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11: 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
12: 10.58 Zeros
§10.58 Zeros
13: 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 , ,
14: 10.44 Sums
§10.44(i) Multiplication Theorem
§10.44(ii) Addition Theorems
Graf’s and Gegenbauer’s Addition Theorems
§10.44(iii) Neumann-Type Expansions
§10.44(iv) Compendia
15: 10.55 Continued Fractions
§10.55 Continued Fractions
16: 30.10 Series and Integrals
For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).
17: 10.39 Relations to Other Functions
Elementary Functions
Parabolic Cylinder Functions
Confluent Hypergeometric Functions
10.39.7 I ν ( z ) = ( 2 z ) - 1 2 M 0 , ν ( 2 z ) 2 2 ν Γ ( ν + 1 ) , 2 ν - 1 , - 2 , - 3 , ,
Generalized Hypergeometric Functions and Hypergeometric Function
18: 10.47 Definitions and Basic Properties
Equation (10.47.1)
Equation (10.47.2)
§10.47(iv) Interrelations
19: 10.18 Modulus and Phase Functions
§10.18(i) Definitions
§10.18(ii) Basic Properties
§10.18(iii) Asymptotic Expansions for Large Argument
The remainder after k terms in (10.18.17) does not exceed the ( k + 1 ) th term in absolute value and is of the same sign, provided that k > ν - 1 2 .
20: 10.2 Definitions
§10.2 Definitions
Bessel Function of the First Kind
Bessel Function of the Second Kind (Weber’s Function)
Bessel Functions of the Third Kind (Hankel Functions)
Branch Conventions