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1: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
10.28.1 𝒲 { I ν ( z ) , I - ν ( z ) } = I ν ( z ) I - ν - 1 ( z ) - I ν + 1 ( z ) I - ν ( z ) = - 2 sin ( ν π ) / ( π z ) ,
10.28.2 𝒲 { K ν ( z ) , I ν ( z ) } = I ν ( z ) K ν + 1 ( z ) + I ν + 1 ( z ) K ν ( z ) = 1 / z .
2: 10.5 Wronskians and Cross-Products
§10.5 Wronskians and Cross-Products
10.5.1 𝒲 { J ν ( z ) , J - ν ( z ) } = J ν + 1 ( z ) J - ν ( z ) + J ν ( z ) J - ν - 1 ( z ) = - 2 sin ( ν π ) / ( π z ) ,
10.5.2 𝒲 { J ν ( z ) , Y ν ( z ) } = J ν + 1 ( z ) Y ν ( z ) - J ν ( z ) Y ν + 1 ( z ) = 2 / ( π z ) ,
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 ) ,
3: 10.50 Wronskians and Cross-Products
§10.50 Wronskians and Cross-Products
𝒲 { j n ( z ) , y n ( z ) } = z - 2 ,
𝒲 { h n ( 1 ) ( z ) , h n ( 2 ) ( z ) } = - 2 i z - 2 .
𝒲 { i n ( 1 ) ( z ) , i n ( 2 ) ( z ) } = ( - 1 ) n + 1 z - 2 ,
𝒲 { i n ( 1 ) ( z ) , k n ( z ) } = 𝒲 { i n ( 2 ) ( z ) , k n ( z ) } = - 1 2 π z - 2 .
4: 1.13 Differential Equations
Wronskian
The Wronskian of w 1 ( z ) and w 2 ( z ) is defined by …(More generally 𝒲 { w 1 ( z ) , , w n ( z ) } = det [ w k ( j - 1 ) ( z ) ] , where 1 j , k n .) …If f ( z ) = 0 , then the Wronskian is constant. The following three statements are equivalent: w 1 ( z ) and w 2 ( z ) comprise a fundamental pair in D ; 𝒲 { w 1 ( z ) , w 2 ( z ) } does not vanish in D ; w 1 ( z ) and w 2 ( z ) are linearly independent, that is, the only constants A and B such that …
5: 14.2 Differential Equations
§14.2(iv) Wronskians and Cross-Products
14.2.3 𝒲 { P ν - μ ( x ) , P ν - μ ( - x ) } = 2 Γ ( μ - ν ) Γ ( ν + μ + 1 ) ( 1 - x 2 ) ,
14.2.7 𝒲 { P ν - μ ( x ) , P ν μ ( x ) } = 𝒲 { P ν - μ ( x ) , P ν μ ( x ) } = 2 sin ( μ π ) π ( 1 - x 2 ) ,
6: 9.11 Products
§9.11(ii) Wronskian
9.11.2 𝒲 { Ai 2 ( z ) , Ai ( z ) Bi ( z ) , Bi 2 ( z ) } = 2 π - 3 .
9.11.6 w 1 w 2 d z = 1 2 ( w 1 w 2 + z 𝒲 { w 1 , w 2 } ) ,
9.11.9 z w 1 w 2 d z = 1 2 w 1 w 2 + 1 4 z 2 𝒲 { w 1 , w 2 } ,
9.11.11 1 w 1 2 f ( w 2 w 1 ) d z = 1 𝒲 { w 1 , w 2 } f ( w 2 w 1 ) .
7: 9.2 Differential Equation
§9.2(iv) Wronskians
9.2.7 𝒲 { Ai ( z ) , Bi ( z ) } = 1 π ,
9.2.8 𝒲 { Ai ( z ) , Ai ( z e 2 π i / 3 ) } = e ± π i / 6 2 π ,
9.2.9 𝒲 { Ai ( z e - 2 π i / 3 ) , Ai ( z e 2 π i / 3 ) } = 1 2 π i .
8: 30.5 Functions of the Second Kind
30.5.4 𝒲 { Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) } = ( n + m ) ! ( 1 - x 2 ) ( n - m ) ! A n m ( γ 2 ) A n - m ( γ 2 ) ( 0 ) ,
9: 12.2 Differential Equations
§12.2(iii) Wronskians
12.2.12 𝒲 { U ( a , z ) , U ( - a , ± i z ) } = i e ± i π ( 1 2 a + 1 4 ) .
10: 28.5 Second Solutions fe n , ge n
Wronskians
28.5.8 𝒲 { ce n , fe n } = ce n ( 0 , q ) fe n ( 0 , q ) ,
28.5.9 𝒲 { se n , ge n } = - se n ( 0 , q ) ge n ( 0 , q ) .