Casorati–Weierstrass theorem explained

In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem.

Formal statement of the theorem

U in the complex plane containing the number z _{0} , and a function f that is holomorphic on U \setminus \{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:

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 .

In the case that

f is an entire function and a = inf ty , the theorem says that the values f(z) approach every complex number and inf ty , as z tends to infinity.It is remarkable that this does not hold for holomorphic maps in higher dimensions, as the famous example of Pierre Fatou shows.^[1]

Examples

The function has an essential singularity at 0, but the function does not (it has a pole at 0).

Consider the function$$f(z)\; =\; e^.$$

This function has the following Laurent series about the essential singular point at 0:$$f(z)\; =\; \backslash sum\_^\backslash fracz^.$$

Because

f'(z) = - (e ^{{1} /{z)/() }} exists for all points we know that is analytic in a punctured neighborhood of . Hence it is an isolated singularity, as well as being an essential singularity. z = re ^{i} our function, becomes:$$f(z)=e^=e^e^.$$

Taking the absolute value of both sides:$$\backslash left|\; f(z)\; \backslash right|\; =\; \backslash left|\; e^\; \backslash right|\; \backslash left|\; e^\; \backslash right\; |\; =e^.$$

Thus, for values of θ such that, we have

f(z) \to inf ty as r \to 0 , and for \cos \theta < 0 , f(z) \to 0 as r \to 0 .

Consider what happens, for example when z takes values on a circle of diameter tangent to the imaginary axis. This circle is given by . Then,$$f(z)\; =\; e^\; \backslash left[\backslash cos\; \backslash left(R\backslash tan\; \backslash theta\; \backslash right)\; -\; i\; \backslash sin\; \backslash left(R\backslash tan\; \backslash theta\; \backslash right)\; \backslash right]$$and$$\backslash left|\; f(z)\; \backslash right|\; =\; e^R.$$

Thus,

\left| f(z) \right| may take any positive value other than zero by the appropriate choice of R. As z \to 0 on the circle, $\backslash theta\; \backslash to\; \backslash frac$ with R fixed. So this part of the equation:$$\backslash left[\backslash cos\; \backslash left(R\; \backslash tan\; \backslash theta\; \backslash right)\; -\; i\; \backslash sin\; \backslash left(R\; \backslash tan\; \backslash theta\; \backslash right)\; \backslash right]$$takes on all values on the unit circle infinitely often. Hence takes on 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:$$g(z)\; =\; \backslash frac$$must be holomorphic on, with zeroes at the poles of f, and bounded by 1/ε. It can therefore be analytically continued (or continuously extended, or holomorphically extended) to all of V by Riemann's analytic continuation theorem. So the original function can be expressed in terms of :$$f(z)\; =\; \backslash frac\; +\; b$$for all arguments z in V \ . Consider the two possible cases for$$\backslash lim\_\; g(z).$$

If the limit is 0, then f has a pole at z₀ . If the limit is not 0, then z₀ is a removable singularity of f . Both possibilities contradict the assumption that the point z₀ is an essential singularity of the function f . Hence the assumption is false and the theorem holds.

History

The history of this important theorem is described by Collingwood and Lohwater.^[2] It was published by Weierstrass in 1876 (in German) and by Sokhotski in 1868 in his Master thesis (in Russian). So it was called Sokhotski's theorem in the Russian literature and Weierstrass's theorem in the Western literature. The same theorem was published by Casorati in 1868, and by Briot and Bouquet in the first edition of their book (1859).^[3] However, Briot and Bouquet removed this theorem from the second edition (1875).

References

1. Fatou . P . Sur les fonctions méromorphes de deux variables . Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris . 1922. 175 . 862–865. 48.0391.02., Fatou . P . Sur certaines fonctions uniformes de deux variables . Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris . 1922. 175 . 1030–1033. 48.0391.03.
2. Book: E. Collingwood. A . Lohwater. The theory of cluster sets. Cambridge University Press. 1966.
3. Book: Ch. Briot. C. Bouquet. Theorie des fonctions doublement periodiques, et en particulier, des fonctions elliptiques. Paris . 1859.

• Section 31, Theorem 2 (pp. 124–125) of