When you use the word analytic, do you mean (1) meromorphic, (2) holomorphic, (3) real analytic, or (4) something else? If (1), this is known as the Levi extension theorem, (2) the Hartogs extension theorem, (3) this is false, (4) you need to be more specific. If you want proof that analytic functions are holomorphic, that depends on how you define analytic. If you define analytic as being solutions to the Cauchy--Riemann equations, and holomorphic as being given locally by convergent complex Taylor series, every book on several complex variables (for example, Krantz, **Function Theory of Several Complex Variables**, section 1.2, p. 29) proves the equivalence. Krantz also considers several other possible definitions of the word holomorphic and proves equivalence of all of them.

separateanalyticity (expandability in convergent power series) in each variable implies analyticity in several variables. – paul garrett May 12 '13 at 14:05