A Property of Real Solutions to Bessel’s Equation

Veltkamp, G.W.

doi:10.1137/1006096

Cited by 1 publication

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“…The other lemma is due essentially to Butlewski [2, p. 41] ; see also [17]. In the special case g(x) = 1 it was proved by Wiman [21].…”

Section: ^2 F(x) ^ G(x)mentioning

confidence: 98%

“…to the equation (3.9) satisfied by g{x)z'(x).Remarks, (i) In case z'(a + ) = 0 and either(4.7) 0< lim [f{x)/g(x)] ^oo, x->a + or (4.8) 0 < lim D x [f(x)/g(x)] ^ oo, lim z'(x)z" (x) = 0, x-$a + x->a +we could include the case Xi = a in the statement of our theorem, because the number / in Lemma 4.1 would exist and equal zero, from FHospitaPs rule, (ii) When n = 0, (4.5) is included in[17].An application of Theorem 4.1 to solutions g(x)y'(x) and g(x)z r (x) of the equation(3.9) gives the following analogue of the Sonin-Butlewski-Pôlya theorem for the case in which f(x)g(x) is decreasing.…”

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confidence: 99%

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Higher Monotonicity Properties of Certain Sturm-Liouville Functions. IV

Lorch

Muldoon

Szego³

1972

Can. j. math.

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The Sturm-Liouville functions considered in this instalment are real (as are all other quantities discussed here) non-trivial solutions of the differential equation1.1Higher monotonicity properties, as defined in § 2, are investigated for a number of sequences (finite or infinite) associated with these functions. One such sequence, discussed in detail later, has the kth term1.2where the constant X > — 1 (to assure convergence of each integral), W(x) possesses higher monotonicity properties and, moreover, is such that, again, each integral converges, and X1, X2, … is a sequence (finite or infinite) of consecutive zeros of a solution of (1.1), which may or may not be linearly independent of y(x), in the interval of definition of the functions under consideration.

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“…The other lemma is due essentially to Butlewski [2, p. 41] ; see also [17]. In the special case g(x) = 1 it was proved by Wiman [21].…”

Section: ^2 F(x) ^ G(x)mentioning

confidence: 98%

mentioning

confidence: 99%