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

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Function (50) is therefore an entire function of λ. 7] 27 FREDHOLM'S DETERMINANT The function D( λ) was obtained from Cramer's denominator by passing to the limit. e. that, on multiplying series (45) by D{X), we get an entire function of A. As a result of the multiplication we get a series, the terms of which are no longer numbers, as in D(X), but functions of (s, t). We introduce the special notation for this series: D(s,t;X) = K(s, *) + J j ( - 1)B-£«*„(«,«) · (53) Both the power series (45) and (50) are convergent in the circle (40).

110,) (1102) B Let us consider, along with transformation (103), the transformation vy (M) = J Ky (M; N) u(N) dco. (Ill) B The continuity of νγ(Μ) is evident from the continuity of the kernel Κγ(Μ; N). We write down inequalities similar to the above. By (110χ), we have the previous inequality for the integral over βδ. I t remains to write an inequality, with fixed positive <5, for the integral Γ \Ky (Μ'; N) - Ky (M; N)\ dcoN . ( Π 2 ) B& This can be done in precisely the same way as above for the integral, except that, when r < γ (or r ' < y), we must replace r (or r') by y in the expression Lx (M't M; N) = r«L(M'\ N) - r'aL(M; N).

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). Such skew-symmetric kernels only have pure imaginary eigenvalues. 4. A further example may be mentioned, of a degenerate kernel having no eigenvalues.