modified Bessel functions
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11: 10.33 Continued Fractions
12: 10.72 Mathematical Applications
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§10.72(i) Differential Equations with Turning Points
►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►If has a double zero , or more generally is a zero of order , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order . …The order of the approximating Bessel functions, or modified Bessel functions, is , except in the case when has a double pole at . … ►Then for large asymptotic approximations of the solutions can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on and ). …13: 10.35 Generating Function and Associated Series
14: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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10.46.2
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►For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).
15: 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 . …16: 10.45 Functions of Imaginary Order
17: 10.34 Analytic Continuation
18: 10.31 Power Series
19: 10.66 Expansions in Series of Bessel Functions
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10.66.1
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