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1: 33.22 Particle Scattering and Atomic and Molecular Spectra
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𝗄 Scaling
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Z Scaling
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i ⁒ 𝗄 Scaling
2: 36.5 Stokes Sets
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36.5.8 16 ⁒ u 5 Y 2 10 ⁒ u + 4 ⁒ u 3 ⁒ sign ⁑ ( z ) 3 10 ⁒ | Y | ⁒ sign ⁑ ( z ) + 4 ⁒ t 5 + 2 ⁒ t 3 ⁒ sign ⁑ ( z ) + | Y | ⁒ t 2 = 0 ,
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36.5.9 t = u + ( | Y | 10 ⁒ u u 2 3 10 ⁒ sign ⁑ ( z ) ) 1 / 2 .
3: 15.12 Asymptotic Approximations
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15.12.5 𝐅 ⁑ ( a + Ξ» , b Ξ» c ; 1 2 1 2 ⁒ z ) = 2 ( a + b 1 ) / 2 ⁒ ( z + 1 ) ( c a b 1 ) / 2 ( z 1 ) c / 2 ⁒ ΞΆ ⁒ sinh ⁑ ΞΆ ⁒ ( Ξ» + 1 2 ⁒ a 1 2 ⁒ b ) 1 c ⁒ ( I c 1 ⁑ ( ( Ξ» + 1 2 ⁒ a 1 2 ⁒ b ) ⁒ ΞΆ ) ⁒ ( 1 + O ⁑ ( Ξ» 2 ) ) + I c 2 ⁑ ( ( Ξ» + 1 2 ⁒ a 1 2 ⁒ b ) ⁒ ΞΆ ) 2 ⁒ Ξ» + a b ⁒ ( ( c 1 2 ) ⁒ ( c 3 2 ) ⁒ ( 1 ΞΆ coth ⁑ ΞΆ ) + 1 2 ⁒ ( 2 ⁒ c a b 1 ) ⁒ ( a + b 1 ) ⁒ tanh ⁑ ( 1 2 ⁒ ΞΆ ) + O ⁑ ( Ξ» 2 ) ) ) ,
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15.12.9 ( z + 1 ) 3 ⁒ Ξ» / 2 ⁒ ( 2 ⁒ Ξ» ) c 1 ⁒ 𝐅 ⁑ ( a + Ξ» , b + 2 ⁒ Ξ» c ; z ) = Ξ» 1 / 3 ⁒ ( e Ο€ ⁒ i ⁒ ( a c + Ξ» + ( 1 / 3 ) ) ⁒ Ai ⁑ ( e 2 ⁒ Ο€ ⁒ i / 3 ⁒ Ξ» 2 / 3 ⁒ Ξ² 2 ) + e Ο€ ⁒ i ⁒ ( c a Ξ» ( 1 / 3 ) ) ⁒ Ai ⁑ ( e 2 ⁒ Ο€ ⁒ i / 3 ⁒ Ξ» 2 / 3 ⁒ Ξ² 2 ) ) ⁒ ( a 0 ⁑ ( ΞΆ ) + O ⁑ ( Ξ» 1 ) ) + Ξ» 2 / 3 ⁒ ( e Ο€ ⁒ i ⁒ ( a c + Ξ» + ( 2 / 3 ) ) ⁒ Ai ⁑ ( e 2 ⁒ Ο€ ⁒ i / 3 ⁒ Ξ» 2 / 3 ⁒ Ξ² 2 ) + e Ο€ ⁒ i ⁒ ( c a Ξ» ( 2 / 3 ) ) ⁒ Ai ⁑ ( e 2 ⁒ Ο€ ⁒ i / 3 ⁒ Ξ» 2 / 3 ⁒ Ξ² 2 ) ) ⁒ ( a 1 ⁑ ( ΞΆ ) + O ⁑ ( Ξ» 1 ) ) ,
4: 8.18 Asymptotic Expansions of I x ⁑ ( a , b )
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8.18.14 I x ⁑ ( a , b ) Q ⁑ ( b , a ⁒ ΞΆ ) ( 2 ⁒ Ο€ ⁒ b ) 1 / 2 Ξ“ ⁑ ( b ) ⁒ ( x x 0 ) a ⁒ ( 1 x 1 x 0 ) b ⁒ k = 0 h k ⁑ ( ΞΆ , ΞΌ ) a k ,
5: 15.7 Continued Fractions
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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 β‹― ,
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6: 15.15 Sums
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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 .
