Laurent Series
The Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. It is named after by Pierre Alphonse Laurent in 1843.
Contents
Creating the Laurent Series
Let such that
for . In order to create the Laurent Series, we need to prove four main theorems.
The First Theorem
Suppose that is holomorphic for . Then
The proof is actually not terrible. Consider some complex valued function such that
where this exists, and . Then it follows that is holomorphic for . Hence,
Now, notice that for all . This means we get
Clearly we have and which means that and by a theorem. This implies the result
The Second Theorem
Suppose that the first theorem holds and let
for all with as long as and . We claim that
Note that from before, we have
for . Now, there must exist an such that for all by a theorem, so we get the inequality
for each once again. But notice: the right hand side is independent of ! So, because we already assumed that we see that the sum
converges. Hence, the sum converges uniformly on the contour . This means we get But we are not finished just yet! Also notice that for each . This gives
and rearrangement gives the desired
The Third Theorem (The Laurent Series Defined)
We define the term "a ring in a wider sense" to mean the following: the set of points between two concentric circles, a disk without its center, and the exterior of a circle not including . Let be a ring in the wider sense, with center and let be holomorphic on . Then there are numbers for each such that for all we have the series
Choose any such that the circle and let
for every . Choose some , and let ,, and denote the same stuff we used earlier. Then it follows that
for of course. The theorem then follows immediately
The Fourth Theorem (Uniqueness of the Laurent Series)
Is such a series unique, however? It certainly should be, or else I would not be writing this all down here. Let be a ring in the wider sense with center of a circle , and let
for every . Then we must prove that
holds for every integer . We have To swap the integrand and the sum, we must note that
Now we can have the following deduction. Crisis avoided! Sort of. We assumed that and converge uniformly on . We can fix this! Let the radius of (remember, is a circle) be . Then there exists some such that with . It follows that the series
holds when so we get
converges. We have to prove uniform convergence though, so back to work. Notice that this result above implies that there is some such that for . Hence we get
for and where . Like before, the RHS is independent of so we get
converges, which implies that
converges uniformly. Repeat the same process for with appropriate tweaks. Now, we finish by noting that
when or we have
when . The result follows from
This shows that the series is unique, done