Download e-book for kindle: A Course of Higher Mathematics. Volume IV by V. I. Smirnov and A. J. Lohwater (Auth.)

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

ISBN-10: 1483167232

ISBN-13: 9781483167237

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Extra resources for A Course of Higher Mathematics. Volume IV

Example text

As above, we can assume t h a t functions (70) and (71) both form orthonormal systems. ,m); q>j(8)=XSK(e,t)S a f(s) = x^K(t,s)f(t)dt-x2W)ÎW)y,(t)dta (74i) 9JÜ)9(t)dt > (75i) Let 99(5) be any solution of equation (74χ).

M). (68) If λ or the kernel is not real, functions (67) must also be assumed complex. We recall t h a t λ = 0 cannot be an eigenvalue [4]. Since any linear combination with constant coefficients of eigenfunctions (67) is also an eigenfunction, we can apply a process of orthogonalization to functions (67). e. b b I

E. is an entire function of λ, and secondly, t h a t we obtain an entire function of λ on multiplying series (45) by series (50). Let us write an inequality for the coefficient dn. A determinant of order n stands under the integral sign in (51), each element of which K(ti, tk) has a modulus not exceeding the positive number M. On applying Hadamard's theorem [IIIu 16] and the usual upper bound for an iterated integral, we obtain: \dn\

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


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