# Question on Hartogs's Extension Theorem

Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)? For Hartogs's Extension Theorem see here:

http://en.wikipedia.org/wiki/Hartogs'_extension_theorem

If yes, is there any reference ?

Thanks bernard

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In several complex variables, as in a single variable, all the various possible definitions of "holomorphic", including "analytic", are proven equivalent. There is even the further theorem of Hartogs, that separate analyticity (expandability in convergent power series) in each variable implies analyticity in several variables. – paul garrett May 12 '13 at 14:05
is there any reference that it actually works? – bernard May 12 '13 at 14:09
The argument that holomorphic = analytic is standard--you use Cauchy's Integral formula one variable at a time. The only difference between the one- and multi-variable cases is that you have more complicated polynomials showing up. (Products of the polynomials you see in the 1-D case.) A standard reference is Griffiths and Harris. – Hiro Lee Tanaka May 12 '13 at 14:21

No. Think about $f(x_1^2+x_2^2+\dots+x_n^2)$ where, for example, $f(t)=e^{1/t}$. The singularity at a single point can be as bad as you like. Maybe if your function satisfies a differential equation, you might have a chance to use some regularity theorem. – Ben McKay May 13 '13 at 6:35