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spherical Bessel functions

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1: 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
2: 10.50 Wronskians and Cross-Products
§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 ,
3: 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 , ,
4: 10.47 Definitions and Basic Properties
Equation (10.47.1)
Equation (10.47.2)
§10.47(iv) Interrelations
5: 10.52 Limiting Forms
§10.52 Limiting Forms
6: 10.49 Explicit Formulas
§10.49(i) Unmodified Functions
§10.49(ii) Modified Functions
§10.49(iii) Rayleigh’s Formulas
§10.49(iv) Sums or Differences of Squares
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: 10.55 Continued Fractions
§10.55 Continued Fractions
8: 10.60 Sums
§10.60 Sums
§10.60(i) Addition Theorems
§10.60(ii) Duplication Formulas
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).
§10.60(iv) Compendia
9: 30.10 Series and Integrals
For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).
10: 10.76 Approximations
Spherical Bessel Functions