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Bessel functions and modified Bessel functions

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1: 10.76 Approximations
§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions
Real Variable; Imaginary Order
2: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
10.46.2 I ν ( z ) = ( 1 2 z ) ν ϕ ( 1 , ν + 1 ; 1 4 z 2 ) .
For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).
3: 10.44 Sums
§10.44(i) Multiplication Theorem
§10.44(ii) Addition Theorems
Graf’s and Gegenbauer’s Addition Theorems
§10.44(iii) Neumann-Type Expansions
§10.44(iv) Compendia
4: 28.27 Addition Theorems
They are analogous to the addition theorems for Bessel functions10.23(ii)) and modified Bessel functions10.44(ii)). …
5: 10.73 Physical Applications
§10.73(i) Bessel and Modified Bessel Functions
Consequently, Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) , are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. … … On separation of variables into cylindrical coordinates, the Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) and K n ( x ) , all appear. …
6: 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
7: 10.1 Special Notation
The main functions treated in this chapter are the Bessel functions J ν ( z ) , Y ν ( z ) ; Hankel functions H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) ; modified Bessel functions I ν ( z ) , K ν ( z ) ; spherical Bessel functions 𝗃 n ( z ) , 𝗒 n ( z ) , 𝗁 n ( 1 ) ( z ) , 𝗁 n ( 2 ) ( z ) ; modified spherical Bessel functions 𝗂 n ( 1 ) ( z ) , 𝗂 n ( 2 ) ( z ) , 𝗄 n ( z ) ; Kelvin functions ber ν ( x ) , bei ν ( x ) , ker ν ( x ) , kei ν ( x ) . For the spherical Bessel functions and modified spherical Bessel functions the order n is a nonnegative integer. … For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
8: 10.74 Methods of Computation
In the case of the modified Bessel function K ν ( z ) see especially Temme (1975). …
§10.74(vii) Integrals
Kontorovich–Lebedev Transform
9: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
10.28.1 𝒲 { I ν ( z ) , I ν ( z ) } = I ν ( z ) I ν 1 ( z ) I ν + 1 ( z ) I ν ( z ) = 2 sin ( ν π ) / ( π z ) ,
10.28.2 𝒲 { K ν ( z ) , I ν ( z ) } = I ν ( z ) K ν + 1 ( z ) + I ν + 1 ( z ) K ν ( z ) = 1 / z .
10: 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. … If f ( z ) has a double zero z 0 , or more generally z 0 is a zero of order m , m = 2 , 3 , 4 , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order 1 / ( m + 2 ) . …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 ). …