View source for Borel theorem

{{MSC|26A93|30E15}}

A class of theorems guaranteeing existence of a smooth functions with any preassigned (eventually diverging) Taylor series and similar statements for complex functions.
===Real version===
For any collection of real numbers $\{c_\alpha:\ \alpha\in\Z_+^n\}$ labeled by [[multi-index notation|multiindices]] there exists a $C^\infty$-smooth function $f:(\R^n,0)\to\R$ such that $c_\alpha=\frac1{\alpha!}\partial^\alpha f(0)$. In other words, any formal series $\sum_{|\alpha|\ge 0} c_\alpha x^\alpha\in\R[[x_1,\dots,x_n]]$ is the Taylor series of a $C^\infty$-smooth function defined in an open neighborhood of the origin.

In this form the Borel theorem is a particular case of the [[Whitney extension theorem]], see {{Cite|N}}.
===Complex version===
Let $S\subset(\C,0)$ be an open sector $\{0<|z|<\rho,\ |\theta_-<\arg z <\theta_+\}$ with the opening angle $\theta_+-\theta_-$ less than $2\pi$ on the complex plane with the vertex at the origin, and $\{c_k:\ k=0,1,2,\dots\}$ a sequence of complex numbers

Then there exists a function $f$ holomorphic in $S$, for which the formal series $\sum c_k z^k$ is an [[asymptotic series]]:
$$
\forall m\in\N\quad \lim_{z\to 0} \frac1{z^m}\Big(f(z)-\sum_1^m c_k z^k\Big)=0\qquad\text{as }z\to 0,\ z\in S.
$$
This theorem is also referred to as the ''[[Borel-Ritt theorem]]'', see {{Cite|W|Sect. 9}}.
====Remark====
One can consider also sectors with opening larger than $2\pi$, but only on the suitable Riemann surface. A (single-valued) function defined in a punctured neighborhood of the origin and admitting an asymptotic series, is necessarily holomorphic, so its asymptotic series must converge.  
===Multidimensional version===
The analog of Borel-Ritt theorem is valid also for formal series in several variables: any such series can be realized as an asymptotic series for a suitable function of several complex variables, holomorphic in a ''proper polysector'' $z\in\{\C^n:\ \theta_{i-}<\arg z_i<\theta_{i+},\ i=1,\dots,n,\ 0<|z|<\rho\}$, provided that $\theta_{i+}-\theta_{i-}<2\pi$. See {{Cite|R}}.
===References===
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|valign="top"|{{Ref|W}}||valign="top"|Wasow, W. ''Asymptotic expansions for ordinary differential equations'', Dover Publications, Inc., New York,  1987. {{MR|0919406}} 
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|valign="top"|{{Ref|N}}||valign="top"|Narasimhan, R.   ''Analysis on real and complex manifolds'', North-Holland Mathematical  Library, '''35'''. North-Holland Publishing Co., Amsterdam,  1985.  {{MR|0832683}}
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|valign="top"|{{Ref|R}}||valign="top"| Ramis, J.-P.  ''À propos du théorème de Borel-Ritt à plusieurs  variables'',  Lecture Notes in Math., '''712''',  Équations différentielles et systèmes de Pfaff dans le champ  complexe (Sem., Inst. Rech. Math. Avancée, Strasbourg, 1975),   pp. 289--292, Springer, Berlin,  1979.  {{MR|0548148}}
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