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1: 36.6 Scaling Relations
§36.6 Scaling Relations
Diffraction Catastrophe Scaling
Indices for k -Scaling of Magnitude of Ψ K or Ψ ( U ) (Singularity Index)
Indices for k -Scaling of Coordinates x m
Indices for k -Scaling of x Hypervolume
2: 15.7 Continued Fractions
15.7.1 F ( a , b ; c ; z ) F ( a , b + 1 ; c + 1 ; z ) = t 0 - u 1 z t 1 - u 2 z t 2 - u 3 z t 3 - ,
3: 15.15 Sums
15.15.1 F ( a , b c ; 1 z ) = ( 1 - z 0 z ) - a s = 0 ( a ) s s ! F ( - s , b c ; 1 z 0 ) ( 1 - z z 0 ) - s .
4: 15.6 Integral Representations
The function F ( a , b ; c ; z ) (not F ( a , b ; c ; z ) ) has the following integral representations:
15.6.1 F ( a , b ; c ; z ) = 1 Γ ( b ) Γ ( c - b ) 0 1 t b - 1 ( 1 - t ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; c > b > 0 .
15.6.2 F ( a , b ; c ; z ) = Γ ( 1 + b - c ) 2 π i Γ ( b ) 0 ( 1 + ) t b - 1 ( t - 1 ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; c - b 1 , 2 , 3 , , b > 0 .
15.6.8 F ( a , b ; c ; z ) = 1 Γ ( c - d ) 0 1 F ( a , b ; d ; z t ) t d - 1 ( 1 - t ) c - d - 1 d t , | ph ( 1 - z ) | < π ; c > d > 0 .
15.6.9 F ( a , b ; c ; z ) = 0 1 t d - 1 ( 1 - t ) c - d - 1 ( 1 - z t ) a + b - λ F ( λ - a , λ - b d ; z t ) F ( a + b - λ , λ - d c - d ; ( 1 - t ) z 1 - z t ) d t , | ph ( 1 - z ) | < π ; λ , c > d > 0 .
5: Possible Errors in DLMF
One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. …
6: 33.22 Particle Scattering and Atomic and Molecular Spectra
k Scaling
The k -scaled variables ρ and η of §33.2 are given by … In these applications, the Z -scaled variables r and ϵ are more convenient.
Z Scaling
i k Scaling
7: 14.3 Definitions and Hypergeometric Representations
14.3.1 P ν μ ( x ) = ( 1 + x 1 - x ) μ / 2 F ( ν + 1 , - ν ; 1 - μ ; 1 2 - 1 2 x ) .
14.3.4 P ν m ( x ) = ( - 1 ) m Γ ( ν + m + 1 ) 2 m Γ ( ν - m + 1 ) ( 1 - x 2 ) m / 2 F ( ν + m + 1 , m - ν ; m + 1 ; 1 2 - 1 2 x ) ;
8: 15.1 Special Notation
9: 15.2 Definitions and Analytical Properties
Except where indicated otherwise principal branches of F ( a , b ; c ; z ) and F ( a , b ; c ; z ) are assumed throughout the DLMF. … The principal branch of F ( a , b ; c ; z ) is an entire function of a , b , and c . …As a multivalued function of z , F ( a , b ; c ; z ) is analytic everywhere except for possible branch points at z = 0 , 1 , and . … (Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does F ( a , b ; c ; z ) , which is analytic at c = 0 , - 1 , - 2 , .) For comparison of F ( a , b ; c ; z ) and F ( a , b ; c ; z ) , with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7. …
10: 15.14 Integrals
15.14.1 0 x s - 1 F ( a , b c ; - x ) d x = Γ ( s ) Γ ( a - s ) Γ ( b - s ) Γ ( a ) Γ ( b ) Γ ( c - s ) , min ( a , b ) > s > 0 .