4.4 The Principal Logarithm
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In the last section we computed the path integral of over the path parameterizing the unit circle and got a non-zero answer
from which we concluded does not have an antiderivative on . (If it did have an antiderivative on the path integral above would have to be zero!)
We can write
for all using our knowledge of the geometric series. This gives a power series representation of on . Let's define
which is the term-by-term antiderivative of our power series representing on . We can check (for example, using the ratio test) that its radius of convergence is . As a power series in its own right, it is holomorphic on and certainly for all . Thus does have an antiderivative on ! We now have two questions.
- Why do we have an antiderivative on but not on ?
- What is the antiderivative of on ?
We will have to wait a little to answer the first question. For the second question, recall first of all from real analysis that
so that our antiderivative has (at least on the real axis) something to do with the natural logarithm. In real analysis the natural logarithm can be defined as the functional inverse of the exponential function. Let's try to define an inverse of our exponential function .
Fix . We would like to solve the equation for . Writing gives
and we see that and that is an argument of . Thus the solutions are
for any .
In particular, every non-zero complex number has a (indeed many) natural logarithms: a complex number such that . The quantity was defined as the argument of lying in the interval . The function is not continuous, so we cannot define a continuous function on all of . However, if we exclude the negative real axis
where the output of jumps, we do get a continuous function.
Definition (Principal Logarithm)
The principal logarithm of is
and this defines a function .
Example
What is ? From we calculate
which is reasonable because .
Having defined a logarithm, we can define complex powers of complex numbers as follows.
Definition (Principal Value)
For with define
to be the principal value of .
Since is continuous on and is continuous on the principal logarithm is continuous on its domain. In fact, more is true.
Theorem
The principal logarithm is holomorphic on its domain and
for all .
Proof:
Fix . We need to evalue the limit
and will do so using the relationship between and . Put and . Since is continuous the limit implies that . We may therefore calculate
as desired.
We have established the following about the function .
- It does not have an antiderivative on .
- It does have an antiderivative on .
What difference between the domains and is responsible for the two statements above? We will uncover the full answer to this question next week.