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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 ) . … 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 ) . … For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
2: 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 ) .
3: 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 ) ,
H ν ( 1 ) ( z ¯ ) = H ν ( 2 ) ( z ) ¯ , H ν ( 2 ) ( z ¯ ) = H ν ( 1 ) ( z ) ¯ .
4: 10.4 Connection Formulas
§10.4 Connection Formulas
Other solutions of (10.2.1) include J - ν ( z ) , Y - ν ( z ) , H - ν ( 1 ) ( z ) , and H - ν ( 2 ) ( z ) . …
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 ) ) ,
5: 10.2 Definitions
Bessel Functions of the Third Kind (Hankel Functions)
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 . …
Branch Conventions
6: 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 ) ,
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. … Many properties of j n ( z ) , y n ( z ) , h n ( 1 ) ( z ) , h n ( 2 ) ( z ) , i n ( 1 ) ( z ) , i n ( 2 ) ( z ) , and k n ( z ) follow straightforwardly from the above definitions and results given in preceding sections of this chapter. 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 . …
10.47.15 h n ( 1 ) ( - z ) = ( - 1 ) n h n ( 2 ) ( z ) , h n ( 2 ) ( - z ) = ( - 1 ) n h n ( 1 ) ( z ) .
7: 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 .
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 ) .
8: 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 ) ) ,
9.6.7 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(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions
9: 10.7 Limiting Forms
§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).
10: 10.3 Graphics
§10.3(ii) Real Order, Complex Variable
See accompanying text
Figure 10.3.10: H 0 ( 1 ) ( x + i y ) , - 10 x 5 , - 2.8 y 4 . … Magnify 3D Help
See accompanying text
Figure 10.3.12: H 1 ( 1 ) ( x + i y ) , - 10 x 5 , - 2.8 y 4 . … Magnify 3D Help
See accompanying text
Figure 10.3.14: H 5 ( 1 ) ( x + i y ) , - 20 x 10 , - 4 y 4 . … Magnify 3D Help
See accompanying text
Figure 10.3.16: H 5.5 ( 1 ) ( x + i y ) , - 20 x 10 , - 4 y 4 . … Magnify 3D Help