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Whittaker functions

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11: 13.26 Addition and Multiplication Theorems
§13.26(i) Addition Theorems for M κ , μ ( z )
The function M κ , μ ( x + y ) has the following expansions: …
§13.26(ii) Addition Theorems for W κ , μ ( z )
The function W κ , μ ( x + y ) has the following expansions: …
§13.26(iii) Multiplication Theorems for M κ , μ ( z ) and W κ , μ ( z )
12: 33.14 Definitions and Basic Properties
§33.14(ii) Regular Solution f ( ϵ , ; r )
33.14.4 f ( ϵ , ; r ) = κ + 1 M κ , + 1 2 ( 2 r / κ ) / ( 2 + 1 ) ! ,
§33.14(iii) Irregular Solution h ( ϵ , ; r )
33.14.7 h ( ϵ , ; r ) = Γ ( + 1 - κ ) π κ ( W κ , + 1 2 ( 2 r / κ ) + ( - 1 ) S ( ϵ , r ) Γ ( + 1 + κ ) 2 ( 2 + 1 ) ! M κ , + 1 2 ( 2 r / κ ) ) ,
33.14.14 ϕ n , ( r ) = ( - 1 ) + 1 + n ( 2 / n 3 ) 1 / 2 s ( - 1 / n 2 , ; r ) = ( - 1 ) + 1 + n n + 2 ( ( n - - 1 ) ! ( n + ) ! ) 1 / 2 ( 2 r ) + 1 e - r / n L n - - 1 ( 2 + 1 ) ( 2 r / n )
13: 13.16 Integral Representations
§13.16(i) Integrals Along the Real Line
13.16.5 W κ , μ ( z ) = z μ + 1 2 2 - 2 μ Γ ( 1 2 + μ - κ ) 1 e - 1 2 z t ( t - 1 ) μ - 1 2 - κ ( t + 1 ) μ - 1 2 + κ d t , μ + 1 2 > κ , | ph z | < 1 2 π ,
§13.16(ii) Contour Integrals
§13.16(iii) Mellin–Barnes Integrals
14: 33.2 Definitions and Basic Properties
The function F ( η , ρ ) is recessive (§2.7(iii)) at ρ = 0 , and is defined by
33.2.3 F ( η , ρ ) = C ( η ) 2 - - 1 ( i ) + 1 M ± i η , + 1 2 ( ± 2 i ρ ) ,
The functions H ± ( η , ρ ) are defined by
33.2.7 H ± ( η , ρ ) = ( i ) e ( π η / 2 ) ± i σ ( η ) W i η , + 1 2 ( 2 i ρ ) ,
15: 12.1 Special Notation
An older notation, due to Whittaker (1902), for U ( a , z ) is D ν ( z ) . …
16: 33.16 Connection Formulas
§33.16(iii) f and h in Terms of W κ , μ ( z ) when ϵ < 0
§33.16(v) s and c in Terms of W κ , μ ( z ) when ϵ < 0
17: 13.17 Continued Fractions
§13.17 Continued Fractions
13.17.1 z M κ , μ ( z ) M κ - 1 2 , μ + 1 2 ( z ) = 1 + u 1 z 1 + u 2 z 1 + ,
13.17.3 W κ , μ ( z ) z W κ - 1 2 , μ - 1 2 ( z ) = 1 + v 1 / z 1 + v 2 / z 1 + ,
18: 13.21 Uniform Asymptotic Approximations for Large κ
§13.21 Uniform Asymptotic Approximations for Large κ
13.21.1 M κ , μ ( x ) = x Γ ( 2 μ + 1 ) κ - μ ( J 2 μ ( 2 x κ ) + env J 2 μ ( 2 x κ ) O ( κ - 1 2 ) ) ,
13.21.6 M - κ , μ ( 4 κ x ) = 2 Γ ( 2 μ + 1 ) κ μ - 1 2 ( x ζ 1 + x ) 1 4 I 2 μ ( 4 κ ζ 1 2 ) ( 1 + O ( κ - 1 ) ) ,
For a uniform asymptotic expansion in terms of Airy functions for W κ , μ ( 4 κ x ) when κ is large and positive, μ is real with | μ | bounded, and x [ δ , ) see Olver (1997b, Chapter 11, Ex. 7.3). …
19: 8.5 Confluent Hypergeometric Representations
For the confluent hypergeometric functions M , M , U , and the Whittaker functions M κ , μ and W κ , μ , see §§13.2(i) and 13.14(i). …
8.5.4 γ ( a , z ) = a - 1 z 1 2 a - 1 2 e - 1 2 z M 1 2 a - 1 2 , 1 2 a ( z ) .
8.5.5 Γ ( a , z ) = e - 1 2 z z 1 2 a - 1 2 W 1 2 a - 1 2 , 1 2 a ( z ) .
20: 33.22 Particle Scattering and Atomic and Molecular Spectra
For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, F ( η , ρ ) and G ( η , ρ ) , or f ( ϵ , ; r ) and h ( ϵ , ; r ) , to determine the scattering S -matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function W - η , + 1 2 ( 2 ρ ) . …