A problem I'm working on requires the application of Cauchy's estimate for the modulus of the coefficients of a holomorphic function's power series representation, but the original functions with which I'm working are real analytic (sometimes in one variable, sometimes in several). I've located in-several-variables analogues of Cauchy's estimates, but I'm searching for theorems that address the extension of real analytic functions in several variables to holomorphic functions.

I've come across the following notes on *Wikipedia* and the *Encyclopedia of Mathematics* that address the univariate and multivariate cases, respectively:

If ƒ is an infinitely differentiable function defined on an open set D ⊂ R, then the following conditions are equivalent.

1) ƒ is real analytic.

2) There is a complex analytic extension of ƒ to an open set G ⊂ C which contains D.

Similarly, for any real analytic function f:U→ℝ and any $x_0 ∈ U$ there is an open neighborhood W in $ℂ^n$ of $x_0+0i$ and an holomorphic map $g:W→ℂ$ such that g coincides with f on $W \cap \{z:Im(z)=0\}$.

(Encyclopedia of Mathematics: Real Analytic Functions)

Unfortunately, searching through the references given at both pages hasn't turned up a name or theorem-to-cite for their claims, nor has digging through a number of texts/articles on real analytic functions or complex analysis in several variables. Is anyone familiar with an authoritative reference for these facts, or a name associated with either of them?