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modified spherical Bessel functions

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1: 10.48 Graphs
§10.48 Graphs
See accompanying text
Figure 10.48.5: 𝗂 0 ( 1 ) ( x ) , 𝗂 0 ( 2 ) ( x ) , 𝗄 0 ( x ) , 0 x 4 . Magnify
See accompanying text
Figure 10.48.6: 𝗂 1 ( 1 ) ( x ) , 𝗂 1 ( 2 ) ( x ) , 𝗄 1 ( x ) , 0 x 4 . Magnify
See accompanying text
Figure 10.48.7: 𝗂 5 ( 1 ) ( x ) , 𝗂 5 ( 2 ) ( x ) , 𝗄 5 ( x ) , 0 x 8 . Magnify
2: 10.52 Limiting Forms
10.52.1 𝗃 n ( z ) , 𝗂 n ( 1 ) ( z ) z n / ( 2 n + 1 ) !! ,
3: 10.47 Definitions and Basic Properties
Equation (10.47.2)
𝗂 n ( 1 ) ( z ) , 𝗂 n ( 2 ) ( z ) , and 𝗄 n ( z ) are the modified spherical Bessel functions. …
10.47.11 𝗄 n ( z ) = ( 1 ) n + 1 1 2 π ( 𝗂 n ( 1 ) ( z ) 𝗂 n ( 2 ) ( z ) ) .
10.47.16 𝗂 n ( 1 ) ( z ) = ( 1 ) n 𝗂 n ( 1 ) ( z ) , 𝗂 n ( 2 ) ( z ) = ( 1 ) n + 1 𝗂 n ( 2 ) ( z ) ,
4: 10.56 Generating Functions
10.56.3 cosh z 2 + 2 i z t z = cosh z z + n = 1 ( i t ) n n ! 𝗂 n 1 ( 1 ) ( z ) ,
10.56.4 sinh z 2 + 2 i z t z = sinh z z + n = 1 ( i t ) n n ! 𝗂 n 1 ( 2 ) ( z ) ,
10.56.5 exp ( z 2 + 2 i z t ) z = e z z + 2 π n = 1 ( i t ) n n ! 𝗄 n 1 ( z ) .
5: 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).
6: 10.49 Explicit Formulas
§10.49(ii) Modified Functions
10.49.8 𝗂 n ( 1 ) ( z ) = 1 2 e z k = 0 n ( 1 ) k a k ( n + 1 2 ) z k + 1 + ( 1 ) n + 1 1 2 e z k = 0 n a k ( n + 1 2 ) z k + 1 .
10.49.12 𝗄 n ( z ) = 1 2 π e z k = 0 n a k ( n + 1 2 ) z k + 1 .
10.49.16 𝗄 n ( z ) = ( 1 ) n 1 2 π z n ( 1 z d d z ) n e z z .
10.49.20 ( 𝗂 n ( 1 ) ( z ) ) 2 ( 𝗂 n ( 2 ) ( z ) ) 2 = ( 1 ) n + 1 k = 0 n ( 1 ) k s k ( n + 1 2 ) z 2 k + 2 .
7: 6.10 Other Series Expansions
§6.10(ii) Expansions in Series of Spherical Bessel Functions
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 ) ) .
8: 10.53 Power Series
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 ) !! .
9: 10.50 Wronskians and Cross-Products
10.50.4 𝗃 0 ( z ) 𝗃 n ( z ) + 𝗒 0 ( z ) 𝗒 n ( z ) = cos ( 1 2 n π ) k = 0 n / 2 ( 1 ) k a 2 k ( n + 1 2 ) z 2 k + 2 + sin ( 1 2 n π ) k = 0 ( n 1 ) / 2 ( 1 ) k a 2 k + 1 ( n + 1 2 ) z 2 k + 3 ,
10: 10.60 Sums
10.60.3 e w w = 2 π n = 0 ( 2 n + 1 ) 𝗂 n ( 1 ) ( v ) 𝗄 n ( u ) P n ( cos α ) , | v e ± i α | < | u | .
10.60.6 𝗄 n ( 2 z ) = 1 π n ! z n + 1 k = 0 n ( 1 ) k 2 n 2 k + 1 k ! ( 2 n k + 1 ) ! 𝗄 n k 2 ( z ) .
10.60.8 e z cos α = n = 0 ( 2 n + 1 ) 𝗂 n ( 1 ) ( z ) P n ( cos α ) ,
10.60.9 e z cos α = n = 0 ( 1 ) n ( 2 n + 1 ) 𝗂 n ( 1 ) ( z ) P n ( cos α ) .
For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). …