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1: 10.73 Physical Applications
§10.73(ii) Spherical Bessel Functions
2: 10.1 Special Notation
For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
3: 10.2 Definitions
§10.2(i) Bessel’s Equation
§10.2(ii) Standard Solutions
The notation 𝒞 ν ( z ) denotes J ν ( z ) , Y ν ( z ) , H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) , or any nontrivial linear combination of these functions, the coefficients in which are independent of z and ν .
§10.2(iii) Numerically Satisfactory Pairs of Solutions
Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.
Pair Interval or Region
4: 10.25 Definitions
§10.25(i) Modified Bessel’s Equation
This equation is obtained from Bessel’s equation (10.2.1) on replacing z by ± i z , and it has the same kinds of singularities. …
§10.25(ii) Standard Solutions
§10.25(iii) Numerically Satisfactory Pairs of Solutions
5: 10.72 Mathematical Applications
§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. …
6: 10.36 Other Differential Equations
§10.36 Other Differential Equations
7: 11.9 Lommel Functions
The inhomogeneous Bessel differential equationFor uniform asymptotic expansions, for large ν and fixed μ = - 1 , 0 , 1 , 2 , , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … …
8: 10.13 Other Differential Equations
§10.13 Other Differential Equations
9: 11.2 Definitions
§11.2(ii) Differential Equations
Modified Struve’s Equation
10: 28.10 Integral Equations
§28.10(ii) Equations with Bessel-Function Kernels