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functions Fℓ(η,ρ),Gℓ(η,ρ),H±ℓ(η,ρ)

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1: 8.22 Mathematical Applications
The so-called terminant function F p ( z ) , defined by
8.22.1 F p ( z ) = Γ ( p ) 2 π z 1 p E p ( z ) = Γ ( p ) 2 π Γ ( 1 p , z ) ,
2: 2.11 Remainder Terms; Stokes Phenomenon
2.11.11 F n + p ( z ) = e z 2 π 0 e z t t n + p 1 1 + t d t = Γ ( n + p ) 2 π E n + p ( z ) z n + p 1 .
Owing to the factor e ρ , that is, e | z | in (2.11.13), F n + p ( z ) is uniformly exponentially small compared with E p ( z ) . … However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the F -functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the F -functions. In this context the F -functions are called terminants, a name introduced by Dingle (1973). …
2.11.20 R n ( 1 ) ( z ) = ( 1 ) n 1 i e ( μ 2 μ 1 ) π i e λ 2 z z μ 2 ( C 1 s = 0 m 1 ( 1 ) s a s , 2 F n + μ 2 μ 1 s ( z ) z s + R m , n ( 1 ) ( z ) ) ,
3: 15.1 Special Notation
4: 32.14 Combinatorics
32.14.1 lim N Prob ( N ( 𝝅 ) 2 N N 1 / 6 s ) = F ( s ) ,
where the distribution function F ( s ) is defined here by
32.14.2 F ( s ) = exp ( s ( x s ) w 2 ( x ) d x ) ,
The distribution function F ( s ) given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of n × n Hermitian matrices; see Tracy and Widom (1994). …
5: 16.1 Special Notation
The main functions treated in this chapter are the generalized hypergeometric function F q p ( a 1 , , a p b 1 , , b q ; z ) , the Appell (two-variable hypergeometric) functions F 1 ( α ; β , β ; γ ; x , y ) , F 2 ( α ; β , β ; γ , γ ; x , y ) , F 3 ( α , α ; β , β ; γ ; x , y ) , F 4 ( α , β ; γ , γ ; x , y ) , and the Meijer G -function G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) . Alternative notations are F q p ( 𝐚 𝐛 ; z ) , F q p ( a 1 , , a p ; b 1 , , b q ; z ) , and F q p ( 𝐚 ; 𝐛 ; z ) for the generalized hypergeometric function, F 1 ( α , β , β ; γ ; x , y ) , F 2 ( α , β , β ; γ , γ ; x , y ) , F 3 ( α , α , β , β ; γ ; x , y ) , F 4 ( α , β ; γ , γ ; x , y ) , for the Appell functions, and G p , q m , n ( z ; 𝐚 ; 𝐛 ) for the Meijer G -function.
6: 33.3 Graphics
§33.3(i) Line Graphs of the Coulomb Radial Functions F ( η , ρ ) and G ( η , ρ )
33.3.1 M ( η , ρ ) = ( F 2 ( η , ρ ) + G 2 ( η , ρ ) ) 1 / 2 = | H ± ( η , ρ ) | .
§33.3(ii) Surfaces of the Coulomb Radial Functions F 0 ( η , ρ ) and G 0 ( η , ρ )
7: 8.14 Integrals
8.14.5 0 x a 1 e s x γ ( b , x ) d x = Γ ( a + b ) b ( 1 + s ) a + b F ( 1 , a + b ; 1 + b ; 1 / ( 1 + s ) ) , s > 0 , ( a + b ) > 0 ,
8.14.6 0 x a 1 e s x Γ ( b , x ) d x = Γ ( a + b ) a ( 1 + s ) a + b F ( 1 , a + b ; 1 + a ; s / ( 1 + s ) ) , s > 1 , ( a + b ) > 0 , a > 0 .
For the hypergeometric function F ( a , b ; c ; z ) see §15.2(i). …
8: 16.2 Definition and Analytic Properties
§16.2(i) Generalized Hypergeometric Series
Equivalently, the function is denoted by F q p ( 𝐚 𝐛 ; z ) or F q p ( 𝐚 ; 𝐛 ; z ) , and sometimes, for brevity, by F q p ( z ) . …
16.2.5 𝐅 q p ( 𝐚 ; 𝐛 ; z ) = F q p ( a 1 , , a p b 1 , , b q ; z ) / ( Γ ( b 1 ) Γ ( b q ) ) = k = 0 ( a 1 ) k ( a p ) k Γ ( b 1 + k ) Γ ( b q + k ) z k k ! ;
When p q + 1 and z is fixed and not a branch point, any branch of 𝐅 q p ( 𝐚 ; 𝐛 ; z ) is an entire function of each of the parameters a 1 , , a p , b 1 , , b q .
9: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
§33.5(i) Small ρ
§33.5(ii) η = 0
For the functions 𝗃 , 𝗒 , J , Y see §§10.47(ii), 10.2(ii). …
§33.5(iii) Small | η |
§33.5(iv) Large
10: 33.11 Asymptotic Expansions for Large ρ
§33.11 Asymptotic Expansions for Large ρ
F ( η , ρ ) = g ( η , ρ ) cos θ + f ( η , ρ ) sin θ ,
F ( η , ρ ) = g ^ ( η , ρ ) cos θ + f ^ ( η , ρ ) sin θ ,
33.11.4 H ± ( η , ρ ) = e ± i θ ( f ( η , ρ ) ± i g ( η , ρ ) ) ,
33.11.7 g ( η , ρ ) f ^ ( η , ρ ) f ( η , ρ ) g ^ ( η , ρ ) = 1 .