Holomorphic Functions of Several Complex Variables

Fix a natural number $m\ge 1$. Complex $m$-space ${\mathbb{C}}^m$ is the set of functions $m\to \mathbb{C}$, so a point $z$ has coordinates $z_k$ for $k<m$, indexed from $0$ exactly as ${\mathbb{R}}^n$ is in this library. With coordinatewise addition and multiplication by complex scalars it is a vector space over the field $\mathbb{C}$ (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 $(a-bi)\mathrm{/}(a^2+b^2)$), with the standard basis $e_0,\dots ,e_{m-1}$ 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 $0$.

The coordinate identification. Writing $z_k=x_k+i{y}_k$ with $x_k,y_k$ real (Real and imaginary parts, complex conjugation, and modulus), define

$\mathrm{\Phi }:{\mathbb{C}}^m\to {\mathbb{R}}^{2m},\mathrm{\Phi }(z)=(x_0,y_0,x_1,y_1,\dots ,x_{m-1},y_{m-1})\mathrm{.}$

The interleaved ordering is the one used throughout this page; the ordering that groups all real parts before all imaginary parts is a different bijection, and nothing below is stated for it. $\mathrm{\Phi }$ is a bijection and is $\mathbb{R}$-linear.

Norms agree. Put $\parallel z\parallel :=({\sum }_{k<m}\mathrm{\mid }{z}_k{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Since $\mathrm{\mid }{z}_k{\mathrm{\mid }}^2=x_k^2+y_k^2$, this is the Euclidean norm $\parallel \mathrm{\Phi }(z){\parallel }_2$ of The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$ and The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and it is a norm on the real vector space underlying ${\mathbb{C}}^m$ in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms. Consequently $\parallel z-w\parallel =\parallel \mathrm{\Phi }(z)-\mathrm{\Phi }(w){\parallel }_2$, so the metric of ${\mathbb{C}}^m$, its balls (Open ball, closed ball and sphere in a metric space), its open sets, its convergent sequences, its Cauchy sequences and its continuous maps are verbatim those of ${\mathbb{R}}^{2m}$ under $\mathrm{\Phi }$. In particular convergence and continuity are coordinatewise (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions), ${\mathbb{C}}^m$ is complete (For $n\ge 1$ a sequence in ${\mathbb{R}}^n$ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and ${\mathbb{R}}^n$ is complete in every norm), and a subset of ${\mathbb{C}}^m$ is compact exactly when it is closed and bounded (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

At $m=1$ this is the published plane dictionary. For $m=1$ the map $\mathrm{\Phi }$ is the bijection of $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves and every clause above reduces to a clause recorded there. Openness, connectedness and real total differentiability on ${\mathbb{C}}^m$ are always read through $\mathrm{\Phi }$, exactly as that remark reads them through its own identification.

What $\mathrm{\Phi }$ does not carry. $\mathrm{\Phi }$ respects the additive and the real scalar structure but not multiplication by $i$ in any way visible to a general $\mathbb{R}$-linear map of ${\mathbb{R}}^{2m}$: an $\mathbb{R}$-linear map of ${\mathbb{C}}^m$ need not be $\mathbb{C}$-linear. That distinction is the whole content of the criterion the page proves next, and it is why "linear" is always qualified below.

Fix $m\ge 1$ and read ${\mathbb{C}}^m$ through Complex $m$-space and its real coordinate dictionary. A polyradius is a function $r:m\to \mathbb{R}$ with $r_k>0$ for every $k<m$; a single positive real $r$ abbreviates the constant polyradius with every $r_k=r$.

For $a\in {\mathbb{C}}^m$ and a polyradius $r$, the open polydisc, the closed polydisc and the distinguished boundary are

${\mathrm{\Delta }}_r(a):=\{z:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }<r_k\text{ for every }k<m\},$ ${\bar{\mathrm{\Delta }}}_r(a):=\{z:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }\le r_k\text{ for every }k<m\},$ ${\mathrm{\Gamma }}_r(a):=\{z:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }=r_k\text{ for every }k<m\}\mathrm{.}$

Thus ${\mathrm{\Gamma }}_r(a)$ is the set of points all of whose coordinates lie on their own circle: it is the product of the $m$ circles $\{\mathrm{\mid }{\zeta }_k-a_k\mathrm{\mid }=r_k\}$.

