Series and residues · Definition

Isolated singularity

complex analysissingularities

Before you start you need to know

It names what happens exactly at a point where a function stops being holomorphic, and it is the entry point to residue calculus.

Definition

An isolated singularity of a function \(f\) is a point \(z_0\) where \(f\) is not holomorphic but is holomorphic on a punctured disc \(0<|z-z_0|<r\) around it.

According to the principal part of the Laurent series on \(0<|z-z_0|<r\), there are three cases. If it has no negative powers, the singularity is removable: \(f\) has a finite limit at \(z_0\) and it suffices to redefine \(f(z_0)\). If it has finitely many, with \(a_{-m}\neq0\) the most negative, it is a pole of order \(m\). If it has infinitely many, it is essential.

There are criteria that avoid the series. If \(|f|\) is bounded near \(z_0\), the singularity is removable (Riemann's theorem). If \(|f(z)|\to\infty\), it is a pole. If neither happens, it is essential, and then \(f\) takes in every neighbourhood all complex values except possibly one (great Picard theorem).

For a pole, \(f(z)=\frac{g(z)}{(z-z_0)^m}\) with \(g\) holomorphic and \(g(z_0)\neq0\). This gives a recipe: factor, simplify, and look at the exponent that remains. Beware of cancellations: \(\frac{z^2-1}{(z-1)^3}\) looks like a pole of order \(3\), but its order is \(2\).

Not every singularity is isolated: \(\frac1{\sin(1/z)}\) has poles accumulating at \(0\), and the branches of \(\sqrt z\) or \(\log z\) fail to be holomorphic along a whole cut. Typical mistake: calling something removable because it tends to \(0\) along the real axis, when it must be bounded along every path.

Think about it

Why does \(\frac{z^2-1}{(z-1)^3}\) have a pole of order \(2\), not \(3\), at \(z=1\)?

Because the numerator also vanishes at \(1\): \(\frac{z^2-1}{(z-1)^3}=\frac{z+1}{(z-1)^2}\).

If \(f\) has a pole at \(z_0\), what happens to \(|f(z)|\) as \(z\to z_0\)?

It tends to infinity, along every path.

Exercises 10

  1. What kind of singularity does \(\frac{\sin z}{z}\) have at \(z=0\)?
  2. What kind of singularity does \(e^{1/z}\) have at \(z=0\)?
  3. What kind of singularity does \(\frac1{z^2+1}\) have at \(z=i\)?
  4. The singularity of \(\frac{\cos z-1}{z^2}\) at \(z=0\) is removable. What value must \(f(0)\) take to remove it?
  5. What is the order of the pole of \(\frac1{z^2(z-1)}\) at \(z=0\)?
  6. If \(|f|\) is bounded on a punctured disc \(0<|z-z_0|<r\), what kind of isolated singularity does \(f\) have at \(z_0\)?
  7. Which of these functions has at \(z=0\) a singular point that is NOT an isolated singularity?
  8. Classify the singularity of \(\frac{z}{\sin z}\) at \(z=\pi\).
  9. Classify the singularity of \(z\sin\frac1z\) at \(z=0\).
  10. What is the order of the pole of \(\frac{z^2-1}{(z-1)^3}\) at \(z=1\)?

What comes next

References

  • Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 3, §1
  • James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. §68
  • Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979. Cap. 4, §3.2