Lectures on Cauchy's Problem in Linear Partial Differential by Jacques Hadamard

By Jacques Hadamard

Excerpt from Lectures on Cauchy's challenge in Linear Partial Differential Equations

Picard's researches - which we will quote of their position - also are crucial in numerous elements of the current paintings. Such can also be the case for Le Roux.

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All this is obviously similar to the classic theory of contact ; and, indeed, the latter is a sub-case of our present considerations : it may be expressed by saying that two curves have a contact of order p at a common point A if their arcs around A have a vicinity of order p, arbitrarily close when the arcs are su fficiently small. The same, of course, applies to functions having a contact of order p for a determinate value of the variable. The extension of all this to functions of several variables is obvious and we need not even formulate it.

Oo , this would not have essentially changed the order of (1 4'), the integral (14") being replaced by. '.. we could assert that it can be of no class lower than a, not only around = 0, but even in any interval whatever: for such a series only changes . . . 2z7r . in its fi rst terms by changing into + bk (w hatever be th e rntegers k y y and l) and the numbers 2;; can approximate as closely as is desired to any given real quantity. Gevrey (Zoe. cit. ) * has shown that functions of class a > 1 remain so through the same general operations as analytic ones, such as multiplication, substitution of one or several functions in another one, integration of differential equations, etc.

Vol. CLII and Thesi's, Paris). DISCUSSION OF CAUCHY' S RESULT 28 [BK I of contemporary geometers have succeeded in pointing out many interesting and important intermediates between the conception of an arbitrary function and that of a continuous function ; more restrictive than the latter is the notion of functions with a limited variation, and again of functions differentiable once, twice, . . , p times. 'I1hen would come functions differentiable to any order. Now, functions satisfying the inequalities (11) show us the use­ fu lness of a distinction which had not been made hitherto : they are intermediate between functions differentiaole to any order and analytic functions.

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