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1: 28.22 Connection Formulas
28.22.7 g o , 2 m + 1 ( h ) = ( 1 ) m 2 π se 2 m + 1 ( 1 2 π , h 2 ) h B 1 2 m + 1 ( h 2 ) ,
where A n m ( h 2 ) , B n m ( h 2 ) are as in §28.4(i), and C m ( h 2 ) , S m ( h 2 ) are as in §28.5(i). …
fe m ( 0 , h 2 ) = 1 2 π C m ( h 2 ) ( g e , m ( h ) ) 2 ce m ( 0 , h 2 ) ,
ge m ( 0 , h 2 ) = 1 2 π S m ( h 2 ) ( g o , m ( h ) ) 2 se m ( 0 , h 2 ) .
Here me ν ( 0 , h 2 ) ( 0 ) is given by (28.14.1) with z = 0 , and M ν ( 1 ) ( 0 , h ) is given by (28.24.1) with j = 1 , z = 0 , and n chosen so that | c 2 n ν ( h 2 ) | = max ( | c 2 ν ( h 2 ) | ) , where the maximum is taken over all integers . …
2: 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 ) ) ,
H ν ( 1 ) ( z ) = e ν π i H ν ( 1 ) ( z ) ,
3: 28.21 Graphics
See accompanying text
Figure 28.21.1: Mc 0 ( 1 ) ( x , h ) for 0 h 3 , 0 x 2 . Magnify 3D Help
See accompanying text
Figure 28.21.2: Mc 1 ( 1 ) ( x , h ) for 0 h 3 , 0 x 2 . Magnify 3D Help
See accompanying text
Figure 28.21.3: Mc 0 ( 2 ) ( x , h ) for 0.1 h 2 , 0 x 2 . Magnify 3D Help
See accompanying text
Figure 28.21.4: Mc 1 ( 2 ) ( x , h ) for 0.2 h 2 , 0 x 2 . Magnify 3D Help
See accompanying text
Figure 28.21.5: Ms 1 ( 1 ) ( x , h ) for 0 h 3 , 0 x 2 . Magnify 3D Help
4: 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 ) .
5: 26.12 Plane Partitions
An equivalent definition is that a plane partition is a finite subset of × × with the property that if ( r , s , t ) π and ( 1 , 1 , 1 ) ( h , j , k ) ( r , s , t ) , then ( h , j , k ) must be an element of π . Here ( h , j , k ) ( r , s , t ) means h r , j s , and k t . It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point ( h , j , k ) π . … A plane partition is symmetric if ( h , j , k ) π implies that ( j , h , k ) π . … A plane partition is cyclically symmetric if ( h , j , k ) π implies ( j , k , h ) π . …
6: 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 ) ¯ .
7: 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 ) . …
8: 28.16 Asymptotic Expansions for Large q
Then as h ( = q ) +
28.16.1 λ ν ( h 2 ) 2 h 2 + 2 s h 1 8 ( s 2 + 1 ) 1 2 7 h ( s 3 + 3 s ) 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
9: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
28.24.2 ε s Mc 2 m ( j ) ( z , h ) = ( 1 ) m = 0 ( 1 ) A 2 2 m ( h 2 ) A 2 s 2 m ( h 2 ) ( J s ( h e z ) 𝒞 + s ( j ) ( h e z ) + J + s ( h e z ) 𝒞 s ( j ) ( h e z ) ) ,
28.24.3 Mc 2 m + 1 ( j ) ( z , h ) = ( 1 ) m = 0 ( 1 ) A 2 + 1 2 m + 1 ( h 2 ) A 2 s + 1 2 m + 1 ( h 2 ) ( J s ( h e z ) 𝒞 + s + 1 ( j ) ( h e z ) + J + s + 1 ( h e z ) 𝒞 s ( j ) ( h e z ) ) ,
28.24.4 Ms 2 m + 1 ( j ) ( z , h ) = ( 1 ) m = 0 ( 1 ) B 2 + 1 2 m + 1 ( h 2 ) B 2 s + 1 2 m + 1 ( h 2 ) ( J s ( h e z ) 𝒞 + s + 1 ( j ) ( h e z ) J + s + 1 ( h e z ) 𝒞 s ( j ) ( h e z ) ) ,
28.24.5 Ms 2 m + 2 ( j ) ( z , h ) = ( 1 ) m = 0 ( 1 ) B 2 + 2 2 m + 2 ( h 2 ) B 2 s + 2 2 m + 2 ( h 2 ) ( J s ( h e z ) 𝒞 + s + 2 ( j ) ( h e z ) J + s + 2 ( h e z ) 𝒞 s ( j ) ( h e z ) ) ,
28.24.6 ε s Ie 2 m ( z , h ) = ( 1 ) s = 0 ( 1 ) A 2 2 m ( h 2 ) A 2 s 2 m ( h 2 ) ( I s ( h e z ) I + s ( h e z ) + I + s ( h e z ) I s ( h e z ) ) ,
10: 28.23 Expansions in Series of Bessel Functions
𝒞 μ ( 3 ) = H μ ( 1 ) ,
𝒞 μ ( 4 ) = H μ ( 2 ) ;
28.23.2 me ν ( 0 , h 2 ) M ν ( j ) ( z , h ) = n = ( 1 ) n c 2 n ν ( h 2 ) 𝒞 ν + 2 n ( j ) ( 2 h cosh z ) ,
28.23.6 Mc 2 m ( j ) ( z , h ) = ( 1 ) m ( ce 2 m ( 0 , h 2 ) ) 1 = 0 ( 1 ) A 2 2 m ( h 2 ) 𝒞 2 ( j ) ( 2 h cosh z ) ,
28.23.7 Mc 2 m ( j ) ( z , h ) = ( 1 ) m ( ce 2 m ( 1 2 π , h 2 ) ) 1 = 0 A 2 2 m ( h 2 ) 𝒞 2 ( j ) ( 2 h sinh z ) ,