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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: 10.2 Definitions
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 )
10.2.6 H ν ( 2 ) ( z ) 2 / ( π z ) e - i ( z - 1 2 ν π - 1 4 π )
Branch Conventions
3: 10.47 Definitions and Basic Properties
10.47.5 h n ( 1 ) ( z ) = 1 2 π / z H n + 1 2 ( 1 ) ( z ) = ( - 1 ) n + 1 i 1 2 π / z H - n - 1 2 ( 1 ) ( z ) ,
10.47.6 h n ( 2 ) ( z ) = 1 2 π / z H n + 1 2 ( 2 ) ( z ) = ( - 1 ) n i 1 2 π / z H - n - 1 2 ( 2 ) ( z ) .
j n ( z ) and y n ( z ) are the spherical Bessel functions of the first and second kinds, respectively; h n ( 1 ) ( z ) and h n ( 2 ) ( z ) are the spherical Bessel functions of the third kind. … For example, z - n j n ( z ) , z n + 1 y n ( z ) , z n + 1 h n ( 1 ) ( z ) , z n + 1 h n ( 2 ) ( z ) , z - n i n ( 1 ) ( z ) , z n + 1 i n ( 2 ) ( z ) , and z n + 1 k n ( z ) are all entire functions of z . …
4: 10.76 Approximations
§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions
5: 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 ) ) .
6: 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 ) .
7: 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 ) .
10.11.7 H n ( 1 ) ( z e m π i ) = ( - 1 ) m n - 1 ( ( m - 1 ) H n ( 1 ) ( z ) + m H n ( 2 ) ( z ) ) ,
8: 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 ) .
9: 9.6 Relations to Other Functions
§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions
9.6.6 Ai ( - z ) = ( z / 3 ) ( J 1 / 3 ( ζ ) + J - 1 / 3 ( ζ ) ) = 1 2 z / 3 ( e π i / 6 H 1 / 3 ( 1 ) ( ζ ) + e - π i / 6 H 1 / 3 ( 2 ) ( ζ ) ) = 1 2 z / 3 ( e - π i / 6 H - 1 / 3 ( 1 ) ( ζ ) + e π i / 6 H - 1 / 3 ( 2 ) ( ζ ) ) ,
§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions
9.6.19 H 1 / 3 ( 2 ) ( ζ ) = e π i / 3 H - 1 / 3 ( 2 ) ( ζ ) = e π i / 6 3 / z ( Ai ( - z ) + i Bi ( - z ) ) ,
9.6.20 H 2 / 3 ( 2 ) ( ζ ) = e 2 π i / 3 H - 2 / 3 ( 2 ) ( ζ ) = e - π i / 6 ( 3 / z ) ( Ai ( - z ) + i Bi ( - z ) ) .
10: 15.14 Integrals
Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …