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1: 16.15 Integral Representations and Integrals
16.15.1 F 1 ( α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( α ) Γ ( γ - α ) 0 1 u α - 1 ( 1 - u ) γ - α - 1 ( 1 - u x ) β ( 1 - u y ) β d u , α > 0 , ( γ - α ) > 0 ,
16.15.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = Γ ( γ ) Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β ) Γ ( γ - β ) 0 1 0 1 u β - 1 v β - 1 ( 1 - u ) γ - β - 1 ( 1 - v ) γ - β - 1 ( 1 - u x - v y ) α d u d v , γ > β > 0 , γ > β > 0 ,
16.15.3 F 3 ( α , α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β - β ) Δ u β - 1 v β - 1 ( 1 - u - v ) γ - β - β - 1 ( 1 - u x ) α ( 1 - v y ) α d u d v , ( γ - β - β ) > 0 , β > 0 , β > 0 ,
16.15.4 F 4 ( α , β ; γ , γ ; x ( 1 - y ) , y ( 1 - x ) ) = Γ ( γ ) Γ ( γ ) Γ ( α ) Γ ( β ) Γ ( γ - α ) Γ ( γ - β ) 0 1 0 1 u α - 1 v β - 1 ( 1 - u ) γ - α - 1 ( 1 - v ) γ - β - 1 ( 1 - u x ) γ + γ - α - 1 ( 1 - v y ) γ + γ - β - 1 ( 1 - u x - v y ) α + β - γ - γ + 1 d u d v , γ > α > 0 , γ > β > 0 .
2: 30.1 Special Notation
x

real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, - 1 < x < 1 .

γ 2

real parameter (positive, zero, or negative).

