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spherical Bessel functions

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11: 10.76 Approximations
Spherical Bessel Functions
12: 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).
13: 6.10 Other Series Expansions
§6.10(ii) Expansions in Series of Spherical Bessel Functions
6.10.4 Si ( z ) = z n = 0 ( 𝗃 n ( 1 2 z ) ) 2 ,
6.10.5 Cin ( z ) = n = 1 a n ( 𝗃 n ( 1 2 z ) ) 2 ,
6.10.6 Ei ( x ) = γ + ln | x | + n = 0 ( 1 ) n ( x a n ) ( 𝗂 n ( 1 ) ( 1 2 x ) ) 2 , x 0 ,
6.10.8 Ein ( z ) = z e z / 2 ( 𝗂 0 ( 1 ) ( 1 2 z ) + n = 1 2 n + 1 n ( n + 1 ) 𝗂 n ( 1 ) ( 1 2 z ) ) .
14: 10.73 Physical Applications
§10.73(ii) Spherical Bessel Functions
The functions 𝗃 n ( x ) , 𝗒 n ( x ) , 𝗁 n ( 1 ) ( x ) , and 𝗁 n ( 2 ) ( x ) arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates ρ , θ , ϕ 1.5(ii)): …Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …
15: 10.54 Integral Representations
§10.54 Integral Representations
10.54.1 𝗃 n ( z ) = z n 2 n + 1 n ! 0 π cos ( z cos θ ) ( sin θ ) 2 n + 1 d θ .
16: 10.58 Zeros
§10.58 Zeros
17: 10.53 Power Series
§10.53 Power Series
10.53.1 𝗃 n ( z ) = z n k = 0 ( 1 2 z 2 ) k k ! ( 2 n + 2 k + 1 ) !! ,
10.53.2 𝗒 n ( z ) = 1 z n + 1 k = 0 n ( 2 n 2 k 1 ) !! ( 1 2 z 2 ) k k ! + ( 1 ) n + 1 z n + 1 k = n + 1 ( 1 2 z 2 ) k k ! ( 2 k 2 n 1 ) !! .
10.53.3 𝗂 n ( 1 ) ( z ) = z n k = 0 ( 1 2 z 2 ) k k ! ( 2 n + 2 k + 1 ) !! ,
10.53.4 𝗂 n ( 2 ) ( z ) = ( 1 ) n z n + 1 k = 0 n ( 2 n 2 k 1 ) !! ( 1 2 z 2 ) k k ! + 1 z n + 1 k = n + 1 ( 1 2 z 2 ) k k ! ( 2 k 2 n 1 ) !! .
18: 7.6 Series Expansions
§7.6(ii) Expansions in Series of Spherical Bessel Functions
7.6.8 erf z = 2 z π n = 0 ( 1 ) n ( 𝗂 2 n ( 1 ) ( z 2 ) 𝗂 2 n + 1 ( 1 ) ( z 2 ) ) ,
7.6.9 erf ( a z ) = 2 z π e ( 1 2 a 2 ) z 2 n = 0 T 2 n + 1 ( a ) 𝗂 n ( 1 ) ( 1 2 z 2 ) , 1 a 1 .
7.6.10 C ( z ) = z n = 0 𝗃 2 n ( 1 2 π z 2 ) ,
7.6.11 S ( z ) = z n = 0 𝗃 2 n + 1 ( 1 2 π z 2 ) .
19: 10.57 Uniform Asymptotic Expansions for Large Order
§10.57 Uniform Asymptotic Expansions for Large Order
10.57.1 𝗃 n ( ( n + 1 2 ) z ) = π 1 2 ( ( 2 n + 1 ) z ) 1 2 J n + 1 2 ( ( n + 1 2 ) z ) π 1 2 ( ( 2 n + 1 ) z ) 3 2 J n + 1 2 ( ( n + 1 2 ) z ) .
20: 10.74 Methods of Computation
Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. … Similar considerations apply to the spherical Bessel functions and Kelvin functions. …
Spherical Bessel Transform