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1: 14.30 Spherical and Spheroidal Harmonics
§14.30(i) Definitions
§14.30(iii) Sums
Distributional Completeness
2: 10.55 Continued Fractions
§10.55 Continued Fractions
For continued fractions for j n + 1 ( z ) / j n ( z ) and i n + 1 ( 1 ) ( z ) / i n ( 1 ) ( z ) see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).
3: 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
4: 10.50 Wronskians and Cross-Products
§10.50 Wronskians and Cross-Products
𝒲 { h n ( 1 ) ( z ) , h n ( 2 ) ( z ) } = - 2 i z - 2 .
𝒲 { i n ( 1 ) ( z ) , i n ( 2 ) ( z ) } = ( - 1 ) n + 1 z - 2 ,
𝒲 { i n ( 1 ) ( z ) , k n ( z ) } = 𝒲 { i n ( 2 ) ( z ) , k n ( z ) } = - 1 2 π z - 2 .
Results corresponding to (10.50.3) and (10.50.4) for i n ( 1 ) ( z ) and i n ( 2 ) ( z ) are obtainable via (10.47.12).
5: 10.47 Definitions and Basic Properties
Equation (10.47.1)
Equation (10.47.2)
§10.47(iv) Interrelations
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: 10.52 Limiting Forms
§10.52 Limiting Forms
10.52.1 j n ( z ) , i n ( 1 ) ( z ) z n / ( 2 n + 1 ) !! ,
h n ( 1 ) ( z ) i - n - 1 z - 1 e i z ,
8: 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 , .
Let g n ( z ) denote i n ( 1 ) ( z ) , i n ( 2 ) ( z ) , or ( - 1 ) n k n ( z ) . Then …
9: 10.57 Uniform Asymptotic Expansions for Large Order
§10.57 Uniform Asymptotic Expansions for Large Order
Asymptotic expansions for j n ( ( n + 1 2 ) z ) , y n ( ( n + 1 2 ) z ) , h n ( 1 ) ( ( n + 1 2 ) z ) , h n ( 2 ) ( ( n + 1 2 ) z ) , i n ( 1 ) ( ( n + 1 2 ) z ) , and k n ( ( n + 1 2 ) z ) as n that are uniform with respect to z can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for i n ( 2 ) ( ( n + 1 2 ) z ) the connection formula (10.47.11) is available. For the corresponding expansion for j n ( ( n + 1 2 ) z ) use
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 ) .
10: 10.53 Power Series
§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 ) !! .
For h n ( 1 ) ( z ) and h n ( 2 ) ( z ) combine (10.47.10), (10.53.1), and (10.53.2). For k n ( z ) combine (10.47.11), (10.53.3), and (10.53.4).