# Borel's lemma

In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations.

## Statement

Suppose U is an open set in the Euclidean space Rn, and suppose that f0, f1 ... is a sequence of smooth functions on U.

If I is an any open interval in R containing 0 (possibly I = R), then there exists a smooth function F(t, x) defined on I×U, such that

$\displaystyle{\left(\frac{\partial^k}{\partial t^k}F\right)(0,x) = f_k(x),}$

for k ≥ 0 and x in U.