The open ball and closed ball of centre $a$ and radius $\rho >0$ are those of the norm $\parallel \cdot \parallel$ of the dictionary, that is the sets $B(a,\rho )=\{z:\parallel z-a\parallel <\rho \}$ and $\bar{B}(a,\rho )=\{z:\parallel z-a\parallel \le \rho \}$ of Open ball, closed ball and sphere in a metric space and Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$.

The distinguished boundary is not the topological boundary when $m\ge 2$. The topological boundary of ${\bar{\mathrm{\Delta }}}_r(a)$ consists of the points where at least one coordinate satisfies $\mathrm{\mid }{z}_k-a_k\mathrm{\mid }=r_k$, whereas ${\mathrm{\Gamma }}_r(a)$ requires every coordinate to do so. For $m=1$ the two coincide. For $m\ge 2$ the inclusion ${\mathrm{\Gamma }}_r(a)\subseteq \mathrm{\partial }{\bar{\mathrm{\Delta }}}_r(a)$ is proper: the point whose first coordinate is $a_0+r_0$ and whose remaining coordinates are $a_k$ lies in the topological boundary and not in ${\mathrm{\Gamma }}_r(a)$.

Polydiscs are open and convex. Openness is coordinatewise: if $\mathrm{\mid }{z}_k-a_k\mathrm{\mid }<r_k$ for every $k$, then $B(z,\rho )\subseteq {\mathrm{\Delta }}_r(a)$ for $\rho ={\min }_{k<m}(r_k-\mathrm{\mid }{z}_k-a_k\mathrm{\mid })>0$, because $\mathrm{\mid }{w}_k-z_k\mathrm{\mid }\le \parallel w-z\parallel$ by the dictionary. Convexity in the sense of A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points is also coordinatewise: for $z,w$ in ${\mathrm{\Delta }}_r(a)$ and $t\in [0,1]$, $\mathrm{\mid }(1-t)z_k+t{w}_k-a_k\mathrm{\mid }\le (1-t)\mathrm{\mid }{z}_k-a_k\mathrm{\mid }+t\mathrm{\mid }{w}_k-a_k\mathrm{\mid }<r_k$ by Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive. Hence a polydisc is star-shaped with respect to each of its points (Star-shaped open subsets of Euclidean space). The same computation gives convexity of the closed polydisc.

Slices are discs. Fixing all coordinates but the $k$th at values $a_j^{\mathrm{\prime }}$ with $\mathrm{\mid }{a}_j^{\mathrm{\prime }}-a_j\mathrm{\mid }<r_j$, the set of $\zeta$ with the resulting point in ${\mathrm{\Delta }}_r(a)$ is exactly the open disc $\{\zeta :\mathrm{\mid }\zeta -a_k\mathrm{\mid }<r_k\}$; this is what makes the one-variable theory applicable one coordinate at a time. Moduli, real and imaginary parts are those of Real and imaginary parts, complex conjugation, and modulus.

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 $L$.

Fix $m\ge 1$, let $U\subseteq {\mathbb{C}}^m$ be open and let $f:U\to \mathbb{C}$. For $a\in U$ and $k<m$ write

$U_{a,k}:=\{\zeta \in \mathbb{C}:(a_0,\dots ,a_{k-1},\zeta ,a_{k+1},\dots ,a_{m-1})\in U\}$

and let $f_{a,k}:U_{a,k}\to \mathbb{C}$ be the $k$th slice $f_{a,k}(\zeta )=f(a_0,\dots ,a_{k-1},\zeta ,a_{k+1},\dots ,a_{m-1})$.

The function $f$ is separately holomorphic on $U$ when for every $a\in U$ and every $k<m$ the slice $f_{a,k}$ is holomorphic on $U_{a,k}$ in the one-variable sense of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions.

Each $U_{a,k}$ is open: if $\zeta \in U_{a,k}$, the corresponding point of $U$ has a ball $B(\cdot ,\rho )\subseteq U$ by 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, and changing only the $k$th coordinate by less than $\rho$ moves the point by less than $\rho$ in the norm of Complex $m$-space and its real coordinate dictionary, so the disc $\{\xi :\mathrm{\mid }\xi -\zeta \mathrm{\mid }<\rho \}$ lies in $U_{a,k}$ (Open ball, closed ball and sphere in a metric space).

