Let $\Omega$ be a (smooth) domain in $\mathbb R^n$, $n\geq2$ and $u$ be a harmonic function in $\Omega$ continuous up to the boundary.
I read somewhere (but unfortunately I don't remember where) that if $B_{\mathbb R^n}(p,r)$ is a ball contained in $\Omega$, then there exists a function $\tilde u$ in $B_{\mathbb C^n}(p, r/\sqrt{2})$ which is holomorphic with respect to each variable and coincides with $u$ in $B_{\mathbb R^n}(p,r/\sqrt{2})$. That is, $\tilde u$ is a function of $z_1, ..., z_n$ such that $\frac {\partial}{\partial \bar z_i} \tilde u = 0$, $1 \leq i \leq n$.
I have two ideas on how to show this (hopefully):
- use that $u(x) = \int_{\partial B(p,r)} u(\xi) P(x,\xi)\,d\xi$ where $P$ is the Poisson kernel, and somehow change the Poisson kernel for an "adequate complex extension".
- use that $u(x) = \sum_{\alpha} a_\alpha P_\alpha(x-p)$ where $P_\alpha$ is a harmonic homogeneous polynomial of degree $\alpha$ (Taylor expansion of $u$ around $p$). Then, somehow "complexify" each harm. homog. pol. to obtain a power series.
Do you know of any reference on how to complete these arguments? Or, ideally, a good reference where I can read about complexification of harmonic functions and applications? Also, I am interested in reference texts on holomorphic functions in multiple variables since I know very little about the topic.
Textbooks on holomorphic functions of several complex variables
I will first answer your request for a reference text on holomorphic functions of several complex variables, since it is simpler to answer. In my opinion the best textbook to start a journey in this realm is the textbook by Krantz [1a] since it tries to give a flavor of all approaches involved i.e. the PDE based approach named after Cauchy and Riemann, the integral representation approach named after Cauchy and (to a lesser extent) the algebraic approach named after Oka and Cartan. In this sense it is a truly introductive text, despite being fully rigorous. As a second reading, I'd advise to have a look at the classical textbook of Range [2a], focused on the integral representation approach: this is however a suggestion biased by my analytic tastes and by the fact that this approach has been flourished in the last 50 years and it seems to go further on.
Holomorphic extension of harmonic functions
It seems to me that you are mixing two different, nevertheless deeply related, statements. The results are precisely the following ones:
Theorem 1 ([1b], §5 Lemma pp. XXV.22-XXV.26). Let $u$ be an harmonic function on the ball $B_{\Bbb R^n}(r)$ centered in $0\in\Bbb R^n$: then $u$ can be continued as an holomorphic function of several variables on the Lie ball $$ \mathscr{B}_{\Bbb C^n}(r) =\Big\{z\in\Bbb C^n : |z|^2+2\sqrt{|x|^2|y|^2-\langle x,y\rangle^2}< R^2\Big\} $$ Aronszajn proves this theorem (originally due to Pierre Lelong, as stated in [5b] p.87) by using the Poisson formula, as does Hayman with the following result.
Theorem 2 ([4b], theorem 1, pp. 155-158). Let $u$ be an harmonic function on the ball $B_{\Bbb R^n}(r)$ centered in $0\in\Bbb R^n$: then the power series expansion of $u$ converges on the interior of the ball $B_{\Bbb R^n}(r/\sqrt{2})$ but in general not on its boundary nor its exterior.
Obviously, the fact that the balls are centered in $0\in\Bbb R^n$ ($0\in\Bbb C^n$) is not a restriction in generality: the result holds for any ball centered around a generic point $p\in\Bbb R^n$ ($p\in\Bbb C^n$) as it can be seen by applying a simple linear change of variable (i.e. the composition of a rotation and a translation) to the base vector space $\Bbb R^n$ ($\Bbb C^n$).
Theorem 1 was further generalized and extended to polyharmonic functions in references [2b] and [3b] by using Almansi's expansion: theorem 2 was later "sharpened" by Khavin in [5b] (where there's also a brief but complete historical note on these researches). I think the best sources on these matters are perhaps the ones I listed below but beware, they are not an easy read.
References
Textbooks on holomorphic functions of several complex variables
[1a] Steven G. Krantz, Function theory of several complex variables. Reprint of the 1992 2nd ed. with corrections. (English) Providence, RI: American Mathematical Society (AMS), AMS Chelsea Publishing (ISBN 0-8218-2724-3/hbk), pp. xvi+564 (2001), MR1846625, Zbl 1087.32001.
[2a] R. Michael Range, Holomorphic functions and integral representations in several complex variables (English), Graduate Texts in Mathematics, 108, Berlin-Heidelberg-New York: Springer-Verlag. pp. XIX+386 (1986), MR0847923, Zbl 0591.32002.
Holomorphic extension of harmonic functions
[1b] Nachman Aronszajn, General Cauchy formulas in $\Bbb C^n$, Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1975-1976), Exposé no. 25, pp. 28, MR481075, Zbl 0337.32002.
[2b] Nachman Aronszajn, Thomas M. Creese, Leonard J. Lipkin, Polyharmonic functions (English), Oxford Mathematical Monographs, Oxford: Oxford University Press (Clarendon Press), pp. X+265 (1983), ISBN: 0-19-853906-1, MR0745128, Zbl 0514.31001.
[3b] Vazgain Avanissian, "Sur les fonctions harmoniques d’ordre quelconque et leur prolongement analytique dans $\Bbb C^N$", (French) Séminaire P. Lelong - H. Skoda, Analyse, Années 1980/81, et: Les fonctions plurisousharmoniques en dimension finie ou infinie, Colloq. Wimereux 1981, Lecture Notes in Mathematics 919, pp. 192-281 (1982), MR0658887, Zbl 0534.31004.
[4b] Walter Kurt Hayman, "Power series expansions for harmonic functions" (English), Bulletin of the London Mathematical Society 2, 152-158 (1970), MR0267114, Zbl 0201.43302.
[5b] Viktor Petrovich Khavin, "A remark on Taylor series of harmonic functions" (English. Russian original) Translations Series 2, American Mathematical Society 146, pp. 85-90 (1990); translation from Multidimensional complex analysis, Work Collection, Krasnoyarsk 1985, 192-197 (1985), MR0899349, Zbl 0714.31001.