Borel's lemma

In mathematics, Borel's lemma is an important result about partial differential equations named after Émile Borel.

Suppose $U$ is an open set in the Euclidean space Rⁿ, and suppose that $f\_0,\; f\_1,\; ...$ is a sequence of smooth, complex-valued functions on $U$. Then there exists a smooth function $F=F(t,x)$ defined on R×$U$ with complex values, such that:$left(frac\{partial^k\}\{partial\; t^k\}F\; ight)(0,x)\; =\; f\_k(x),$for all $k=0,1,...$, and $x$ in $U.$

A constructive proof of this result is given in Golubitsky and Guillemin (1974).

References

* M. Golubitsky, V. Guillemin (1974). "Stable mappings and their singularities". Springer-Verlag, Graduate texts in Mathematics: Vol. 14. ISBN 0-387-90072-1.