in Bessel functions or modified Bessel functions
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11: 33.9 Expansions in Series of Bessel Functions
§33.9 Expansions in Series of Bessel Functions
►§33.9(i) Spherical Bessel Functions
… ►where the function is as in §10.47(ii), , , and … ►§33.9(ii) Bessel Functions and Modified Bessel Functions
►In this subsection the functions , , and are as in §§10.2(ii) and 10.25(ii). …12: 10.35 Generating Function and Associated Series
§10.35 Generating Function and Associated Series
… ►
10.35.1
►Jacobi–Anger expansions: for ,
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10.35.2
…
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10.35.4
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13: 33.20 Expansions for Small
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§33.20(i) Case
… ►For the functions , , , and see §§10.2(ii), 10.25(ii). … ►The functions and are as in §§10.2(ii), 10.25(ii), and the coefficients are given by , , and … ►The functions and are as in §§10.2(ii), 10.25(ii), and the coefficients are given by (33.20.6). ►§33.20(iv) Uniform Asymptotic Expansions
…14: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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►
10.46.2
►For asymptotic expansions of as
in various sectors of the complex -plane for fixed real values of and fixed real or complex values of , see Wright (1935) when , and Wright (1940b) when .
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►The Laplace transform of can be expressed in terms of the Mittag-Leffler function:
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►For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).
15: 13.24 Series
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§13.24(ii) Expansions in Series of Bessel Functions
… ►
13.24.1
,
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►Additional expansions in terms of Bessel functions are given in Luke (1959).
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16: 10.25 Definitions
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►Its solutions are called modified Bessel functions or Bessel functions
of imaginary argument.
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§10.25(ii) Standard Solutions
… ►Branch Conventions
… ►Corresponding to the symbol introduced in §10.2(ii), we sometimes use to denote , , or any nontrivial linear combination of these functions, the coefficients in which are independent of and . …17: 10.31 Power Series
18: 10.34 Analytic Continuation
19: 10.45 Functions of Imaginary Order
20: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
… ►Also, with and denoting the modified Bessel functions (§10.25(ii)), and again with , … ►
28.24.10
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28.24.11
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►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).