Flammer (1957) and Abramowitz and Stegun (1964) use λ m n ( γ ) for λ n m ( γ 2 ) + γ 2 , R m n ( j ) ( γ , z ) for S n m ( j ) ( z , γ ) , and
S m n ( 1 ) ( γ , x ) = d m n ( γ ) Ps n m ( x , γ 2 ) ,
S m n ( 2 ) ( γ , x ) = d m n ( γ ) Qs n m ( x , γ 2 ) ,
where d m n ( γ ) is a normalization constant determined by …
3: 30.5 Functions of the Second Kind
Other solutions of (30.2.1) with μ = m , λ = λ n m ( γ 2 ) , and z = x are
30.5.1 Qs n m ( x , γ 2 ) , n = m , m + 1 , m + 2 , .
30.5.2 Qs n m ( - x , γ 2 ) = ( - 1 ) n - m + 1 Qs n m ( x , γ 2 ) ,
30.5.4 𝒲 { Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) } = ( n + m ) ! ( 1 - x 2 ) ( n - m ) ! A n m ( γ 2 ) A n - m ( γ 2 ) ( 0 ) ,
with A n ± m ( γ 2 ) as in (30.11.4). …
4: 5.22 Tables
Abramowitz and Stegun (1964, Chapter 6) tabulates Γ ( x ) , ln Γ ( x ) , ψ ( x ) , and ψ ( x ) for x = 1 ( .005 ) 2 to 10D; ψ ′′ ( x ) and ψ ( 3 ) ( x ) for x = 1 ( .01 ) 2 to 10D; Γ ( n ) , 1 / Γ ( n ) , Γ ( n + 1 2 ) , ψ ( n ) , log 10 Γ ( n ) , log 10 Γ ( n + 1 3 ) , log 10 Γ ( n + 1 2 ) , and log 10 Γ ( n + 2 3 ) for n = 1 ( 1 ) 101 to 8–11S; Γ ( n + 1 ) for n = 100 ( 100 ) 1000 to 20S. Zhang and Jin (1996, pp. 67–69 and 72) tabulates Γ ( x ) , 1 / Γ ( x ) , Γ ( - x ) , ln Γ ( x ) , ψ ( x ) , ψ ( - x ) , ψ ( x ) , and ψ ( - x ) for x = 0 ( .1 ) 5 to 8D or 8S; Γ ( n + 1 ) for n = 0 ( 1 ) 100 ( 10 ) 250 ( 50 ) 500 ( 100 ) 3000 to 51S. … Abramov (1960) tabulates ln Γ ( x + i y ) for x = 1 ( .01 ) 2 , y = 0 ( .01 ) 4 to 6D. Abramowitz and Stegun (1964, Chapter 6) tabulates ln Γ ( x + i y ) for x = 1 ( .1 ) 2 , y = 0 ( .1 ) 10 to 12D. …Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of Γ ( x + i y ) , ln Γ ( x + i y ) , and ψ ( x + i y ) for x = 0.5 , 1 , 5 , 10 , y = 0 ( .5 ) 10 to 8S.
5: 30.6 Functions of Complex Argument
The solutions
Ps n m ( z , γ 2 ) ,
Qs n m ( z , γ 2 ) ,
30.6.3 𝒲 { Ps n m ( z , γ 2 ) , Qs n m ( z , γ 2 ) } = ( - 1 ) m ( n + m ) ! ( 1 - z 2 ) ( n - m ) ! A n m ( γ 2 ) A n - m ( γ 2 ) ,
with A n ± m ( γ 2 ) as in (30.11.4). …
6: 16.16 Transformations of Variables
16.16.3 F 2 ( α ; β , β ; γ , α ; x , y ) = ( 1 - y ) - β F 1 ( β ; α - β , β ; γ ; x , x 1 - y ) ,
16.16.4 F 3 ( α , γ - α ; β , β ; γ ; x , y ) = ( 1 - y ) - β F 1 ( α ; β , β ; γ ; x , y y - 1 ) ,
16.16.7 F 4 ( α , β ; γ , γ ; x ( 1 - y ) , y ( 1 - x ) ) = k = 0 ( α ) k ( β ) k ( α + β - γ - γ + 1 ) k ( γ ) k ( γ ) k k ! x k y k F 1 2 ( α + k , β + k γ + k ; x ) F 1 2 ( α + k , β + k γ + k ; y ) ;
16.16.9 F 2 ( α ; β , β ; γ , γ ; x , y ) = ( 1 - x ) - α F 2 ( α ; γ - β , β ; γ , γ ; x x - 1 , y 1 - x ) ,
16.16.10 F 4 ( α , β ; γ , γ ; x , y ) = Γ ( γ ) Γ ( β - α ) Γ ( γ - α ) Γ ( β ) ( - y ) - α F 4 ( α , α - γ + 1 ; γ , α - β + 1 ; x y , 1 y ) + Γ ( γ ) Γ ( α - β ) Γ ( γ - β ) Γ ( α ) ( - y ) - β F 4 ( β , β - γ + 1 ; γ , β - α + 1 ; x y , 1 y ) .
7: 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 .
If w ( a , z ) = γ ( a , z ) or Γ ( a , z ) , then …
8.8.8 γ ( a , z ) = Γ ( a ) Γ ( a - n ) γ ( a - n , z ) - z a - 1 e - z k = 0 n - 1 Γ ( a ) Γ ( a - k ) z - k ,
8.8.10 Γ ( a , z ) = Γ ( a ) Γ ( a - n ) Γ ( a - n , z ) + z a - 1 e - z k = 0 n - 1 Γ ( a ) Γ ( a - k ) z - k ,
8: 8.2 Definitions and Basic Properties
The general values of the incomplete gamma functions γ ( a , z ) and Γ ( a , z ) are defined by …
8.2.3 γ ( a , z ) + Γ ( a , z ) = Γ ( a ) , a 0 , - 1 , - 2 , .
In this subsection the functions γ and Γ have their general values. The function γ * ( a , z ) is entire in z and a . … If w = γ ( a , z ) or Γ ( a , z ) , then …
9: 5.8 Infinite Products
5.8.1 Γ ( z ) = lim k k ! k z z ( z + 1 ) ( z + k ) , z 0 , - 1 , - 2 , ,
5.8.2 1 Γ ( z ) = z e γ z k = 1 ( 1 + z k ) e - z / k ,
5.8.3 | Γ ( x ) Γ ( x + i y ) | 2 = k = 0 ( 1 + y 2 ( x + k ) 2 ) , x 0 , - 1 , .
5.8.5 k = 0 ( a 1 + k ) ( a 2 + k ) ( a m + k ) ( b 1 + k ) ( b 2 + k ) ( b m + k ) = Γ ( b 1 ) Γ ( b 2 ) Γ ( b m ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a m ) ,
10: 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 , .
For an expansion for γ ( a , i x ) in series of Bessel functions J n ( x ) that converges rapidly when a > 0 and x ( 0 ) is small or moderate in magnitude see Barakat (1961).