The logarithm function
Ln(z)
is an inverse of the exponential function.
Because the complex exponential function
is
a periodic function
Proof
an equation
f(z)=
has
infinitely many solutions in a case
of complex variable, and the complex
logarithm function Ln(z)
is a multi-valued function.
Ln(z)
= ln(|z|)
+ i[arg(z)
+
],
k
= 0, ±1, ±2, ...
If
k = 0
we have a principal logarithm ln(z)
or principal branch of the logarithm:
ln(z)
= ln(|z|)
+ i[arg(z)]
Example
1
Find the values
of
1)
ln(-5)
2)
Ln(-5),
where ln(-5)
is a principal logarithm.
1)
ln(-5)
= ln(|-5|)
+ i[arg(-5)]
= ln(5)
+
2)
Ln(-5)
= ln(|-5|)
+ i[arg
(-5)] +
 =
= ln(5)
+ (2k
+ 1),
where k
– any integer.
|
The logarithm function
Ln(z)
has a singularity at z
=
0. If the non-zero complex number
z
is expressed in polar coordinates as
with r
>
0 and 
,
then
Ln(z)
= ln(r)
+ i(
+),
where k
is any integer and ln(r)
is the usual natural logarithm of a
real number.
A fact that the complex
logarithm function is the multi-valued
function explains Paradox
of Bernoulli and Leibniz
The paradox of Bernoulli
and Leibniz is not an “exclusive” case
for the complex logarithm function.
Let us look at Example 2.
Let us consider several properties of
the logarithm function familiar from
the real logarithm. It is necessary
to remember about “”
to make them always valid for the complex
extension.
k,
k1,
k2
are integers.
by
Tetyana Butler
+
i); w
= (-1 +
i
i]
+ [Ln(2)
+ 
i]
=
Ln(4) + i
)
= nLn(z)
)
= 
Ln(z)
