Theorem of Borel

Either $(a\_n)\; \backslash ,$une continuation of complex numbers, then there exists a function $f\; \backslash \; in\; C^\; \{\backslash \; infty\}$, defined in the vicinity of 0, such as:

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\},\; \backslash ;\; f^\; \{(N)\}(0)\; =a\_n$

A consequence of this theorem is that there exist functions different from their Taylor series on any vicinity of zero, it is indeed enough to take an associated foncion F after $(N!)$.

Category: analyzes Borel

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