No continuity in the remaining variables is asked. The condition constrains each slice separately and says nothing about how the slices fit together: a separately holomorphic function is not assumed continuous as a function on $U$, and on this page the two theorems that supply joint regularity — Osgood's lemma under continuity, and the locally bounded theorem — are what close that gap.

The slice through a point of a polydisc is a disc. If $U={\mathrm{\Delta }}_r(a^{\mathrm{\prime }})$ is a polydisc (Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$) and $a\in U$, then $U_{a,k}$ is the disc $\{\zeta :\mathrm{\mid }\zeta -a_k^{\mathrm{\prime }}\mathrm{\mid }<r_k\}$, which is what lets the one-variable theory be applied one coordinate at a time with the others held fixed.

Fix $m\ge 1$, let $U\subseteq {\mathbb{C}}^m$ be open and let $f:U\to \mathbb{C}$. Read ${\mathbb{C}}^m$ as ${\mathbb{R}}^{2m}$ through Complex $m$-space and its real coordinate dictionary, with real coordinates $x_k,y_k$ for $k<m$ given by $z_k=x_k+i{y}_k$ (Real and imaginary parts, complex conjugation, and modulus), and let ${\mathrm{\partial }}_{x_k}f$ and ${\mathrm{\partial }}_{y_k}f$ be the partial derivatives of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ applied to the two real components of $f$ and recombined.

At a point where all $2m$ of these partial derivatives exist, define the Wirtinger operators

${\mathrm{\partial }}_{z_k}f:=\frac{1}{2}({\mathrm{\partial }}_{x_k}f-i{\mathrm{\partial }}_{y_k}f),{\mathrm{\partial }}_{{\bar{z}}_k}f:=\frac{1}{2}({\mathrm{\partial }}_{x_k}f+i{\mathrm{\partial }}_{y_k}f)(k<m)\mathrm{.}$

The differential identity. Suppose in addition that $f$ is real totally differentiable at a point $a\in U$, so that $Df(a)$ is the $\mathbb{R}$-linear map with $Df(a)h={\sum }_{k<m}\left(\right({\mathrm{\partial }}_{x_k}f(a)){\xi }_k+({\mathrm{\partial }}_{y_k}f(a)\left){\eta }_k\right)$ for $h_k={\xi }_k+i{\eta }_k$, by A total derivative computes every directional derivative, and its matrix is the Jacobian read in the standard basis (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 $0$). Substituting ${\xi }_k=\frac{1}{2}(h_k+\overline{h_k})$ and ${\eta }_k=\frac{1}{2i}(h_k-\overline{h_k})$ and collecting the coefficients of $h_k$ and $\overline{h_k}$ using finite sums in the additive commutative monoid of $\mathbb{C}$ (A finite sum in a commutative monoid indexed by an arbitrary finite set) and distributivity in the complex 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 $(a-bi)\mathrm{/}(a^2+b^2)$) gives

$Df(a)h={\sum }_{k<m}\left(\right({\mathrm{\partial }}_{z_k}f(a))h_k+({\mathrm{\partial }}_{{\bar{z}}_k}f(a)\left)\overline{h_k}\right)\mathrm{.}$

Indeed the coefficient of $h_k$ is $\frac{1}{2}{\mathrm{\partial }}_{x_k}f(a)+\frac{1}{2i}{\mathrm{\partial }}_{y_k}f(a)=\frac{1}{2}({\mathrm{\partial }}_{x_k}f(a)-i{\mathrm{\partial }}_{y_k}f(a))$, and the coefficient of $\overline{h_k}$ is $\frac{1}{2}{\mathrm{\partial }}_{x_k}f(a)-\frac{1}{2i}{\mathrm{\partial }}_{y_k}f(a)=\frac{1}{2}({\mathrm{\partial }}_{x_k}f(a)+i{\mathrm{\partial }}_{y_k}f(a))$.

