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1: 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).
2: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
The function ϕ ( ρ , β ; z ) is defined by
10.46.1 ϕ ( ρ , β ; z ) = k = 0 z k k ! Γ ( ρ k + β ) , ρ > - 1 .
The Laplace transform of ϕ ( ρ , β ; z ) can be expressed in terms of the Mittag-Leffler function: …
3: 10.18 Modulus and Phase Functions
§10.18(i) Definitions
10.18.1 M ν ( x ) e i θ ν ( x ) = H ν ( 1 ) ( x ) ,
10.18.2 N ν ( x ) e i ϕ ν ( x ) = H ν ( 1 ) ( x ) ,
where M ν ( x ) ( > 0 ) , N ν ( x ) ( > 0 ) , θ ν ( x ) , and ϕ ν ( x ) are continuous real functions of ν and x , with the branches of θ ν ( x ) and ϕ ν ( x ) fixed by …
§10.18(ii) Basic Properties
4: 35.5 Bessel Functions of Matrix Argument
§35.5 Bessel Functions of Matrix Argument
§35.5(i) Definitions
35.5.3 B ν ( T ) = Ω etr ( - ( T X + X - 1 ) ) | X | ν - 1 2 ( m + 1 ) d X , ν , T Ω .
§35.5(ii) Properties
§35.5(iii) Asymptotic Approximations
5: 10.24 Functions of Imaginary Order
§10.24 Functions of Imaginary Order
J ~ ν ( x ) = sech ( 1 2 π ν ) ( J i ν ( x ) ) ,
Y ~ ν ( x ) = sech ( 1 2 π ν ) ( Y i ν ( x ) ) ,
6: 10.25 Definitions
Its solutions are called modified Bessel functions or Bessel functions of imaginary argument. …
10.25.2 I ν ( z ) = ( 1 2 z ) ν k = 0 ( 1 4 z 2 ) k k ! Γ ( ν + k + 1 ) .
10.25.3 K ν ( z ) π / ( 2 z ) e - z ,
Branch Conventions
7: 10.45 Functions of Imaginary Order
§10.45 Functions of Imaginary Order
10.45.2 I ~ ν ( x ) = ( I i ν ( x ) ) , K ~ ν ( x ) = K i ν ( x ) .
K ~ ν ( x ) = ( π / ( 2 x ) ) 1 2 e - x ( 1 + O ( x - 1 ) ) .
8: 10.2 Definitions
§10.2 Definitions
10.2.2 J ν ( z ) = ( 1 2 z ) ν k = 0 ( - 1 ) k ( 1 4 z 2 ) k k ! Γ ( ν + k + 1 ) .
10.2.3 Y ν ( z ) = J ν ( z ) cos ( ν π ) - J - ν ( z ) sin ( ν π ) .
10.2.5 H ν ( 1 ) ( z ) 2 / ( π z ) e i ( z - 1 2 ν π - 1 4 π )
10.2.6 H ν ( 2 ) ( z ) 2 / ( π z ) e - i ( z - 1 2 ν π - 1 4 π )
9: 10.47 Definitions and Basic Properties
10.47.3 j n ( z ) = 1 2 π / z J n + 1 2 ( z ) = ( - 1 ) n 1 2 π / z Y - n - 1 2 ( z ) ,
10.47.4 y n ( z ) = 1 2 π / z Y n + 1 2 ( z ) = ( - 1 ) n + 1 1 2 π / z J - n - 1 2 ( z ) ,
10.47.7 i n ( 1 ) ( z ) = 1 2 π / z I n + 1 2 ( z )
10.47.8 i n ( 2 ) ( z ) = 1 2 π / z I - n - 1 2 ( z )
10.47.9 k n ( z ) = 1 2 π / z K n + 1 2 ( z ) = 1 2 π / z K - n - 1 2 ( z ) .
10: 2.8 Differential Equations with a Parameter
For example, u can be the order of a Bessel function or degree of an orthogonal polynomial. … Solutions are Bessel functions, or modified Bessel functions, of order ± ( 1 + 4 ρ ) 1 / 2 (§§10.2, 10.25). … Define … For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). …