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1: 10.25 Definitions
§10.25(ii) Standard Solutions
The defining property of the second standard solution K ν ( z ) of (10.25.1) is …
2: 14.21 Definitions and Basic Properties
Standard solutions: the associated Legendre functions P ν μ ( z ) , P ν - μ ( z ) , Q ν μ ( z ) , and Q - ν - 1 μ ( z ) . … …
3: 10.2 Definitions
§10.2(ii) Standard Solutions
The notation 𝒞 ν ( z ) denotes J ν ( z ) , Y ν ( z ) , H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) , or any nontrivial linear combination of these functions, the coefficients in which are independent of z and ν . …
4: 14.2 Differential Equations
§14.2(i) Legendre’s Equation
Standard solutions: P ν ( ± x ) , Q ν ( ± x ) , Q - ν - 1 ( ± x ) , P ν ( ± x ) , Q ν ( ± x ) , Q - ν - 1 ( ± x ) . …
§14.2(ii) Associated Legendre Equation
Standard solutions: P ν μ ( ± x ) , P ν - μ ( ± x ) , Q ν μ ( ± x ) , Q - ν - 1 μ ( ± x ) , P ν μ ( ± x ) , P ν - μ ( ± x ) , Q ν μ ( ± x ) , Q - ν - 1 μ ( ± x ) . …
5: 13.2 Definitions and Basic Properties
Standard Solutions
The first two standard solutions are: … Another standard solution of (13.2.1) is U ( a , b , z ) , which is determined uniquely by the property …
13.2.7 U ( - m , b , z ) = ( - 1 ) m ( b ) m M ( - m , b , z ) = ( - 1 ) m s = 0 m ( m s ) ( b + s ) m - s ( - z ) s .
13.2.8 U ( a , a + n + 1 , z ) = ( - 1 ) n ( 1 - a - n ) n z a + n M ( - n , 1 - a - n , z ) = z - a s = 0 n ( n s ) ( a ) s z - s .
6: 10.47 Definitions and Basic Properties
§10.47(ii) Standard Solutions
7: 12.2 Differential Equations
§12.2 Differential Equations
Standard solutions are U ( a , ± z ) , V ( a , ± z ) , U ¯ ( a , ± x ) (not complex conjugate), U ( - a , ± i z ) for (12.2.2); W ( a , ± x ) for (12.2.3); D ν ( ± z ) for (12.2.4), where …
8: 13.14 Definitions and Basic Properties
Standard Solutions
Standard solutions are: …
9: 14.3 Definitions and Hypergeometric Representations
14.3.8 P ν m ( x ) = Γ ( ν + m + 1 ) 2 m Γ ( ν - m + 1 ) ( x 2 - 1 ) m / 2 F ( ν + m + 1 , m - ν ; m + 1 ; 1 2 - 1 2 x ) .
As standard solutions of (14.2.2) we take the pair P ν - μ ( x ) and Q ν μ ( x ) , where …
10: 9.2 Differential Equation
Standard solutions are: …