The Cauchy–Kowalski theorem for Cauchy initial value problems essentially states that if the terms in a partial differential equation are all made up of analytic functions and a certain transversality condition is satisfied (the hyperplane or more generally hypersurface where the initial data are posed must be noncharacteristic with respect to the partial differential operator), then on certain regions, there necessarily exist solutions which are as well analytic functions. This is a fundamental result in the study of analytic partial differential equations. Surprisingly, the theorem does not hold in the setting of smooth functions; an example discovered by Hans Lewy in 1957 consists of a linear partial differential equation whose coefficients are smooth (i.e., have derivatives of all orders) but not analytic for which no solution exists. So the Cauchy-Kowalevski theorem is necessarily limited in its scope to analytic functions.