7: 15.2 Definitions and Analytical Properties
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15.2.2 𝐅 ⁑ ( a , b ; c ; z ) = s = 0 ( a ) s ⁒ ( b ) s Ξ“ ⁑ ( c + s ) ⁒ s ! ⁒ z s , | z | < 1 ,
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15.2.3 𝐅 ⁑ ( a , b c ; x + i ⁒ 0 ) 𝐅 ⁑ ( a , b c ; x i ⁒ 0 ) = 2 ⁒ Ο€ ⁒ i Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ ( x 1 ) c a b ⁒ 𝐅 ⁑ ( c a , c b c a b + 1 ; 1 x ) , x > 1 .
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15.2.3_5 lim c n F ⁑ ( a , b ; c ; z ) Ξ“ ⁑ ( c ) = 𝐅 ⁑ ( a , b ; n ; z ) = ( a ) n + 1 ⁒ ( b ) n + 1 ( n + 1 ) ! ⁒ z n + 1 ⁒ F ⁑ ( a + n + 1 , b + n + 1 ; n + 2 ; z ) , n = 0 , 1 , 2 , .
8: 15.1 Special Notation
9: 15.6 Integral Representations
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15.6.1 𝐅 ⁑ ( a , b ; c ; z ) = 1 Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c b ) ⁒ 0 1 t b 1 ⁒ ( 1 t ) c b 1 ( 1 z ⁒ t ) a ⁒ d t , | ph ⁑ ( 1 z ) | < Ο€ ; ⁑ c > ⁑ b > 0 .
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15.6.2 𝐅 ⁑ ( a , b ; c ; z ) = Ξ“ ⁑ ( 1 + b c ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( b ) ⁒ 0 ( 1 + ) t b 1 ⁒ ( t 1 ) c b 1 ( 1 z ⁒ t ) a ⁒ d t , | ph ⁑ ( 1 z ) | < Ο€ ; c b 1 , 2 , 3 , , ⁑ b > 0 .
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15.6.2_5 𝐅 ⁑ ( a , b ; c ; z ) = 1 Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c b ) ⁒ 0 t b 1 ⁒ ( t + 1 ) a c ( t z ⁒ t + 1 ) a ⁒ d t , | ph ⁑ ( 1 z ) | < Ο€ ; ⁑ c > ⁑ b > 0 .
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15.6.6 𝐅 ⁑ ( a , b ; c ; z ) = 1 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ i ⁒ i ⁒ Ξ“ ⁑ ( a + t ) ⁒ Ξ“ ⁑ ( b + t ) ⁒ Ξ“ ⁑ ( t ) Ξ“ ⁑ ( c + t ) ⁒ ( z ) t ⁒ d t , | ph ⁑ ( z ) | < Ο€ ; a , b 0 , 1 , 2 , .
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15.6.8 𝐅 ⁑ ( a , b ; c ; z ) = 1 Ξ“ ⁑ ( c d ) ⁒ 0 1 𝐅 ⁑ ( a , b ; d ; z ⁒ t ) ⁒ t d 1 ⁒ ( 1 t ) c d 1 ⁒ d t , | ph ⁑ ( 1 z ) | < Ο€ ; ⁑ c > ⁑ d > 0 .
10: 15.14 Integrals
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15.14.1 0 x s 1 ⁒ 𝐅 ⁑ ( a , b c ; x ) ⁒ d x = Ξ“ ⁑ ( s ) ⁒ Ξ“ ⁑ ( a s ) ⁒ Ξ“ ⁑ ( b s ) Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c s ) , min ⁑ ( ⁑ a , ⁑ b ) > ⁑ s > 0 .