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

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System (89χ) becomes ( l — Λ - ^ - ) a^ = Λϊ There are two eigenvalues λν eigenfunctions are 2 (ΐ+λ-2-)^-/,· = ± 2 / π , and the corresponding normalized 9t (*) = 1/ — cos 5, φ2 (s) = l· — sin 8. 2. '> — +*"' In this case ρχ(5) = a^s) = 5; ρ2(β) = σ2(β) = s 2 and 2 2 _n. zu — - ^ - , α12 — α21 — υ, α22 — - ^ - There are two eigenvalues λχ = 3/2 and λ2 = 5/2, the corresponding eigenfunctions being Ψι (*) = ^ - y *; /~γs2- P2 («) = In both examples the kernel K(s, t) has been real and has satisfied the condition K(t, s) = K(s, t).

Theorem 1 gives the complete answer regarding the solution of equation (33) in the case when λ is not an eigenvalue. We are concerned in the present section with the problem when λ is an eigenvalue. Let λ be an eigenvalue, and let non-homogeneous equation (33) have a solution 99(5). We multiply both sides of (33) by a solution 10] 35 THE CASE OF AN EIGENVALUE tp(s) of the adjoint homogeneous equation (65) and integrate witk respect to s: b b b b j φ{8) ψ(8) άβ = j f{8) V(8) ds + j [A j K(8, t) ψ(8) dß]

We shall assume t h a t the kernel K(M; N) tends to infinity only when the points M and N coincide. This is the type of kernel most commonly encountered in mathematical physics. Thus we take a kernel of the form K(M; N) = L(M Ja N) , (100) where L(M, N) is a continuous function of the pair of points (M, N) in the bounded closed domain B} r is the distance between points M 56 [17 INTEGRAL EQUATIONS and Nt and the number a satisfies the condition 0 < a < 2. We shall describe this type of kernel as polar.

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