By V. I. Smirnov and A. J. Lohwater (Auth.)
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Extra resources for A Course of Higher Mathematics. Volume IV
55) which enables us to evaluate the coefficients dn(s, t) successively; here, we have to take d0(s, t) = K(s, t). We observe t h a t series (53) is always absolutely and uniformly convergent with respect to (s, t) under condition (40), since the terms of the cross-multiplied series (45) and (50) are less than the positive terms of a convergent numerical series. This makes it possible for us to integrate term by term in the right-hand side of formula (54). e. having regard to the notation (49): 6 a With n = 2, formula (55) gives: 6 da(*.
We have shown t h a t every zero A0 of the function Ό(λ) is a pole of the resolvent. Let λ0 be a pole of multiplicity r. 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 , 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±.
I t m a y happen, in addition, t h a t these positive numbers indefinitely approach zero with a given ε. I t now becomes impossible to choose an η' which is t h e same for all the f(x) of ©. For instance, in t h e case of t h e functions / n ( # ) = s i n nx (n=l, 2 , . . ) , given ε, t h e number η evidently tends t o zero on indefinite increase of n. This follows a t once from t h e fact that, when t h e independent variable x varies b y an amount <5, t h e argument of the sine changes by ηδ.
A Course of Higher Mathematics. Volume IV by V. I. Smirnov and A. J. Lohwater (Auth.)