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Bessel functions

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21: 10.18 Modulus and Phase Functions
§10.18(i) Definitions
§10.18(ii) Basic Properties
§10.18(iii) Asymptotic Expansions for Large Argument
In (10.18.17) and (10.18.18) the remainder after n terms does not exceed the ( n + 1 ) th term in absolute value and is of the same sign, provided that n > ν 1 2 for (10.18.17) and 3 2 ν 3 2 for (10.18.18).
22: 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
23: 10.4 Connection Formulas
§10.4 Connection Formulas
10.4.5 J ν ( z ) = csc ( ν π ) ( Y ν ( z ) Y ν ( z ) cos ( ν π ) ) .
10.4.7 H ν ( 1 ) ( z ) = i csc ( ν π ) ( e ν π i J ν ( z ) J ν ( z ) ) = csc ( ν π ) ( Y ν ( z ) e ν π i Y ν ( z ) ) ,
10.4.8 H ν ( 2 ) ( z ) = i csc ( ν π ) ( J ν ( z ) e ν π i J ν ( z ) ) = csc ( ν π ) ( Y ν ( z ) e ν π i Y ν ( z ) ) .
24: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
10.28.1 𝒲 { I ν ( z ) , I ν ( z ) } = I ν ( z ) I ν 1 ( z ) I ν + 1 ( z ) I ν ( z ) = 2 sin ( ν π ) / ( π z ) ,
10.28.2 𝒲 { K ν ( z ) , I ν ( z ) } = I ν ( z ) K ν + 1 ( z ) + I ν + 1 ( z ) K ν ( z ) = 1 / z .
25: 10.3 Graphics
§10.3 Graphics
§10.3(i) Real Order and Variable
See accompanying text
Figure 10.3.4: θ 5 ( x ) , ϕ 5 ( x ) , 0 x 15 . Magnify
§10.3(iii) Imaginary Order, Real Variable
See accompanying text
Figure 10.3.19: J ~ 5 ( x ) , Y ~ 5 ( x ) , 0.01 x 10 . Magnify
26: 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 ,
27: 10.23 Sums
§10.23(i) Multiplication Theorem
Neumann’s Addition Theorem
§10.23(iii) Series Expansions of Arbitrary Functions
Fourier–Bessel Expansion
28: 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
29: 10.66 Expansions in Series of Bessel Functions
§10.66 Expansions in Series of Bessel Functions
10.66.1 ber ν x + i bei ν x = k = 0 e ( 3 ν + k ) π i / 4 x k J ν + k ( x ) 2 k / 2 k ! = k = 0 e ( 3 ν + 3 k ) π i / 4 x k I ν + k ( x ) 2 k / 2 k ! .
30: 10.16 Relations to Other Functions
Elementary Functions
Parabolic Cylinder Functions
Confluent Hypergeometric Functions
10.16.6 H ν ( 1 ) ( z ) H ν ( 2 ) ( z ) } = 2 π 1 2 i e ν π i ( 2 z ) ν e ± i z U ( ν + 1 2 , 2 ν + 1 , 2 i z ) .
Generalized Hypergeometric Functions