In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of meromorphic functions near essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati.
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Formal statement of the theorem
Start with some open subset U in the complex plane containing the number z_{0}, and a function f that is holomorphic on U \ {z_{0}}, but has an essential singularity at z_{0} . The Casorati–Weierstrass theorem then states that
This can also be stated as follows:
Or in still more descriptive terms:
This form of the theorem also applies if f is only meromorphic.
The theorem is considerably strengthened by Picard's great theorem, which states, in the notation above, that f assumes every complex value, with one possible exception, infinitely often on V.
Examples
The function f(z) = exp(1/z) has an essential singularity at z_{0} = 0, but the function g(z) = 1/z^{3} does not (it has a pole at 0).
Consider the function
This function has the following Laurent series about the essential singular point at z_{0}:
Because exists for all points z ≠ 0 we know that ƒ(z) is analytic in the punctured neighborhood of z = 0. Hence it is an isolated singularity, as well as being an essential singularity.
Using a change of variable to polar coordinates z = re^{iθ} our function, ƒ(z) = e^{1/z} becomes:
Taking the absolute value of both sides:
Thus, for values of θ such that cos θ > 0, we have as , and for cosθ < 0, as .
Consider what happens, for example when z takes values on a circle of diameter 1/R tangent to the imaginary axis. This circle is given by r = (1/R) cos θ. Then,
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