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Let $f$ be a holomorphic function with multiple variables.

$f: {\mathbb C}^n \to {\mathbb C}$

Does $f$ have an infinite radius of convergence for its Taylor series?

If so, is the function equal everywhere to its Taylor series?

I think the convergence of Taylor series could be extended from this 1D case. But I'm not sure how to give a formal prove whether it's equal to the original function or not.

Thank you.

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Yuting Yang
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  • If it's multiple variables, it's not on the complex plane, it's on $\mathbb C^n$ for some $n > 1$. – Robert Israel Mar 07 '17 at 20:43
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    Also the word entire looses it's meaning. Do you mean differentiable? Analytic? A solution to a particular set of PDEs? –  Mar 07 '17 at 20:46
  • I think you mean that a holomorphic function $f: {\mathbb C}^n\to {\mathbb C}$ is represented by its Taylor series whose radius of convergence is infinite. This is indeed the case and should be covered in your textbook (if you have one: if not, you should get one). The claim is usually prove via Cauchy integral formula. – Moishe Kohan Mar 08 '17 at 00:53
  • Yes, holomorphic function is exactly what I'm asking. I can find some discussions about the 1D case online, but I want to make sure I understand the formal prove for the multidimensional case. @MoisheCohen Do you have any recommendation of textbooks that cover the multidimensional case? Thank you. – Yuting Yang Mar 08 '17 at 21:12
  • You can find several suggestions here: http://math.stackexchange.com/questions/112237/sources-on-several-complex-variables?rq=1, http://mathoverflow.net/questions/31395/functions-of-several-complex-variables-book-recommendations; also: Gunning and Rossi "Analytic functions in several complex variables"; Malgrange's book is free online: http://www.math.tifr.res.in/~publ/ln/tifr13.pdf – Moishe Kohan Mar 08 '17 at 21:47
  • Thanks, that's very helpful! – Yuting Yang Mar 09 '17 at 02:28

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