Lectures on Ordinary Differential Equations (Dover Books on Mathematics) by Witold Hurewicz

Author:Witold Hurewicz [Hurewicz, Witold]
Language: eng
Format: azw3, epub
Publisher: Dover Publications
Published: 2014-07-21T04:00:00+00:00

is given by

Proof. By Lemma 1 the integral (6) exists. Since the derivatives of K(s, t) with respect to t up to the (n – 2)nd order are continuous in s, we may differentiate under the integral sign (n – 1) times with respect to t, obtaining

The fact that has a single discontinuity at s = t does not of course impair the validity of the last step. Since , we may also write

Since the integrand is now continuous over the whole range of integration, we may differentiate again by the usual rule, obtaining

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