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scaled gamma function

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11: 15.6 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 .
15.6.6 F ( a , b ; c ; z ) = 1 2 π i Γ ( a ) Γ ( b ) - i i Γ ( a + t ) Γ ( b + t ) Γ ( - t ) Γ ( c + t ) ( - z ) t d t , | ph ( - z ) | < π ; a , b 0 , - 1 , - 2 , .
15.6.7 F ( a , b ; c ; z ) = 1 2 π i Γ ( a ) Γ ( b ) Γ ( c - a ) Γ ( c - b ) - i i Γ ( a + t ) Γ ( b + t ) Γ ( c - a - b - t ) Γ ( - t ) ( 1 - z ) t d t , | ph ( 1 - z ) | < π ; a , b , c - a , c - b 0 , - 1 , - 2 , .
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 .
12: 15.8 Transformations of Variable
15.8.2 sin ( π ( b - a ) ) π F ( a , b c ; z ) = ( - z ) - a Γ ( b ) Γ ( c - a ) F ( a , a - c + 1 a - b + 1 ; 1 z ) - ( - z ) - b Γ ( a ) Γ ( c - b ) F ( b , b - c + 1 b - a + 1 ; 1 z ) , | ph ( - z ) | < π .
15.8.3 sin ( π ( b - a ) ) π F ( a , b c ; z ) = ( 1 - z ) - a Γ ( b ) Γ ( c - a ) F ( a , c - b a - b + 1 ; 1 1 - z ) - ( 1 - z ) - b Γ ( a ) Γ ( c - b ) F ( b , c - a b - a + 1 ; 1 1 - z ) , | ph ( - z ) | < π .
15.8.4 sin ( π ( c - a - b ) ) π F ( a , b c ; z ) = 1 Γ ( c - a ) Γ ( c - b ) F ( a , b a + b - c + 1 ; 1 - z ) - ( 1 - z ) c - a - b Γ ( a ) Γ ( b ) F ( c - a , c - b c - a - b + 1 ; 1 - z ) , | ph z | < π , | ph ( 1 - z ) | < π .
15.8.5 sin ( π ( c - a - b ) ) π F ( a , b c ; z ) = z - a Γ ( c - a ) Γ ( c - b ) F ( a , a - c + 1 a + b - c + 1 ; 1 - 1 z ) - ( 1 - z ) c - a - b z a - c Γ ( a ) Γ ( b ) F ( c - a , 1 - a c - a - b + 1 ; 1 - 1 z ) , | ph z | < π , | ph ( 1 - z ) | < π .
15.8.10 F ( a , b a + b + m ; z ) = 1 Γ ( a + m ) Γ ( b + m ) k = 0 m - 1 ( a ) k ( b ) k ( m - k - 1 ) ! k ! ( z - 1 ) k - ( z - 1 ) m Γ ( a ) Γ ( b ) k = 0 ( a + m ) k ( b + m ) k k ! ( k + m ) ! ( 1 - z ) k ( ln ( 1 - z ) - ψ ( k + 1 ) - ψ ( k + m + 1 ) + ψ ( a + k + m ) + ψ ( b + k + m ) ) , | z - 1 | < 1 , | ph ( 1 - z ) | < π ,
13: 15.2 Definitions and Analytical Properties
15.2.2 F ( a , b ; c ; z ) = s = 0 ( a ) s ( b ) s Γ ( c + s ) s ! z s , | z | < 1 ,
15.2.3 F ( a , b c ; x + i 0 ) - F ( a , b c ; x - i 0 ) = 2 π i Γ ( a ) Γ ( b ) ( x - 1 ) c - a - b F ( c - a , c - b c - a - b + 1 ; 1 - x ) , x > 1 .
15.2.3_5 lim c - n F ( a , b ; c ; z ) Γ ( 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 , .
14: 10.43 Integrals
10.43.26 0 K μ ( a t ) J ν ( b t ) t λ d t = b ν Γ ( 1 2 ν - 1 2 λ + 1 2 μ + 1 2 ) Γ ( 1 2 ν - 1 2 λ - 1 2 μ + 1 2 ) 2 λ + 1 a ν - λ + 1 F ( ν - λ + μ + 1 2 , ν - λ - μ + 1 2 ; ν + 1 ; - b 2 a 2 ) , ( ν + 1 - λ ) > | μ | , a > | b | .
15: 14.19 Toroidal (or Ring) Functions
14.19.2 P ν - 1 2 μ ( cosh ξ ) = Γ ( 1 2 - μ ) π 1 / 2 ( 1 - e - 2 ξ ) μ e ( ν + ( 1 / 2 ) ) ξ F ( 1 2 - μ , 1 2 + ν - μ ; 1 - 2 μ ; 1 - e - 2 ξ ) , μ 1 2 , 3 2 , 5 2 , .
16: 15.9 Relations to Other Functions
15.9.16 F ( a , b 2 b ; z ) = π Γ ( b ) z - b + ( 1 / 2 ) ( 1 - z ) ( b - a - ( 1 / 2 ) ) / 2 P a - b - ( 1 / 2 ) - b + ( 1 / 2 ) ( 2 - z 2 1 - z ) , b 0 , - 1 , - 2 , , | ph ( 1 - z ) | < π and | 1 - z | < 1 .
15.9.22 F ( a , b 1 2 ; z ) = 2 a + b - ( 3 / 2 ) π Γ ( a + 1 2 ) Γ ( b + 1 2 ) ( z - 1 ) ( - a - b + ( 1 / 2 ) ) / 2 ( e ± π i ( a + b - ( 1 / 2 ) ) P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ( - z ) + P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ( z ) ) , a , b - 1 2 , - 3 2 , - 5 2 , , 0 < | ph z | < π ,
15.9.23 F ( a , b 3 2 ; z ) = 2 a + b - ( 5 / 2 ) π z Γ ( a - 1 2 ) Γ ( b - 1 2 ) ( z - 1 ) ( - a - b + ( 3 / 2 ) ) / 2 ( e ± π i ( a + b - ( 3 / 2 ) ) P a - b - ( 1 / 2 ) - a - b + ( 3 / 2 ) ( - z ) - P a - b - ( 1 / 2 ) - a - b + ( 3 / 2 ) ( z ) ) , a , b 1 2 , - 1 2 , - 3 2 , , 0 < | ph z | < π ,
17: 15.1 Special Notation
18: 3.10 Continued Fractions
For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … However, this may be unstable; also overflow and underflow may occur when evaluating A n and B n (making it necessary to re-scale from time to time). … In contrast to the preceding algorithms in this subsection no scaling problems arise and no a priori information is needed. In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). … Again, no scaling problems arise and no a priori information is needed. …
19: 13.4 Integral Representations
13.4.15 U ( a , b , z ) Γ ( c ) Γ ( c - b + 1 ) = z 1 - c 2 π i - ( 0 + ) e z t t - c F 1 2 ( a , c ; a + c - b + 1 ; 1 - 1 t ) d t , | ph z | < 1 2 π .
20: 13.10 Integrals
13.10.3 0 e - z t t b - 1 M ( a , c , k t ) d t = Γ ( b ) z - b F 1 2 ( a , b ; c ; k / z ) , b > 0 , z > max ( k , 0 ) ,
13.10.7 0 e - z t t b - 1 U ( a , c , t ) d t = Γ ( b ) Γ ( b - c + 1 ) z - b F 1 2 ( a , b ; a + b - c + 1 ; 1 - 1 z ) , b > max ( c - 1 , 0 ) , z > 0 .