On inflection points of Bessel functions of the second kind of positive order

Let jν, 1, jν,2, … denote the positive zeros of the Bessel function Jν(x), and similarly, let j'v,1, j'v,2, … denote the positive zeros of J'v(x), which are the positive critical points of Jv(x). It is well-known that when v is positive, both jν ,k. it and j'ν k are increasing functions of ν; see, e.g., [12, pp. 246 and 248]. Recently, Lorch and Szego [6] have attempted to show that the same is true for the positive zeros j″v,1, j″v,2, … of j″v(x), which are the positive inflection points of Jv(x).

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On the Points of Inflection of Bessel Functions of Positive Order, I

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-049-x ◽

1990 ◽

Vol 42 (5) ◽

pp. 933-948 ◽

Cited By ~ 5

Author(s):

Lee Lorch ◽

Peter Szego

Keyword(s):

Bessel Function ◽

Bessel Functions ◽

Primary Concern ◽

Second Derivative ◽

Positive Order ◽

Fixed Rank

The primary concern addressed here is the variation with respect to the order v > 0 of the zeros jʺvk of fixed rank of the second derivative of the Bessel function Jv(x) of the first kind. It is shown that jʺv1 increases 0 < v < ∞ (Theorem 4.1) and that jʺvk increases in 0 < v ≤ 3838 for fixed k = 2, 3,… (Theorem 10.1).

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Liouville-Green expansions of exponential form, with an application to modified Bessel functions

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.117 ◽

2019 ◽

Vol 150 (3) ◽

pp. 1289-1311 ◽

Cited By ~ 2

Author(s):

T. M. Dunster

Keyword(s):

Error Bounds ◽

Asymptotic Expansions ◽

Turning Point ◽

Bessel Functions ◽

Alternative Form ◽

Modified Bessel Functions ◽

Exponential Form ◽

Positive Order ◽

Second Order Differential Equations ◽

Computable Error Bounds

AbstractLinear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z).

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Further on the Points of Inflection of Bessel Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1996-027-7 ◽

1996 ◽

Vol 39 (2) ◽

pp. 216-218

Author(s):

Lee Lorch ◽

Peter Szego

Keyword(s):

Bessel Functions ◽

Positive Order

AbstractWe offer here a substantial simplification and shortening of a proof of the monotonicity of the abscissae of the points of inflection of Bessel functions of the first kind and positive order.

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Inflection Points of Bessel Functions of Negative Order

Canadian Journal of Mathematics ◽

10.4153/cjm-1991-075-5 ◽

1991 ◽

Vol 43 (6) ◽

pp. 1309-1322 ◽

Cited By ~ 4

Author(s):

Lee Lorch ◽

Martin E. Muldoon ◽

Peter Szego

Keyword(s):

Bessel Function ◽

Bessel Functions ◽

Positive Zero ◽

Second Derivative ◽

Monotonicity Properties ◽

Negative Order ◽

Inflection Points

AbstractWe consider the positive zeros j″vk, k = 1, 2,…, of the second derivative of the Bessel function Jν(x). We are interested first in how many zeros there are on the interval (0,jν1), where jν1 is the smallest positive zero of Jν(x). We show that there exists a number ƛ = —0.19937078… such that and . Moreover, j″v1 decreases to 0 and j″ν2 increases to j″01 as ν increases from ƛ to 0. Further, j″vk increases in —1 < ν< ∞, for k = 3,4,… Monotonicity properties are established also for ordinates, and the slopes at the ordinates, of the points of inflection when — 1 < ν < 0.

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