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

ISBN-10: 1483167232

ISBN-13: 9781483167237

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**Extra info for A Course of Higher Mathematics. Volume IV**

**Example text**

We show t h a t the ranks of the eigenvalues of these equations are the same. THEOREM 9. e. the ranks of coincident eigenvalues are the same. We use reductio ad absurdum. Let the rank of equation (61) be m, and the rank of equation (65) be n, and let m < n. We prove a contradiction. >i(*)»

The theory of integral equations with symmetric kernels will be given below. Such equations have wide applications in mathematical physics. 3. We now give an example of a degenerate real kernel with imaginary eigenvalues. Let ( K(8,t)=8-t 0 < i < 1 J. Here we can take so that Qi (*) = 8', 1 1; tf! (0 = 1; <*t(t) = t, QM 1; 1 3 a2 ; 2 " We obtain the following equation for the eigenvalues: ι—τ-ι 4* > + 4 = _A 2 +1= 0 > 46 INTEGRAL EQUATIONS [14 which has pure imaginary roots. In this example the real kernel satisfies the condition K(t, s) = —K(8, t).

F(8) at (s) ds = 0 (i = 1, 2 , . , n). a (93) If λ is not an eigenvalue in this case, system (91x) gives us only a zero solution, and, by (88), we get φ(β) = f(s). This solution can be checked by substituting it directly in (87), if we take (93) into account. Degenerate kernels are used for the approximate solution of integral equations, the given kernel being replaced by a degenerate kernel close to it, then the resulting degenerate equation solved with the aid of the above algebraic method. This method of approximate solution of integral equations is described, with other methods, in Approximation Methods of Advanced Analysis (Priblizhennye metody vysshego analiza) by L.

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