Bessel functions and modified Bessel functions
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21: 10.40 Asymptotic Expansions for Large Argument
§10.40 Asymptotic Expansions for Large Argument
►§10.40(i) Hankel’s Expansions
… ►Products
… ► … ►§10.40(iv) Exponentially-Improved Expansions
…22: 10.42 Zeros
§10.42 Zeros
… ►23: 33.9 Expansions in Series of Bessel Functions
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►
§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). … ►
33.9.3
,
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►
33.9.6
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24: 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).
25: 10.32 Integral Representations
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§10.32(i) Integrals along the Real Line
… ►Basset’s Integral
… ►§10.32(ii) Contour Integrals
… ►§10.32(iii) Products
… ►§10.32(iv) Compendia
…26: 10.27 Connection Formulas
§10.27 Connection Formulas
… ►
10.27.1
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10.27.2
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10.27.3
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►Many properties of modified Bessel functions follow immediately from those of ordinary Bessel functions by application of (10.27.6)–(10.27.8).
27: 10.30 Limiting Forms
28: 10.48 Graphs
29: 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 … ►
33.20.8
,
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►The functions
and are as in §§10.2(ii), 10.25(ii), and the coefficients are given by (33.20.6).
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