${\mathrm{\partial }}_z^{\beta }(\text{sum})(a)=\beta !c_{\beta }$ (An absolutely convergent multi-indexed power series is holomorphic and differentiates termwise).

A holomorphic function of several variables is continuous and separately holomorphic, with $Df(a)h={\sum }_{k<m}({\mathrm{\partial }}_{z_k}f(a))h_k$ (A holomorphic function of several variables is continuous and separately holomorphic).

${\mathrm{\partial }}_{z_k}f=\frac{1}{2}({\mathrm{\partial }}_{x_k}f-i{\mathrm{\partial }}_{y_k}f)$ and ${\mathrm{\partial }}_{{\bar{z}}_k}f=\frac{1}{2}({\mathrm{\partial }}_{x_k}f+i{\mathrm{\partial }}_{y_k}f)$ (Wirtinger operators in ${\mathbb{C}}^m$).

An $\mathbb{R}$-linear $T$ has the unique representation $T(h)={\sum }_k{c}_k^{\mathrm{\prime }}{h}_k+{\sum }_k{d}_k^{\mathrm{\prime }}\overline{h_k}$ and is $\mathbb{C}$-linear exactly when every $d_k^{\mathrm{\prime }}=0$; for a real totally differentiable $f$ these coefficients are ${\mathrm{\partial }}_{z_k}f$ and ${\mathrm{\partial }}_{{\bar{z}}_k}f$ (A real-linear functional on ${\mathbb{C}}^m$ is complex linear exactly when its antiholomorphic part vanishes).

If a property holds at $0$ and passes from $n$ to $n+1$, it holds for every natural number (The principle of mathematical induction).

Complex differentiability at $a$ gives real total differentiability at $a$ with the same differential (Holomorphic functions on an open subset of ${\mathbb{C}}^m$).

$f$ is of class $C^n$ on an open subset of ${\mathbb{R}}^{2m}$ when every iterated coordinate partial derivative of order at most $n$ exists and is continuous ($C^k$ maps and multi-index derivative notation in Euclidean space), and $\alpha !={\prod }_{k<m}{\alpha }_k!$ with $0!=1$ (The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$).

A complex differentiable function is continuous (Complex differentiability at a point implies continuity there).

${\mathrm{\Delta }}_r(a)$ is defined coordinatewise by $\mathrm{\mid }{z}_k-a_k\mathrm{\mid }<r_k$ (Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$).