120
VIII – Cauchy Theory
L(z) = log |z| + i Arg(z) with − π < Arg(z) ≤ +π
(14.9’)
on C
∗ , and so z = exp [L(z)] for all z ∈ C
∗ . The L function is not holomorphic
on all of C
∗ ; it is discontinuous at every point of R
∗
− . To be precise, let us
consider a sequence of points z n ∈ C
∗ converging to some z ∈ C
∗ . If z ∈ C + ,
clearly Arg(z) = lim Arg(z n ), and as a result L(z) = lim L(z n ). But if z ∈ R − ,
then Arg(z) = +π by convention whereas the arguments of the z n are, for
large n, near +π or to −π according to the values of n; hence there are
integers k n ∈ {−1, 0} such that
Arg(z) = lim [Arg(z n ) + 2k n πi] .
In conclusion, in all cases, there exist k n such that such that
L(lim z n ) = lim [L(z n ) + 2k n πi] .
(14.10)
In such a case, L(z n ) may approach a limit. The same then holds for 2k n πi,
which is, therefore, constant for large n; as a result,
L(lim z n ) = lim L(z n ) mod2πi if lim L(z n ) exists.
(14.10’)
The functional equation of the logarithm can be generalized with some
precaution to the function L(s). With definition (9’) for the argument,
clearly
66
z = z 1 . . . z n =⇒ Arg(z) =
Arg(z p ) mod 2π ,
so that
L(z 1 . . . z n ) =
L(z p ) mod 2πi ,
(14.11)
and
L(z 1 . . . z n ) =
L(z p ) ⇐⇒ −π <
Arg(z p ) ≤ +π .
(14.12)
For real x > 0 and s = σ + it ∈ C, x
s = x
σ exp(it log x); Since the
argument of x ∈ R
∗
+ is zero, (12) shows that
L(x
s ) = L (x
σ ) + L [exp(it log x)]
and as the argument of exp(it log x) is equal to t log x mod 2π,
L(x
s ) = sL(x) mod2πi for x ∈ R
∗
+ , s ∈ C ;
(14.13)
the formula L(x
s ) = sL(x) holds if s ∈ R since calculations are then in R
∗
+ .
66 In what follows, write a = b mod 2π or mod 2πi to mean that a − b is a multiple
of 2π or of 2πi as the case may be. The traditional notation is the sign ≡, which
is unnecessary if it is followed by an indication such as “ mod 25 ”.