Isolated Singularities

Removable Singularities, Poles, and Essential Singularities:

We shall see momentarily that, if previous case holds, then f has a limit at P that extends f so that it is holomorphic on all of U (this is not at all obvious; it is a theorem of Riemann). It is commonly said in this circumstance that f has a removable singularity at P. In  previous case, we will say that f has a pole at P. In  previous case, f will be said to have an essential singularity at P. Our goal in this and the next two subsections is to understand – Past case in some further detail.

Riemann Removable Singularities Theorem:

Let f : D(P, r) \ { P } → C be holomorphic and bounded. Then

1. lim_z→P f(z) exists.

2. The function b : D(P, r) → C defined by

is holomorphic.

For the proof, take P = 0 and consider the function g (z) = z² .f (z). One may verify directly that g is C¹ and satisfies the Cauchy-Riemann equations on all of D(P, r) (the boundedness hypothesis is used to check that both g and its first derivative have limits at 0). Thus g is holomorphic on the disc, and it vanishes to second order at 0. It follows then that f (z) = g (z)/z² is a bona fide holomorphic function on all of D(P, r).

Casorati-Weierstrass Theorem:

If f : D(P, r₀) \ { P } → C is holomorphic and P is an essential singularity of f, then f(D(P, r) \ { P }) is dense in C for any 0 < r < r₀ .

For the reason, suppose that the assertion is not true. So there is a complex value μ and a positive number ε so that the image of D(P, r) \ { P }  under f does not contain the disc But then the function g(z) = 1/[f(z) − μ] is bounded and non-vanishing near P, hence has a removable singularity. We see then that f is bounded near P, and that contradicts that P is an essential singularity. Now we have seen that, at a removable singularity P, a holomorphic function f on D(P, r₀) \ { P } can be continued to be holomorphic on all of D(P, r₀). And, near an essential singularity at P, a holomorphic function g on D(P, r₀) \ { P } has image that is dense in C. The third possibility, that h has a pole at P, has yet to be described. This case will be examined further in the coming sections.