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1: 10.4 Connection Formulas
H n ( 1 ) ( z ) = ( 1 ) n H n ( 1 ) ( z ) ,
H n ( 2 ) ( z ) = ( 1 ) n H n ( 2 ) ( z ) .
J ν ( z ) = 1 2 ( H ν ( 1 ) ( z ) + H ν ( 2 ) ( z ) ) ,
H ν ( 1 ) ( z ) = e ν π i H ν ( 1 ) ( z ) ,
H ν ( 2 ) ( z ) = e ν π i H ν ( 2 ) ( z ) .
2: 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 ) .
3: 10.11 Analytic Continuation
H ν ( 1 ) ( z e π i ) = e ν π i H ν ( 2 ) ( z ) ,
H ν ( 2 ) ( z e π i ) = e ν π i H ν ( 1 ) ( z ) .
H ν ( 1 ) ( z ¯ ) = H ν ( 2 ) ( z ) ¯ , H ν ( 2 ) ( z ¯ ) = H ν ( 1 ) ( z ) ¯ .
4: 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 ) . … Abramowitz and Stegun (1964): j n ( z ) , y n ( z ) , h n ( 1 ) ( z ) , h n ( 2 ) ( z ) , for 𝗃 n ( z ) , 𝗒 n ( z ) , 𝗁 n ( 1 ) ( z ) , 𝗁 n ( 2 ) ( z ) , respectively, when n 0 . Jeffreys and Jeffreys (1956): Hs ν ( z ) for H ν ( 1 ) ( z ) , Hi ν ( z ) for H ν ( 2 ) ( z ) , Kh ν ( z ) for ( 2 / π ) K ν ( z ) . …
5: 10.2 Definitions
These solutions of (10.2.1) are denoted by H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) , and their defining properties are given by … The principal branches of H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) are two-valued and discontinuous on the cut ph z = ± π . … For fixed z ( 0 ) each branch of H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) is entire in ν . … 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. …
Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.
Pair Interval or Region
H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) neighborhood of in | ph z | π
6: 10.52 Limiting Forms
𝗁 n ( 1 ) ( z ) i n 1 z 1 e i z ,
𝗁 n ( 2 ) ( z ) i n + 1 z 1 e i z ,
7: 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. …
10.47.15 𝗁 n ( 1 ) ( z ) = ( 1 ) n 𝗁 n ( 2 ) ( z ) , 𝗁 n ( 2 ) ( z ) = ( 1 ) n 𝗁 n ( 1 ) ( z ) .
8: 9.6 Relations to Other 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 ) ) ,
9.6.7 Ai ( z ) = ( z / 3 ) ( J 2 / 3 ( ζ ) J 2 / 3 ( ζ ) ) = 1 2 ( z / 3 ) ( e π i / 6 H 2 / 3 ( 1 ) ( ζ ) + e π i / 6 H 2 / 3 ( 2 ) ( ζ ) ) = 1 2 ( z / 3 ) ( e 5 π i / 6 H 2 / 3 ( 1 ) ( ζ ) + e 5 π i / 6 H 2 / 3 ( 2 ) ( ζ ) ) ,
9.6.8 Bi ( z ) = z / 3 ( J 1 / 3 ( ζ ) J 1 / 3 ( ζ ) ) = 1 2 z / 3 ( e 2 π i / 3 H 1 / 3 ( 1 ) ( ζ ) + e 2 π i / 3 H 1 / 3 ( 2 ) ( ζ ) ) = 1 2 z / 3 ( e π i / 3 H 1 / 3 ( 1 ) ( ζ ) + e π i / 3 H 1 / 3 ( 2 ) ( ζ ) ) ,
9.6.9 Bi ( z ) = ( z / 3 ) ( J 2 / 3 ( ζ ) + J 2 / 3 ( ζ ) ) = 1 2 ( z / 3 ) ( e π i / 3 H 2 / 3 ( 1 ) ( ζ ) + e π i / 3 H 2 / 3 ( 2 ) ( ζ ) ) = 1 2 ( z / 3 ) ( e π i / 3 H 2 / 3 ( 1 ) ( ζ ) + e π i / 3 H 2 / 3 ( 2 ) ( ζ ) ) .
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: 10.16 Relations to Other Functions
H 1 2 ( 1 ) ( z ) = i H 1 2 ( 1 ) ( z ) = i ( 2 π z ) 1 2 e i z ,
H 1 2 ( 2 ) ( z ) = i H 1 2 ( 2 ) ( z ) = i ( 2 π z ) 1 2 e i z .
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: 10.7 Limiting Forms
For H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) when ν > 0 combine (10.4.6) and (10.7.7). For H i ν ( 1 ) ( z ) and H i ν ( 2 ) ( z ) when ν and ν 0 combine (10.4.3), (10.7.3), and (10.7.6). … For the corresponding results for H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) see (10.2.5) and (10.2.6).