Various representations for the Gamma function
The integral representation of the Gamma function
Γ(x) =
∫∞
0
ξx−^1 e−ξdξ x > 0 (12.101)
can be transformed to many alternative representations for use in special situations.
(i) The substitution ξ= ln(
1
y)or y=e
−ξ, converts the equation (12.101) to the form
Γ(x) =
∫ 1
0
(
ln
1
y
)x− 1
dy (12.102)
which is the form Euler originally studied.
(ii) The substitution ξ=zt converts equation (12.101) to the form
Γ(x) =
∫∞
0
zx−^1 tx−^1 e−zt z dt
and replacing xby x+ 1 there results
Γ(x+ 1) = zx+1
∫∞
0
txe−zt dt (12.103)
The above change of variables are just a sampling of forms for obtaining alter-
native integral representations of the Gamma function.
Euler’s constant γ, defined by the limit
γ= limn→∞
[
1
1
+^1
2
+^1
3
+^1
4
+···+^1
n− 1
+^1
n
−ln(n)
]
= 0. 5772156649 ··· (12.104)
occurs in many alternative representations of the Gamma function. When γ is
represented as a continued fraction (See chapter 4, page 337 ) one finds the list
notation given by
γ= [0 ; 1 , 1 , 2 , 1 , 2 , 1 , , 4 , 3 , 13 , 5 , 1 , 1 , 8 , 1 , 2 , 4 , 40 , 1 ,.. .]
The first nine convergents are
γ 1 =1
γ 2 =^1
2
= 0. 5
γ 3 =^3
5
= 0. 6
γ 4 =^4
7
= 0. 571428571
γ 5 =
11
19 = 0.^578947368
γ 6 =^15
26
= 0. 5769233077
γ 7 =^71
123
= 0. 577235772
γ 8 =
228
395 = 0.^577215190
γ 9 =^3035
5258
= 0. 577215671
(12.105)
where γ 9 is accurate to seven decimal places.
