Read e-book online A Course of Higher Mathematics. Volume IV PDF

By V. I. Smirnov and A. J. Lohwater (Auth.)

ISBN-10: 1483167232

ISBN-13: 9781483167237

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Ds , a a a whence it follows, since λ Φ 0, t h a t b J i

I ) = K ( 8 , t) j 6 6 6 j K ( ^ y a«, di 2 - 2 j * j z ( « , y * ( ^ ' j di x d*2. di ll . (56) α α a We proved above t h a t this formula holds with n = 1. We write d*(s, £) for the right-hand side of (56). We have, from what has been said: d\{s, t) = d±(s} t). We now show t h a t d*(s, t) satisfies the same relationship as dn(s, t) 6 d*(s, t) = K(s, t)dn-n$ a K(s, tx) d*_Sv *) <"i · (^ι) By (55) and (55x), dn(sy t) and d%(s, t) (n = 2, 3, . . n(s, £) for any n. The proof of (56) therefore reduces to the proof of (55x), where d*(sf t) is the right-hand side of (56).

4]. When proving the fundamental theorems, we have had occasion to change the order of integration in the iterated integrals. This is justifiable with the assumptions made above regarding the kernel. Everything t h a t has been said still holds in the theory of integral equations with multiple integrals. Corresponding to discontinuities on the diagonal s = t, we now have discontinuities of the kernel K(M, N) when points M and N coincide. The problem becomes far more difficult if the kernel is unbounded.

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A Course of Higher Mathematics. Volume IV by V. I. Smirnov and A. J. Lohwater (Auth.)

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