Lemma 21.9.400432B20A966

Proof. Fix $j$ and the variables other than $z_ j$. The resulting function $g(\zeta )$ is holomorphic on $\mathbb C^*$, and for $k\in \mathbb Z$ the integral $\frac1{2\pi }\int _0^{2\pi }g(\varrho \mathrm e^{\mathrm i\theta })\varrho ^{-k}\mathrm e^{-\mathrm ik\theta }\, \mathrm d\theta =\frac1{2\pi \mathrm i}\int _{|\zeta |=\varrho }g(\zeta )\zeta ^{-k-1}\, \mathrm d\zeta $ does not depend on $\varrho >0$, by Cauchy's theorem on the annulus between two such circles. Integrating over the other angles, by Fubini's theorem, we see that $c_ m(r)$ does not depend on $r_ j$; hence it does not depend on $r$. The definition gives the Cauchy estimates $|c_ m|\leq C(r)r^{-m}$, where $C(r)$ is the maximum of $|f|$ on the torus $\{ |z_ j|=r_ j\} $. Let $K\subset (\mathbb C^*)^ n$ be compact, and choose $0<a\leq b$ with $a\leq |z_ j|\leq b$ on $K$ for all $j$. Given $m$, take $r_ j=2b$ if $m_ j\geq 0$ and $r_ j=a/2$ if $m_ j<0$. Then, for $z\in K$,

where $C_ K$ is the largest of the numbers $C(r)$ for the $2^ n$ tori with $r_ j\in \{ a/2,2b\} $. Since $\sum _{m\in \mathbb Z^ n}2^{-(|m_1|+\cdots +|m_ n|)}=3^ n$, the series converges normally on $K$.

We show by induction on $n$ that its sum is $f$. For $n=1$ this is the Laurent theorem of one-variable theory, with the coefficients given by the integral above. Let $n\geq 2$, and write $z=(z',z_ n)$ with $z'\in (\mathbb C^*)^{n-1}$. For fixed $z'$, the one-variable theorem gives $f(z',z_ n)=\sum _ ka_ k(z')z_ n^ k$ with $a_ k(z')=\frac1{2\pi }\int _0^{2\pi }f(z',\mathrm e^{\mathrm i\theta })\mathrm e^{-\mathrm ik\theta }\, \mathrm d\theta $. Differentiation under the integral sign shows that $a_ k$ is smooth and satisfies the Cauchy–Riemann equations, so it is holomorphic on $(\mathbb C^*)^{n-1}$, and by Fubini's theorem its Laurent coefficients are the numbers $c_{(m',k)}$, $m'\in \mathbb Z^{n-1}$. By the induction hypothesis, $a_ k(z')=\sum _{m'}c_{(m',k)}z'^{m'}$, so $f(z)=\sum _ k\sum _{m'}c_{(m',k)}z'^{m'}z_ n^ k$. This double series converges absolutely, by the previous paragraph, so it may be summed as $\sum _ mc_ mz^ m$.

For the uniqueness, integrate the normally convergent series $\sum _{m''}c'_{m''}r^{m''}\mathrm e^{\mathrm i\langle m'',\theta \rangle }=f(r_1\mathrm e^{\mathrm i\theta _1},\ldots )$ term by term against $(2\pi )^{-n}r^{-m}\mathrm e^{-\mathrm i\langle m,\theta \rangle }\, \mathrm d\theta $; since $\int _0^{2\pi }\mathrm e^{\mathrm il\theta }\, \mathrm d\theta =0$ for every integer $l\neq 0$, only the term $m''=m$ survives, and we obtain $c'_ m=c_ m(r)=c_ m$. Finally, if $f$ is holomorphic on $\mathbb C^ n$, its Taylor series at the origin converges normally on every polydisc by (5.2.1), in particular on every torus, so by uniqueness the coefficients $c_ m$ are the Taylor coefficients for $m\in \mathbb Z_{\geq 0}^ n$ and vanish for all other $m$. $\square$