At $m=1$ these are the published Wirtinger derivatives. The two displayed formulas are literally those of The Wirtinger derivatives ${\mathrm{\partial }}_{z}f$ and ${\mathrm{\partial }}_{\bar{z}}f$, and antiholomorphic functions with $x_0,y_0$ written $x,y$, and the differential identity reduces to the identity $Df(h)=({\mathrm{\partial }}_{z}f)h+({\mathrm{\partial }}_{\bar{z}}f)\bar{h}$ recorded there.

These are operators on real-differentiable functions, not on holomorphic ones. Nothing above assumes any complex differentiability: the definition needs only the $2m$ real partial derivatives, and the differential identity needs only real total differentiability. Which functions have all ${\mathrm{\partial }}_{{\bar{z}}_k}f=0$ is the question the next lemma and the Cauchy–Riemann characterisation answer.

Fix $m\ge 1$ and let $T:{\mathbb{C}}^m\to \mathbb{C}$ be $\mathbb{R}$-linear, that is additive with $T(\lambda h)=\lambda T(h)$ for every real $\lambda$. Then there are unique $c_0,\dots ,c_{m-1}$ and $d_0,\dots ,d_{m-1}$ in $\mathbb{C}$ with

$T(h)={\sum }_{k<m}{c}_k{h}_k+{\sum }_{k<m}{d}_k\overline{h_k}(h\in {\mathbb{C}}^m),$

namely $c_k=\frac{1}{2}(T(e_k)-iT(i{e}_k))$ and $d_k=\frac{1}{2}(T(e_k)+iT(i{e}_k))$. Moreover $T$ is $\mathbb{C}$-linear if and only if $d_k=0$ for every $k<m$.

Let $U\subseteq {\mathbb{C}}^m$ be open and let $f:U\to \mathbb{C}$ be holomorphic (Holomorphic functions on an open subset of ${\mathbb{C}}^m$). Then $f$ is continuous on $U$ and separately holomorphic on $U$ (Separately holomorphic functions). Moreover, for $a\in U$ and $k<m$ the slice $f_{a,k}$ is complex differentiable at $a_k$ with derivative $Df(a)e_k$, and

$Df(a)e_k={\mathrm{\partial }}_{z_k}f(a),Df(a)h={\sum }_{k<m}({\mathrm{\partial }}_{z_k}f(a))h_k\mathrm{.}$

Fix $m\ge 1$ and read ${\mathbb{C}}^m$ through Complex $m$-space and its real coordinate dictionary. A multi-index is $\alpha \in {\mathbb{N}}^m$, with $\mathrm{\mid }\alpha \mathrm{\mid }={\sum }_{k<m}{\alpha }_k$ and $\alpha !={\prod }_{k<m}{\alpha }_k!$ as in $C^k$ maps and multi-index derivative notation in Euclidean space, every index running over $k<m$ from $0$. For $w\in {\mathbb{C}}^m$ the complex monomial is

$w^{\alpha }:={\prod }_{k<m}{w}_k^{{\alpha }_k},$

a finite product in the multiplicative commutative monoid of $\mathbb{C}$ (The product $g_0{g}_1\cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n=0$) equal to the identity, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$) of natural powers in $\mathbb{C}$ (Integer powers in the complex field); for the zero multi-index $w^0=1$.

Enumerating the index set. $\mathbb{N}$ is countable and Every finite power of an at most countable set is at most countable makes ${\mathbb{N}}^m$ at most countable; it is infinite, so there is a bijection $\sigma :\mathbb{N}\to {\mathbb{N}}^m$ (Injection, surjection, bijection, Finite, countably infinite, countable, uncountable).

Let $c:{\mathbb{N}}^m\to \mathbb{C}$ and $a,z\in {\mathbb{C}}^m$. The multi-indexed power series ${\sum }_{\alpha }{c}_{\alpha }(z-a{)}^{\alpha }$ converges absolutely at $z$ when the complex series ${\sum }_n{c}_{\sigma (n)}(z-a{)}^{\sigma (n)}$ converges absolutely (Complex series, absolute convergence, complex power series, and radius of convergence) for one bijection $\sigma$, equivalently for every one. The two conditions agree, and the sums agree, because for bijections $\sigma ,\tau$ the series along $\tau$ is a rearrangement of the series along $\sigma$: applying Every absolutely convergent complex series converges, and rearrangements preserve its sum to the nonnegative series of moduli transfers convergence, and applying it again to the series itself transfers the sum. That common value is written ${\sum }_{\alpha }{c}_{\alpha }(z-a{)}^{\alpha }$ and no other notion of unordered sum is introduced.

