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Bessel functions

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11: 4.46 Tables
This handbook also includes lists of references for earlier tables, as do Fletcher et al. (1962) and Lebedev and Fedorova (1960). … (These roots are zeros of the Bessel function J 3 / 2 ( x ) ; see §10.21.) …
12: 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. … The order of the approximating Bessel functions, or modified Bessel functions, is 1 / ( λ + 2 ) , except in the case when g ( z ) has a double pole at z 0 . … Then for large u asymptotic approximations of the solutions w can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on u and α ). …
13: 10.48 Graphs
§10.48 Graphs
See accompanying text
Figure 10.48.7: 𝗂 5 ( 1 ) ( x ) , 𝗂 5 ( 2 ) ( x ) , 𝗄 5 ( x ) , 0 x 8 . Magnify
14: 10.58 Zeros
§10.58 Zeros
15: 10.51 Recurrence Relations and Derivatives
Let f n ( z ) denote any of 𝗃 n ( z ) , 𝗒 n ( z ) , 𝗁 n ( 1 ) ( z ) , or 𝗁 n ( 2 ) ( z ) . …
n f n 1 ( z ) ( n + 1 ) f n + 1 ( z ) = ( 2 n + 1 ) f n ( z ) , n = 1 , 2 , ,
f n ( z ) = f n + 1 ( z ) + ( n / z ) f n ( z ) , n = 0 , 1 , .
Then …
n g n 1 ( z ) + ( n + 1 ) g n + 1 ( z ) = ( 2 n + 1 ) g n ( z ) , n = 1 , 2 , ,
16: 10.55 Continued Fractions
§10.55 Continued Fractions
17: 30.10 Series and Integrals
For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).
18: 10.39 Relations to Other Functions
Elementary Functions
Parabolic Cylinder Functions
Confluent Hypergeometric Functions
10.39.7 I ν ( z ) = ( 2 z ) 1 2 M 0 , ν ( 2 z ) 2 2 ν Γ ( ν + 1 ) , 2 ν 1 , 2 , 3 , ,
Generalized Hypergeometric Functions and Hypergeometric Function
19: 10.52 Limiting Forms
§10.52 Limiting Forms
10.52.1 𝗃 n ( z ) , 𝗂 n ( 1 ) ( z ) z n / ( 2 n + 1 ) !! ,
20: 10.47 Definitions and Basic Properties
Equation (10.47.1)
Equation (10.47.2)
§10.47(iv) Interrelations