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1: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
2: 10.63 Recurrence Relations and Derivatives
§10.63(ii) Cross-Products
10.63.7 p ν s ν = r ν 2 + q ν 2 .
3: 10.6 Recurrence Relations and Derivatives
§10.6(iii) Cross-Products
10.6.10 p ν s ν - q ν r ν = 4 / ( π 2 a b ) .
4: 10.5 Wronskians and Cross-Products
§10.5 Wronskians and Cross-Products
5: 28.28 Integrals, Integral Representations, and Integral Equations
28.28.25 sinh z π 2 0 2 π cos t me ν ( t , h 2 ) me - ν - 2 m - 1 ( t , h 2 ) sinh 2 z + sin 2 t d t = ( - 1 ) m + 1 i h α ν , m ( 0 ) D 0 ( ν , ν + 2 m + 1 , z ) ,
28.28.30 sinh z π 2 0 2 π cos t me ν ( t , h 2 ) me - ν - 2 m - 1 ( t , h 2 ) sinh 2 z + sin 2 t d t = ( - 1 ) m i h α ν , m ( 1 ) D 1 ( ν , ν + 2 m + 1 , z ) ,
§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
28.28.49 α ^ n , m ( c ) = 1 2 π 0 2 π cos t ce n ( t , h 2 ) ce m ( t , h 2 ) d t = ( - 1 ) p + 1 2 i π ce n ( 0 , h 2 ) ce m ( 0 , h 2 ) h Dc 0 ( n , m , 0 ) .
6: 10.50 Wronskians and Cross-Products
§10.50 Wronskians and Cross-Products
7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).
8: 1.6 Vectors and Vector-Valued Functions
Cross Product (or Vector Product)
1.6.9 a × b = | i j k a 1 a 2 a 3 b 1 b 2 b 3 | = ( a 2 b 3 - a 3 b 2 ) i + ( a 3 b 1 - a 1 b 3 ) j + ( a 1 b 2 - a 2 b 1 ) k = a b ( sin θ ) n ,
See accompanying text
Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify
9: 10.65 Power Series
§10.65(iii) Cross-Products and Sums of Squares
10: 14.2 Differential Equations
§14.2(iv) Wronskians and Cross-Products
14.2.5 P ν + 1 μ ( x ) Q ν μ ( x ) - P ν μ ( x ) Q ν + 1 μ ( x ) = Γ ( ν + μ + 1 ) Γ ( ν - μ + 2 ) ,
14.2.11 P ν + 1 μ ( x ) Q ν μ ( x ) - P ν μ ( x ) Q ν + 1 μ ( x ) = e μ π i Γ ( ν + μ + 1 ) Γ ( ν - μ + 2 ) .