Borel lemma

1. Nov 9, 2007

Castilla

There is a theorem (the "Borel lemma") that says: Let (A_n) by any sequence of real numbers. We can built a function "F", indefinitely differentiable, such that if G is the n-derivative of f, G(0) = a_n.

Does someone knows a proof or where can I find it? The theorem appears in wikipedia, under the name "Borel Lemma", but all the wiki information is that this theorem is sometimes useful in PDE...

2. Nov 9, 2007

HallsofIvy

Staff Emeritus
At first glance I thought you were talking about Taylor's series. But Borel's Lemma is more general than that: it says that if have a sequence of smooth, complex valued functions, fn(x), on an open subset of Rn, then there exist F(x,t) such that
$$\frac{\partial^k f}{\partial t^k}= f^k(0, x)$$.

3. Nov 9, 2007

Castilla

Yes, that is the theorem. Only that the version in my book is simplified from complex to real numbers and from partial derivatives to simple derivatives.

Sadly, I can't follow all the steps in my book, that is why I ask if you know some source where I can find the proof.