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11: 30.16 Methods of Computation
If λ n m ( γ 2 ) is known, then we can compute Ps n m ( x , γ 2 ) (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions w ( 0 ) = 1 , w ( 0 ) = 0 if n - m is even, or w ( 0 ) = 0 , w ( 0 ) = 1 if n - m is odd. …
12: 10.74 Methods of Computation
In the case of J n ( x ) , the need for initial values can be avoided by application of Olver’s algorithm (§3.6(v)) in conjunction with Equation (10.12.4) used as a normalizing condition, or in the case of noninteger orders, (10.23.15). …
13: 2.11 Remainder Terms; Stokes Phenomenon
For notational convenience assume that the original differential equation (2.7.1) is normalized so that λ 2 - λ 1 = 1 . …
14: 14.30 Spherical and Spheroidal Harmonics
In the quantization of angular momentum the spherical harmonics Y l , m ( θ , ϕ ) are normalized solutions of the eigenvalue equations
15: 31.9 Orthogonality
31.9.2 ζ ( 1 + , 0 + , 1 - , 0 - ) t γ - 1 ( 1 - t ) δ - 1 ( t - a ) ϵ - 1 w m ( t ) w k ( t ) d t = δ m , k θ m .
The normalization constant θ m is given by
31.9.3 θ m = ( 1 - e 2 π i γ ) ( 1 - e 2 π i δ ) ζ γ ( 1 - ζ ) δ ( ζ - a ) ϵ f 0 ( q , ζ ) f 1 ( q , ζ ) q 𝒲 { f 0 ( q , ζ ) , f 1 ( q , ζ ) } | q = q m ,
and the integration paths 1 , 2 are Pochhammer double-loop contours encircling distinct pairs of singularities { 0 , 1 } , { 0 , a } , { 1 , a } . For further information, including normalization constants, see Sleeman (1966a). …
16: 8.12 Uniform Asymptotic Expansions for Large Parameter
8.12.5 e ± π i a 2 i sin ( π a ) Q ( - a , z e ± π i ) = ± 1 2 erfc ( ± i η a / 2 ) - i T ( a , η ) ,
8.12.16 e ± π i a 2 i sin ( π a ) Q ( - a , a e ± π i ) ± 1 2 - i 2 π a k = 0 c k ( 0 ) ( - a ) - k , | ph a | π - δ ,
8.12.18 Q ( a , z ) P ( a , z ) } z a - 1 2 e - z Γ ( a ) ( d ( ± χ ) k = 0 A k ( χ ) z k / 2 k = 1 B k ( χ ) z k / 2 ) ,
17: 28.31 Equations of Whittaker–Hill and Ince
§28.31 Equations of Whittaker–Hill and Ince
§28.31(i) Whittaker–Hill Equation
§28.31(ii) Equation of Ince; Ince Polynomials
The normalization is given by … They satisfy the differential equation
18: 9.13 Generalized Airy Functions
§9.13(i) Generalizations from the Differential Equation
Equations of the form … In Olver (1977a, 1978) a different normalization is used. … Another normalization of (9.13.17) is used in Smirnov (1960), given by …
19: 3.7 Ordinary Differential Equations
The eigenvalues λ k are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …
20: 36.10 Differential Equations
In terms of the normal form (36.2.1) the Ψ K ( x ) satisfy the operator equationIn terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( x ) satisfy the following operator equations