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Euler integral

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21: 8.8 Recurrence Relations and Derivatives
22: 5.9 Integral Representations
5.9.3 c - z Γ ( z ) = - | t | 2 z - 1 e - c t 2 d t , c > 0 , z > 0 ,
5.9.4 Γ ( z ) = 1 t z - 1 e - t d t + k = 0 ( - 1 ) k ( z + k ) k ! , z 0 , - 1 , - 2 , .
§5.9(ii) Psi Function, Euler’s Constant, and Derivatives
5.9.16 ψ ( z ) + γ = 0 e - t - e - z t 1 - e - t d t = 0 1 1 - t z - 1 1 - t d t .
23: 35.7 Gaussian Hypergeometric Function of Matrix Argument
24: 6.10 Other Series Expansions
6.10.6 Ei ( x ) = γ + ln | x | + n = 0 ( - 1 ) n ( x - a n ) ( i n ( 1 ) ( 1 2 x ) ) 2 , x 0 ,
25: 8.19 Generalized Exponential Integral
8.19.1 E p ( z ) = z p - 1 Γ ( 1 - p , z ) .
Most properties of E p ( z ) follow straightforwardly from those of Γ ( a , z ) . …
8.19.4 E p ( z ) = z p - 1 e - z Γ ( p ) 0 t p - 1 e - z t 1 + t d t , | ph z | < 1 2 π , p > 0 .
8.19.10 E p ( z ) = z p - 1 Γ ( 1 - p ) - k = 0 ( - z ) k k ! ( 1 - p + k ) ,
8.19.11 E p ( z ) = Γ ( 1 - p ) ( z p - 1 - e - z k = 0 z k Γ ( 2 - p + k ) ) ,
26: 6.4 Analytic Continuation
27: 25.12 Polylogarithms
25.12.14 F s ( x ) = 1 Γ ( s + 1 ) 0 t s e t - x + 1 d t , s > - 1 ,
25.12.15 G s ( x ) = 1 Γ ( s + 1 ) 0 t s e t - x - 1 d t , s > - 1 , x < 0 ; or s > 0 , x 0 ,
28: 29.18 Mathematical Applications
0 γ 4 K ,
β = K + i β , 0 β 2 K , 0 γ 4 K ,
29: 13.6 Relations to Other Functions
13.6.6 U ( a , a , z ) = z 1 - a U ( 1 , 2 - a , z ) = z 1 - a e z E a ( z ) = e z Γ ( 1 - a , z ) .
30: 13.4 Integral Representations
13.4.1 M ( a , b , z ) = 1 Γ ( a ) Γ ( b - a ) 0 1 e z t t a - 1 ( 1 - t ) b - a - 1 d t , b > a > 0 ,
13.4.3 M ( a , b , - z ) = z 1 2 - 1 2 b Γ ( a ) 0 e - t t a - 1 2 b - 1 2 J b - 1 ( 2 z t ) d t , a > 0 .
13.4.4 U ( a , b , z ) = 1 Γ ( a ) 0 e - z t t a - 1 ( 1 + t ) b - a - 1 d t , a > 0 , | ph z | < 1 2 π ,
13.4.9 M ( a , b , z ) = Γ ( 1 + a - b ) 2 π i Γ ( a ) 0 ( 1 + ) e z t t a - 1 ( t - 1 ) b - a - 1 d t , b - a 1 , 2 , 3 , , a > 0 .
13.4.16 M ( a , b , - z ) = 1 2 π i Γ ( a ) - i i Γ ( a + t ) Γ ( - t ) Γ ( b + t ) z t d t , | ph z | < 1 2 π ,