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1: 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 older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
2: 10.76 Approximations
§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions
3: 10.5 Wronskians and Cross-Products
§10.5 Wronskians and Cross-Products
10.5.3 𝒲 { J ν ( z ) , H ν ( 1 ) ( z ) } = J ν + 1 ( z ) H ν ( 1 ) ( z ) - J ν ( z ) H ν + 1 ( 1 ) ( z ) = 2 i / ( π z ) ,
10.5.4 𝒲 { J ν ( z ) , H ν ( 2 ) ( z ) } = J ν + 1 ( z ) H ν ( 2 ) ( z ) - J ν ( z ) H ν + 1 ( 2 ) ( z ) = - 2 i / ( π z ) ,
10.5.5 𝒲 { H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) } = H ν + 1 ( 1 ) ( z ) H ν ( 2 ) ( z ) - H ν ( 1 ) ( z ) H ν + 1 ( 2 ) ( z ) = - 4 i / ( π z ) .
4: 10.2 Definitions
§10.2(i) Bessel’s Equation
Bessel Functions of the Third Kind (Hankel Functions)
10.2.5 H ν ( 1 ) ( z ) 2 / ( π z ) e i ( z - 1 2 ν π - 1 4 π )
Branch Conventions
Except where indicated otherwise, it is assumed throughout the DLMF that the symbols J ν ( z ) , Y ν ( z ) , H ν ( 1 ) ( z ) , and H ν ( 2 ) ( z ) denote the principal values of these functions. …
5: 10.11 Analytic Continuation
§10.11 Analytic Continuation
10.11.3 sin ( ν π ) H ν ( 1 ) ( z e m π i ) = - sin ( ( m - 1 ) ν π ) H ν ( 1 ) ( z ) - e - ν π i sin ( m ν π ) H ν ( 2 ) ( z ) ,
10.11.4 sin ( ν π ) H ν ( 2 ) ( z e m π i ) = e ν π i sin ( m ν π ) H ν ( 1 ) ( z ) + sin ( ( m + 1 ) ν π ) H ν ( 2 ) ( z ) .
6: 10.4 Connection Formulas
§10.4 Connection Formulas
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 ) ) .
7: 10.3 Graphics
§10.3 Graphics
§10.3(i) Real Order and Variable
For the modulus and phase functions M ν ( x ) , θ ν ( x ) , N ν ( x ) , and ϕ ν ( x ) see §10.18. …
§10.3(ii) Real Order, Complex Variable
§10.3(iii) Imaginary Order, Real Variable
8: 10.74 Methods of Computation
The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument x or z is sufficiently small in absolute value. … For z the function H ν ( 1 ) ( z ) , for example, can always be computed in a stable manner in the sector 0 ph z π by integrating along rays towards the origin. … For evaluation of the Hankel functions H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) for complex values of ν and z based on the integral representations (10.9.18) see Remenets (1973). …
§10.74(vi) Zeros and Associated Values
9: 10.16 Relations to Other Functions
H 1 2 ( 2 ) ( z ) = i H - 1 2 ( 2 ) ( z ) = i ( 2 π z ) 1 2 e - i z .
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 ) .
10: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).