Dunster’s
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11: Bibliography B
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Ramanujan’s Notebooks. Part II.
Springer-Verlag, New York.
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Ramanujan’s Notebooks. Part III.
Springer-Verlag, Berlin-New York.
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Waves and Thom’s theorem.
Advances in Physics 25 (1), pp. 1–26.
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Uniform asymptotic smoothing of Stokes’s discontinuities.
Proc. Roy. Soc. London Ser. A 422, pp. 7–21.
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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions.
SIAM J. Math. Anal. 17 (2), pp. 422–450.
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12: 10.45 Functions of Imaginary Order
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►With , and replaced by , the modified Bessel’s equation (10.25.1) becomes
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10.45.8
►where again denotes Euler’s constant (§5.2(ii)).
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►For properties of and , including uniform asymptotic expansions for large and zeros, see Dunster (1990a).
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13: 10.24 Functions of Imaginary Order
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►With and replaced by , Bessel’s equation (10.2.1) becomes
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10.24.9
►where denotes Euler’s constant §5.2(ii).
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►For mathematical properties and applications of and , including zeros and uniform asymptotic expansions for large , see Dunster (1990a).
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14: 14.23 Values on the Cut
15: 15.12 Asymptotic Approximations
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15.12.2
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15.12.3
►where and , , are defined by the generating function
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15.12.4
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►See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1).
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16: 8.20 Asymptotic Expansions of
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8.20.1
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8.20.2
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8.20.3
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►For further information, including extensions to complex values of and , see Temme (1994b, §4) and Dunster (1996b, 1997).
17: Errata
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►We now include Markov’s Theorem.
In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem.
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Subsection 17.9(iii)
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Equation (10.23.11)
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Equation (25.2.4)
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The title of the paragraph which was previously “Gasper’s -Analog of Clausen’s Formula” has been changed to “Gasper’s -Analog of Clausen’s Formula (16.12.2)”.
10.23.11
Originally the contour of integration written incorrectly as , has been corrected to be .
Reported by Mark Dunster on 2021-03-22
The original constraint, , was removed because, as stated after (25.2.1), is meromorphic with a simple pole at , and therefore is an entire function.
Suggested by John Harper.