§ 3. Some Applications of Cauchy’s Method
121
We will also need the relation
L(1 + z) = z − z
2 /2 + z
3 /3 . . . for |z| < 1 ;
(14.14)
it is true for real z since L(1 + z) = log(1 + z) then, and so by analytic
extension also on the unit disc; besides, in effect (14) only deals with the Log
function.
Let us now specify what should be understood by Log Γ (s). The Γ function is holomorphic and never zero (infinite product expansion) on the simply
connected domain C + . Hence the equation exp [f (s)] = Γ (s) has holomorphic
solutions in C + , namely (§ 1, n
◦ 3, Corollary 2 of Theorem 3) some primitives
of Γ
(s)/Γ (s). Choose the function
Log Γ (s) =
s
1
Γ
(z)
Γ (z)
dz ,
(14.15)
where integration is along a path connecting 1 to s in C + , the simplest being
the line segment. With this definition,
Log Γ (s) = log Γ (s) for s ∈ R
∗
+
(14.16)
since then the real function Γ
(x)/Γ (x) is integrated over the interval (1, s).
It may be though that Log Γ (s) = L [Γ (s)]. This would be the case if
s ∈ C + =⇒ Γ (s) ∈ C + was known to hold, for then the right hand side,
consisting of two holomorphic functions on C + , would, like the left hand one,
also be so. The obviously exact formula on R
∗
+ , would then hold in all of C + .
But the assumption on which this argument is based does not seem to be
exact.
67 Since, by definition,
exp [Log Γ (s)] = Γ (s) = exp {L [Γ (s)]}
the relation
Log Γ (s) = L [Γ (s)] mod 2πi
(14.17)
is exact and for us, this suffices.
Exercise. Using the infinite product, show that
−Γ
(s)/Γ (s) = C + 1/s +
n≥1
1
s + n
−
1
n
(14.18)
if s is not an integer ≤ 0, then that
Log Γ (s) = −Cs − Log s +
[s/n − Log(1 + s/n)] for s ∈ C + .
(14.19)
67 Remmert 2 alludes discretely to it on p. 42 but, unfortunately, does not prove it,
and I do not see how to do so.