Isolated Singularities

Isolated Singularities

Isolated Singularities

If and , but for some , then is called an isolated singularity of .

e.g. has isolated singularities at for , and is not an isolated singularly.

Corollary

Suppose , and is an isolated singularity of . Then we can form the Laurent expansion: converging on any .

Proof Since is an isolated singularity, there is some such that . Then , so we can apply the Laurent expansion theorem.

Removable Singularity, Pole, Essential Singularity

Suppose has an isolated singularity , with Laurent expansion . Then the singularity is called

• removable singularity, if for all .
• pole, if for a nonempty finite set of .
• essential singularity, if for infinitely many .

The series is called the principal part of the Laurent expansion.

Characterise Removable Singularities

Characterization of Removable Singularities

Let , and suppose that has an isolated singularity . Then the following are equivalent.

1. has a removable singularity at .
2. has a holomorphic extension to .
3. exists.
4. is bounded on some .

Proof : If has a removable singularity at , then has a Laurent expansion with for all . That is, , which is a power series, converging normally on some disk to a holomorphic function . Clearly agrees with on . Then we define byThen is complex differentiable on since it agrees with in a neighborhood of any , and is -differentiable at since it agrees with on . Therefore , as required. : The holomorphic extension is continuous at , so . : If exists, then is bounded on a neighborhood of , thus bounded on some . This can be rigorously showed by - argument. : Suppose on , and is the Laurent series. Fix some , for any , by ML estimate, we haveAs it holds for all , note that , we can take to get . Therefore the singularity is removable.

Characterise Poles and Zeroes

Lemma

Let be a domain, , and . Suppose either or is a pole of . Then there is a unique and such that for all and .

Proof First consider existence. Suppose are coefficients of the Laurent series of centered at . Since , there is a smallest such that . i.e. So we can write . Notice that converges pointwise on for some , thus it converges normally to a holomorphic function by Abel’s theorem. Now we define byThen is complex differentiable on since it equals to . And is complex differentiable on since it agrees with on . Therefore , as required. Now consider uniqueness. Suppose with and , . Without loss of generality, assume . Then Since are holomorphic, thus continuity at implies that yielding a contradiction. Therefore , and for all . Thus on by continuity.

Characterization of Poles

Let , and is an isolated singularity of . Then is a pole iff .

Proof Suppose has a pole at . Write for some and with . Then Conversely, suppose . Then on some . That is . Therefore is bounded on , so has a removable singularity at . Let be the holomorphic extension of to . Then Hence . Since , by the lemma, we can write where with . Then for , we have As , is holomorphic on some , by Taylor expansion, we have , so This shows that is a pole of by definition.

Remark

Another way to think about poles is that extends to a holomorphic function from into the Riemann sphere .

Order of Pole and Zero

If has a pole at , then we can write for some and with . Such is called the order of the pole at . Similarly, if and with , then is called the order of the zero at if for some and with . We call zero of order a simple zero, pole of order a simple pole.

The order of a zero is telling you how quickly approaches as , and similarly the order of a pole tells you how quickly approaches as .

Corollary

If has a pole/zero of order at , then has a zero/pole of order .

Corollary

1. If has a zero of order at and has a zero of order at , then has a zero of order at .
2. Similarly, if has a pole of order at and has a pole of order at , then has a pole of order at .
3. If has a zero of order at and has a pole of order at , then
4. If , then has a pole of order at .
5. If , then has a removable singularity at , and if then (once we remove the singularity) it becomes a zero of order .

Meromorphic Function

A function is called meromorphic on if is holomorphic on except for some isolated singularities which are poles.

Characterise Essential Singularities

Casorati-Weierstrass Theorem

Let , and is an essential singularity of . Then for any , there is a sequence with and .

Proof Case . Since the singularity is not removable, is unbounded on every , so we may choose such that . Then and . Next consider and there exists a sequence with and for all . The result is immediate. Now if and there is no such sequence, then there exists , such that for all . Consider . has an essential singularity at , because adding only affects the constant term of the Laurent series. Notice that has an isolated singularity at . If it had a pole or removable singularity, then its reciprocal, , would have a pole or removable singularity at , yielding a contradiction. Therefore has an essential singularity at . Now apply case 1 to to get a sequence with and . This implies that .

Great Picard Theorem

If has an essential singularity at , then for any the set is either or for some .