Box partial sums. For $N\in \mathbb{N}$ put $B_N:=\{\alpha \in {\mathbb{N}}^m:{\alpha }_k\le N\text{ for every }k<m\}$, a finite set, and let $S_N(z):={\sum }_{\alpha \in B_N}{c}_{\alpha }(z-a{)}^{\alpha }$ be the corresponding finite sum (A finite sum in a commutative monoid indexed by an arbitrary finite set). If the series converges absolutely at $z$ with sum $S$, then $S_N(z)\to S$. Given $\epsilon >0$, absolute convergence supplies $n_0$ with ${\sum }_{n\ge n_0}\mathrm{\mid }{c}_{\sigma (n)}(z-a{)}^{\sigma (n)}\mathrm{\mid }<\epsilon$ and $\mid {\sum }_{n<n_0}{c}_{\sigma (n)}(z-a{)}^{\sigma (n)}-S\mid \le \epsilon$; taking $N$ large enough that $B_N$ contains $\sigma (0),\dots ,\sigma (n_0-1)$, every index of $B_N$ outside that finite list is $\sigma (n)$ for some $n\ge n_0$, so $\mathrm{\mid }{S}_N(z)-S\mathrm{\mid }\le 2\epsilon$. The same argument bounds $\mathrm{\mid }{S}_N(z)\mathrm{\mid }$ and the tail of the series by the corresponding tails of the series of moduli.

The series converges absolutely and uniformly on a set $S\subseteq {\mathbb{C}}^m$ when there are reals $M_{\alpha }\ge 0$ with $\mathrm{\mid }{c}_{\alpha }(z-a{)}^{\alpha }\mathrm{\mid }\le M_{\alpha }$ for every $z\in S$ and every $\alpha$, and with ${\sum }_n{M}_{\sigma (n)}$ convergent. By Weierstrass M-test for complex-valued function series the partial sums along $\sigma$ then converge uniformly on $S$ (Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary) and the series converges absolutely at every point of $S$.

Why a bijection is fixed rather than an unordered sum defined. The library already has one theory of complex series and one rearrangement theorem, and the clause above uses exactly those. Introducing a separate notion of summation over ${\mathbb{N}}^m$ would create a second convergence notion that every later statement would have to be matched against; instead every multi-indexed sum below means the sum of the one-variable series along any enumeration, which the rearrangement theorem makes unambiguous.

Where the series live. The natural regions here are the polydiscs of Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$ rather than balls. If every $\mathrm{\mid }{z}_k-a_k\mathrm{\mid }$ is positive, absolute convergence at $z$ controls the series on the closed polydisc with that polyradius. If some coordinate is zero, the same coordinatewise domination holds on the corresponding degenerate product set, but that radius vector is not called a polyradius. This is exactly the shape the kernel expansion and the Cauchy estimates on this page produce.

Fix $m\ge 1$, a point $a\in {\mathbb{C}}^m$, a polyradius $r$ and a real $\theta$ with $0\le \theta <1$. For $\zeta \in {\mathrm{\Gamma }}_r(a)$ and $z\in {\bar{\mathrm{\Delta }}}_{\theta r}(a)$, that is $\mathrm{\mid }{\zeta }_k-a_k\mathrm{\mid }=r_k$ and $\mathrm{\mid }{z}_k-a_k\mathrm{\mid }\le \theta r_k$ for every $k<m$,

${\prod }_{k<m}\frac{1}{{\zeta }_k-z_k}={\sum }_{\alpha }(z-a{)}^{\alpha }{\prod }_{k<m}({\zeta }_k-a_k{)}^{-{\alpha }_k-1},$

the multi-indexed series converging absolutely (Multi-indexed power series in ${\mathbb{C}}^m$ and their absolute convergence). Each term is dominated by

$M_{\alpha }:={\prod }_{k<m}\frac{{\theta }^{{\alpha }_k}}{r_k},{\sum }_{\alpha }{M}_{\alpha }={\prod }_{k<m}\frac{1}{r_k(1-\theta )},$