In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the remarkable behavior of meromorphic functions near essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati.
Formal statement of the theorem.
Start with some open subset in the complex plane containing the number , and a function that is holomorphic on , but has an essential singularity at . The Casorati–Weierstrass theorem then states that
if is any neighborhood of contained in , then is dense in .
This can also be stated as follows
for any and any complex number , there exists a complex number in with and .
Or in still more descriptive terms
comes arbitrarily close to any complex value in every neighbourhood of .
This form of the theorem also applies if is only meromorphic. The theorem is considerably strengthened by Picard’s great theorem, which states, in the notation above, that assumes every complex value, with one possible exception, infinitely often on .
The function has an essential singularity at , but the function does not (it has a pole at ). Consider the function
This function has the following Laurent series about the essential singular point at
exists for all points we know that is analytic in the neighborhood of . Hence it is an isolated singularity, as well as being and essential singularity. Using a change of variable to polar coordinates our function, becomes
Taking the absolute value of both sides
Thus, for values of such that , we have as , and for , as .
Consider what happens, for example when takes values on a circle of diameter tangent to the imaginary axis. This circle is given by . Then,
Thus, may take any positive value other than zero by the appropriate choice of . As on the circle, with fixed. So this part of the equation
takes on all values on the unit circle infinitely often. Hence takes on all the value of every number in the complex plane except for zero infinitely often.
Proof of the theorem.
A short proof of the theorem is as follows: Take as given that function is meromorphic on some punctured neighborhood , and that is an essential singularity. Assume by way of contradiction that some value exists that the function can never get close to; that is: assume that there is some complex value and some such that for all in at which is defined.
Then the new function
must be holomorphic on , with zeroes at the poles of , and bounded by . It can therefore be analytically continued (or continuously extended, or holomorphically extended) to all of by Riemann’s analytic continuation theorem. So the original function can be expressed in terms of
for all arguments in . Consider the two possible cases for . If the limit is , then has a pole at . If the limit is not , then is a removable singularity of . Both possibilities contradict the assumption that the the point is an essential singularity of the function . Hence the assumption is false and the theorem holds.