Several variables contour integral

Question

Suppose I have an entire function of $n$ complex variables $f \colon \mathbb C^n \to \mathbb C$ and $f$ is rapidly decreasing (or enough conditions to make the following work). I want to show that $$\int_{\mathbb R^n} f(x) \, dx = \int_{\mathbb R^n} f(x+i\eta) \, dx$$ for any $\eta \in \mathbb R^n$. In the case $n=1$ I can just apply Cauchy's theorem integrating along a rectangle and show that the integral along the vertical lines vanishes. But what if $n>1$? I thought about applying Stokes' theorem, but does this work and what shape do I parametrize? Or is applying Cauchy's theorem to every variable seperately possible?

Answer 1

You could use Stokes' on the strip $S=\mathbb{R}^n\times[0,1]$ viewed as a noncompact(*) submanifold-with-boundary of $\mathbb{C}^n$ via $(x,t)\mapsto x+it\eta$, with the $n$-form $\omega=f\,\mathrm{d}z^1\wedge\dots\wedge\mathrm{d}z^n$ on $\mathbb{C}^n$. Note that $\partial f=0$ for degree reasons, and $\bar{\partial}f=0$ since $f$ is holomorphic, so $\mathrm{d}f=0$.

(*) Note that $f$ being holomorphic and rapidly decreasing on each translate of $\mathbb{R}^n$ implies $\omega$ is a $C^k$-limit and hence an $L^1_k$-limit of compactly supported forms on a neighbourhood of $S$, hence Stokes' can be applied.

If instead, you want to use Riemann integration and the compact version of Stokes', then consider the solid cylinder $D_R\times[0,1]$. In the limit $R\to\infty$ the two parallel faces combined to give $$ \int_{\mathbb{R}^n} f(x+i\eta)\,\mathrm{d}x-\int_{\mathbb{R}^n} f(x)\,\mathrm{d}x $$ so it suffices to show the contribution from the curved face $S'=(\partial D_R)\times[0,1]$ vanish in the limit. By the M-L estimate, its contribution is at most $$ \operatorname{Area}(\partial D_R)\cdot\sup\{|f(x+it\eta)|:x+it\eta\in S'\}=cR^{n-1}\sup_{S'}|f| $$ where $c$ is a constant (the area of the unit $(n-1)$-dimensional sphere, which is $\dfrac{2\pi^{n/2}}{\Gamma(n/2)}$). Now since $f$ is holomorphic, $|f|$ is pluriharmonic and hence harmonic, so Harnack's inequality allows you to bound $|f(x+it\eta)|$ by a constant multiple of $|f(x)|$ independent of $x\in D_R$, $t\in[0,1]$ for fixed $\eta\in\mathbb{R}^n$ (you can make this constant independent of $x,R$ by translation in $\mathbb{C}^n$ and treating each imaginary line segment separately). Finally, $f$ is rapidly decreasing on $\mathbb{R}^n$ so $R^{n-1}\sup_{D_R}|f|$ can be made as small as you like for large enough $R$.