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1: 33.13 Complex Variable and Parameters
33.13.1 C ( η ) = 2 e i σ ( η ) ( π η / 2 ) Γ ( + 1 i η ) / Γ ( 2 + 2 ) ,
33.13.2 R = ( 2 + 1 ) C ( η ) / C 1 ( η ) .
2: 30.1 Special Notation
where d m n ( γ ) is a normalization constant determined by …
3: 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 ,
For further information, including normalization constants, see Sleeman (1966a). …
4: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
33.5.6 C ( 0 ) = 2 ! ( 2 + 1 ) ! = 1 ( 2 + 1 ) !! .
33.5.9 C ( η ) e π η / 2 ( 2 + 1 ) !! e π η / 2 e 2 ( 2 ) + 1 .
5: 33.7 Integral Representations
33.7.2 H ( η , ρ ) = e i ρ ρ ( 2 + 1 ) ! C ( η ) 0 e t t i η ( t + 2 i ρ ) + i η d t ,
33.7.3 H ( η , ρ ) = i e π η ρ + 1 ( 2 + 1 ) ! C ( η ) 0 ( exp ( i ( ρ tanh t 2 η t ) ) ( cosh t ) 2 + 2 + i ( 1 + t 2 ) exp ( ρ t + 2 η arctan t ) ) d t ,
33.7.4 H + ( η , ρ ) = i e π η ρ + 1 ( 2 + 1 ) ! C ( η ) 1 i e i ρ t ( 1 t ) i η ( 1 + t ) + i η d t .
6: 33.2 Definitions and Basic Properties
33.2.3 F ( η , ρ ) = C ( η ) 2 1 ( i ) + 1 M ± i η , + 1 2 ( ± 2 i ρ ) ,
33.2.5 C ( η ) = 2 e π η / 2 | Γ ( + 1 + i η ) | ( 2 + 1 ) ! .
F ( η , ρ ) is a real and analytic function of ρ on the open interval 0 < ρ < , and also an analytic function of η when < η < . The normalizing constant C ( η ) is always positive, and has the alternative form
33.2.6 C ( η ) = 2 ( ( 2 π η / ( e 2 π η 1 ) ) k = 1 ( η 2 + k 2 ) ) 1 / 2 ( 2 + 1 ) ! .
7: 33.9 Expansions in Series of Bessel Functions
33.9.3 F ( η , ρ ) = C ( η ) ( 2 + 1 ) ! ( 2 η ) 2 + 1 ρ k = 2 + 1 b k t k / 2 I k ( 2 t ) , η > 0 ,
33.9.4 F ( η , ρ ) = C ( η ) ( 2 + 1 ) ! ( 2 | η | ) 2 + 1 ρ k = 2 + 1 b k t k / 2 J k ( 2 t ) , η < 0 .
33.9.6 G ( η , ρ ) ρ ( + 1 2 ) λ ( η ) C ( η ) k = 2 + 1 ( 1 ) k b k t k / 2 K k ( 2 t ) ,
8: 33.6 Power-Series Expansions in ρ
33.6.1 F ( η , ρ ) = C ( η ) k = + 1 A k ( η ) ρ k ,
33.6.2 F ( η , ρ ) = C ( η ) k = + 1 k A k ( η ) ρ k 1 ,
9: 33.10 Limiting Forms for Large ρ or Large | η |
10: 33.16 Connection Formulas