The Logarithmic Function
Consider any nonzero complex number. We would like to solve for the equation
If with then and can be written
as follows
Then equation () becomes
Thus, we have
where Since is the same as it follows that equation
() is satisfied if and only if has one of the values
Therefore, the (multiple-valued) logarithmic function of a nonzero complex variable
is defined by the formula
Example 1: Calculate for
Solution: If then and Hence
with
The principal value of is the value obtained from equation () when
and is denoted by Thus
The function is well defined and single-valued when and that
This is reduced to the usual logarithm in calculus when is a positive real number.
Example 2: Calculate and
Solution: From expression ()
and
Notice that and
Expression () is also equivalent to the following:
Some basic properties of the function are the following:
-
-
- There may hold
Branches of logarithms
From definition () let (), so we can write
Now, let be any real number. If we restrict the value of so that , then the function
with components
is a
single-value and continuous function in the stated domain.
A branch of a multiple-valued function is any single-valued function that is
analytic in some domain at each point of which the value is one of the values of The
requirement of analyticity, of course, prevents from taking on a random selection of the values
of
Observe that for each fixed the single-valued function () is a branch of the
multiple-valued function (). The function
is called the principal branch.
A branch cut is a portion of a line or curve that is introduced in order to define a branch
of a multiple-valued function Points on the branch cut for are singular points
of and any point that is common to all branch cuts of is called a branch point.
The origin and the ray make up the branch cut for the branch () of the
logarithmic function. The branch cut for the principal branch () consists of the origin
and the ray The origin is evidently a branch point for branches of the multiple-valued
logarithmic function.
We can visualize the multiple-valued nature of by
using Riemann
surfaces.
The following interactive images show the real and imaginary
components of
Imaginary component of : Each branch is
identified with a different color.
Special care must be taken in using branches of the logarithmic function, especially since expected
identities involving logarithms do not always carry over from calculus.