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1: 8.17 Incomplete Beta Functions
Throughout §§8.17 and 8.18 we assume that a > 0 , b > 0 , and 0 x 1 . …
8.17.4 I x ( a , b ) = 1 - I 1 - x ( b , a ) .
With a > 0 , b > 0 , and 0 < x < 1 , …
8.17.13 ( a + b ) I x ( a , b ) = a I x ( a + 1 , b ) + b I x ( a , b + 1 ) ,
8.17.14 ( a + b x ) I x ( a , b ) = x b I x ( a - 1 , b + 1 ) + a I x ( a + 1 , b ) ,
2: 8.10 Inequalities
8.10.1 x 1 - a e x Γ ( a , x ) 1 , x > 0 , 0 < a 1 ,
8.10.2 γ ( a , x ) x a - 1 a ( 1 - e - x ) , x > 0 , 0 < a 1 .
8.10.3 x 1 - a e x Γ ( a , x ) = 1 + a - 1 x ϑ ,
8.10.4 0 < ϑ 1 , x > 0 , a 2 .
8.10.11 ( 1 - e - α a x ) a P ( a , x ) ( 1 - e - β a x ) a , x 0 , a > 0 ,
3: 8.8 Recurrence Relations and Derivatives
8.8.1 γ ( a + 1 , z ) = a γ ( a , z ) - z a e - z ,
8.8.2 Γ ( a + 1 , z ) = a Γ ( a , z ) + z a e - z .
8.8.3 w ( a + 2 , z ) - ( a + 1 + z ) w ( a + 1 , z ) + a z w ( a , z ) = 0 .
8.8.5 P ( a + 1 , z ) = P ( a , z ) - z a e - z Γ ( a + 1 ) ,
8.8.6 Q ( a + 1 , z ) = Q ( a , z ) + z a e - z Γ ( a + 1 ) .
4: Bibliography U
  • F. Ursell (1972) Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.
  • F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.
  • F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
  • 5: 8.5 Confluent Hypergeometric Representations
    8.5.1 γ ( a , z ) = a - 1 z a e - z M ( 1 , 1 + a , z ) = a - 1 z a M ( a , 1 + a , - z ) , a 0 , - 1 , - 2 , .
    8.5.2 γ * ( a , z ) = e - z M ( 1 , 1 + a , z ) = M ( a , 1 + a , - z ) .
    8.5.3 Γ ( a , z ) = e - z U ( 1 - a , 1 - a , z ) = z a e - z U ( 1 , 1 + a , z ) .
    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 ) .
    6: 8.21 Generalized Sine and Cosine Integrals
    8.21.20 si ( a , z ) = f ( a , z ) cos z + g ( a , z ) sin z ,
    8.21.21 ci ( a , z ) = - f ( a , z ) sin z + g ( a , z ) cos z .
    8.21.22 f ( a , z ) = 0 sin t ( t + z ) 1 - a d t ,
    8.21.23 g ( a , z ) = 0 cos t ( t + z ) 1 - a d t .
    8.21.24 f ( a , z ) = z a 2 0 ( ( 1 + i t ) a - 1 + ( 1 - i t ) a - 1 ) e - z t d t ,
    7: 31.3 Basic Solutions
    31.3.2 a γ c 1 - q c 0 = 0 ,
    8: 8.2 Definitions and Basic Properties
    8.2.2 Γ ( a , z ) = z t a - 1 e - t d t ,
    8.2.3 γ ( a , z ) + Γ ( a , z ) = Γ ( a ) , a 0 , - 1 , - 2 , .
    8.2.12 d 2 w d z 2 + ( 1 + 1 - a z ) d w d z = 0 .
    8.2.13 d 2 w d z 2 - ( 1 + 1 - a z ) d w d z + 1 - a z 2 w = 0 .
    9: 8.7 Series Expansions
    8.7.1 γ * ( a , z ) = e - z k = 0 z k Γ ( a + k + 1 ) = 1 Γ ( a ) k = 0 ( - z ) k k ! ( a + k ) .
    8.7.2 γ ( a , x + y ) - γ ( a , x ) = Γ ( a , x ) - Γ ( a , x + y ) = e - x x a - 1 n = 0 ( 1 - a ) n ( - x ) n ( 1 - e - y e n ( y ) ) , | y | < | x | .
    8.7.3 Γ ( a , z ) = Γ ( a ) - k = 0 ( - 1 ) k z a + k k ! ( a + k ) = Γ ( a ) ( 1 - z a e - z k = 0 z k Γ ( a + k + 1 ) ) , a 0 , - 1 , - 2 , .
    8.7.4 γ ( a , x ) = Γ ( a ) x 1 2 a e - x n = 0 e n ( - 1 ) x 1 2 n I n + a ( 2 x 1 / 2 ) , a 0 , - 1 , - 2 , .
    8.7.6 Γ ( a , x ) = x a e - x n = 0 L n ( a ) ( x ) n + 1 , x > 0 .
    10: 8.14 Integrals
    8.14.1 0 e - a x γ ( b , x ) Γ ( b ) d x = ( 1 + a ) - b a , a > 0 , b > - 1 ,
    8.14.2 0 e - a x Γ ( b , x ) d x = Γ ( b ) 1 - ( 1 + a ) - b a , a > - 1 , b > - 1 .
    8.14.3 0 x a - 1 γ ( b , x ) d x = - Γ ( a + b ) a , a < 0 , ( a + b ) > 0 ,
    8.14.4 0 x a - 1 Γ ( b , x ) d x = Γ ( a + b ) a , a > 0 , ( a + b ) > 0 ,
    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 ,