Borel's lemma

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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

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

for all $k = 0,1,...$ , and $x$ in $U .$

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

[edit] References

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

This article incorporates material from Borel lemma on PlanetMath, which is licensed under the GFDL.