4.1 Laurent Series
The Laurent series generalizes the Taylor series to holomorphic functions with isolated singularities. While Taylor series are valid within a disk centered at a point of holomorphy, Laurent series apply to annular regions surrounding a singularity, making them essential for studying functions near non-removable singularities (refer to Theorem 3.2.6).
We now introduce a fundamental result in complex analysis due to Weierstrass, which formalizes the conditions under which the limit of a sequence of holomorphic functions is itself holomorphic. This theorem not only guarantees the holomorphy of the limit function but also the uniform convergence of its derivatives (its statement was used in the proof of Theorem 3.3.5).
Theorem 4.1.1 (Weierstrass).
Let be a sequence of holomorphic functions on an open region that converges uniformly to on every compact subset of . Then is holomorphic on , and , the sequence uniformly converges to on all compact subsets of .
Proof.
By Morera’s Theorem (Theorem 3.2.4) and the uniform convergence of , the holomorphy of follows (refer to (4.1.2) and preceding explanations).
Following the same logic, by Corollary 3.2.1, and for all compact and open relatively compact in there exists a finite constant such that
Since is uniformly convergent, the limit on the right-hand side vanishes. Then,
and therefore uniformly converges on all compact subsets of .
The condition of uniform convergence on every compact subset can also be significantly loosened, by the fact demonstrated below:
Proposition 4.1.1.
Let be an open bounded region, and let be holomorphic on . Let be compact. If on , then on .
Proof.
By the converse statement of the Cauchy Criterion (Theorem 2.3.1), , such that ,
By the Maximum Modulus Principle (Theorem 3.4.1) on ,
It follows that on by Theorem 2.3.1.
Remark.
From the above result, the uniform convergence on every compact subset in Theorem 4.1.1 can therefore be loosened to the uniform convergence on every simple closed curve.
We will now study Laurent series. Let and be constants. A series in the form of
is a Laurent series at the point . The series can be separated into a power series with non-negative exponents,
and a power series with negative exponents,
(4.1.1) is said to be convergent at if the two power series are both convergent. Let the convergence radius of (4.1.2) be
by the Cauchy–Hadamard Theorem (Theorem 2.3.4). It follows that is holomorphic on . Let . Then (4.1.3) becomes
This series converges when
Let . Then converges when
or when .
If , then is convergent on the annulus and divergent on . If , the series diverges possibly everywhere but on . Similar to power series with positive exponents, the convergence on the boundary varies. For example,
where , converges (absolutely) on , whereas
diverges on all of , while
converges (conditionally) on all of and diverges at . If , then the series is divergent on all of . The region is known as the annulus of convergence. in (4.1.1) is holomorphic over this annulus. The series is known as the holomorphic part of , and is known as the principal part of the Laurent series. The properties of the convergence disk in Abel’s Theorem (Theorem 2.3.3) can be generalized to Laurent series. In other words, is absolutely convergent on the annulus and is uniformly convergent on every compact subset of it.
Theorem 4.1.2.
Let for some . Let be holomorphic on . Then has the unique Laurent expansion
for any simple closed curve enclosing . Moreover, the series converges absolutely on and uniformly on all compact subsets of .
Proof.
By the openness of , there exist two circles with radius and with radius centered at such that encloses and encloses both without intersection. Let and let be arbitrary. By the Cauchy–Goursat Formula (Theorem 3.1.8),
For all (or ), and therefore, . It follows that
is uniformly convergent with respect to . Similarly, for all ,
and it follows that
is uniformly convergent with respect to . By the boundedness of on and from holomorphy on a compact set, the uniform convergence from the Weierstrass -Test (Theorem 2.3.2), gives that
By the Cauchy–Goursat Theorem (Theorem 3.1.7), for a given ,
In other words, the integrals in (4.1.7) are the same as on . Hence, we obtain the absolutely convergent expansion
which converges uniformly on compact sets of . The constants are also unique in the expansion. For the sake of contradiction, assume there exists another set of constants such that
where and the series is uniformly convergent on . Let be arbitrary. By Cauchy–Goursat (Theorem 3.2.1),
Multiplying (4.1.8) by and from integrating over , we get that
implying that
which is a contradiction, implying uniqueness.
Remark.
Unlike Taylor series, Laurent series are not necessarily unique up to the point of expansion. Depending on the chosen annulus, the expansion may differ.