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11: 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 ) ,
12: 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).
13: 30.8 Expansions in Series of Ferrers Functions
Then the set of coefficients a n , k m ( γ 2 ) , k = R , R + 1 , R + 2 , is the solution of the difference equation …
30.8.7 k 2 a n , k m ( γ 2 ) a n , k 1 m ( γ 2 ) = γ 2 16 + O ( 1 k ) ,
30.8.8 λ n m ( γ 2 ) B k A k a n , k m ( γ 2 ) a n , k 1 m ( γ 2 ) = 1 + O ( 1 k 4 ) .
The coefficients a n , k m ( γ 2 ) satisfy (30.8.4) for all k when we set a n , k m ( γ 2 ) = 0 for k < N . …
30.8.10 A N 1 a n , N 2 m ( γ 2 ) + ( B N 1 λ n m ( γ 2 ) ) a n , N 1 m ( γ 2 ) + C a n , N m ( γ 2 ) = 0 ,
14: 5.3 Graphics
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Figure 5.3.1: Γ ( x ) and 1 / Γ ( x ) . x 0 = 1.46 , Γ ( x 0 ) = 0.88 ; see §5.4(iii). Magnify
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Figure 5.3.2: ln Γ ( x ) . … Magnify
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Figure 5.3.4: | Γ ( x + i y ) | . Magnify 3D Help
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Figure 5.3.5: 1 / | Γ ( x + i y ) | . Magnify 3D Help
15: 16.13 Appell Functions
16.13.1 F 1 ( α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
16.13.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m ( β ) n ( γ ) m ( γ ) n m ! n ! x m y n , | x | + | y | < 1 ,
16.13.3 F 3 ( α , α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m ( α ) n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
16.13.4 F 4 ( α , β ; γ , γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m + n ( γ ) m ( γ ) n m ! n ! x m y n , | x | + | y | < 1 .
Here and elsewhere it is assumed that neither of the bottom parameters γ and γ is a nonpositive integer. …
16: 8.1 Special Notation
x real variable.
Γ ( z ) gamma function (§5.2(i)).
ψ ( z ) Γ ( z ) / Γ ( z ) .
The functions treated in this chapter are the incomplete gamma functions γ ( a , z ) , Γ ( a , z ) , γ ( a , z ) , P ( a , z ) , and Q ( a , z ) ; the incomplete beta functions B x ( a , b ) and I x ( a , b ) ; the generalized exponential integral E p ( z ) ; the generalized sine and cosine integrals si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) . Alternative notations include: Prym’s functions P z ( a ) = γ ( a , z ) , Q z ( a ) = Γ ( a , z ) , Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); ( a , z ) ! = γ ( a + 1 , z ) , [ a , z ] ! = Γ ( a + 1 , z ) , Dingle (1973); B ( a , b , x ) = B x ( a , b ) , I ( a , b , x ) = I x ( a , b ) , Magnus et al. (1966); Si ( a , x ) Si ( 1 a , x ) , Ci ( a , x ) Ci ( 1 a , x ) , Luke (1975).
17: 5.6 Inequalities
5.6.2 1 Γ ( x ) + 1 Γ ( 1 / x ) 2 ,
5.6.3 1 ( Γ ( x ) ) 2 + 1 ( Γ ( 1 / x ) ) 2 2 ,
5.6.4 x 1 s < Γ ( x + 1 ) Γ ( x + s ) < ( x + 1 ) 1 s , 0 < s < 1 .
5.6.6 | Γ ( x + i y ) | | Γ ( x ) | ,
5.6.8 | Γ ( z + a ) Γ ( z + b ) | 1 | z | b a .
18: 8.3 Graphics
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Figure 8.3.1: Γ ( a , x ) , a = 0. … Magnify
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Figure 8.3.2: γ ( a , x ) , a = 0. … Magnify
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Figure 8.3.3: γ ( a , x ) , a = 1, 2, 2. … Magnify
Some monotonicity properties of γ ( a , x ) and Γ ( a , x ) in the four quadrants of the ( a , x )-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). …
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Figure 8.3.8: Γ ( 0.25 , x + i y ) , 3 x 3 , 3 y 3 . …When x = y = 0 , Γ ( 0.25 , 0 ) = Γ ( 0.25 ) = 3.625 . Magnify 3D Help
19: 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 ,
20: 30.16 Methods of Computation
For m = 2 , n = 4 , γ 2 = 10 , … If | γ 2 | is large, then we can use the asymptotic expansions referred to in §30.9 to approximate 𝖯𝗌 n m ( x , γ 2 ) . … If λ n m ( γ 2 ) is known, then 𝖯𝗌 n m ( x , γ 2 ) can be found by summing (30.8.1). The coefficients a n , r m ( γ 2 ) are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). … The coefficients a n , k m ( γ 2 ) calculated in §30.16(ii) can be used to compute S n m ( j ) ( z , γ ) , j = 1 , 2 , 3 , 4 from (30.11.3) as well as the connection coefficients K n m ( γ ) from (30.11.10) and (30.11.11). …