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

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

If condition (79) is satisfied, the equation has an infinite set of solutions, all of which are given by formula (82). Note 1. To test condition (79), it is sufficient to replace ip(s) by a complete set of linearly independent solutions ip^s), ip2(s), . ·> 1/^(5), of equation (80), since every other solution is a linear com- 11] FREDHOLM MINORS 41 bination of them. Thus, if condition (79) is fulfilled with ^J(S) = ipk(s) (k = 1, 2, . . , m), it is also fulfilled for any solution ip(s) of equation (80).

In the neighbourhood of λ = λ0 we have an expansion of the form: (A-V T (A- A / " 1 ^ Λ-Α0 oo + ^α Ι ·(Μ)(λ-Α 0 )', where the coefficient a_ r (s, i) is not identically zero in Jc0. It follows from what was said at the end of [7], t h a t ak(s, t) are continuous functions in the square Jc0. On substituting this last expansion in the first of equations (47), multiplying both sides by (λ — A0)r and then putting λ = λ0, we get: b a_r(s, t) = λ0 J Z(5, ^) α_Γ(ίχ, ί) àt±. α I t thus turns out t h a t the coefficient a„r(s, t), considered as a function of s, is a solution of the homogeneous equation b <ρ(*)=λ0 JZ(M)

The entire functions Ό(λ) and D(s, t; λ) can be expanded throughout the λ plane in non-negative integral powers of (A — λ 0 ), where λ0 is any fixed complex number. For instance, D(s, t;X) = D(s,t; A0) + J ^ ΐ Μ 4 ψ . , where dkD{ X'X) 8AK =2{-l)n-^-dn{s,t) n »/c ( n — fc)! (0! = 1). I t follows at once from the inequalities for the dn(s, t) t h a t the latter series is uniformly convergent in k0 for any A, and we can assert t h a t the coefficients of the expansion of D(s, t; λ) in powers of (λ — λ0) are also continuous functions in k0.

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


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