Casorati-Sokhotskii-Weierstrass theorem

Weierstrass theorem, Weierstrass–Sokhotskii–Casorati theorem

Let be an essential singular point of an analytic function of a complex variable . Given any complex number (including ), there is a sequence converging to such that

This theorem was the first result characterizing the cluster set of an analytic function at an essential singularity . According to the theorem, is total, that is, it coincides with the extended plane of the variable . The theorem was proved by Yu.V. Sokhotskii [1] (see also [2]). K. Weierstrass stated this theorem in 1876 (see [3]). Further information about the behaviour of an analytic function in a neighbourhood of an essential singularity is contained in the Picard theorem.

This theorem does not admit a direct generalization to the case of analytic mappings for (see [5]).

References

[1]	Yu.V. Sokhotskii, "Theory of integral residues with some applications" , St. Petersburg (1868) (In Russian)
[2]	F. Casorati, "Teoria delle funzioni di variabili complesse" , Pavia (1868)
[3]	K. Weierstrass, "Zur Theorie der eindeutigen analytischen Funktionen" , Math. Werke , 2 , Mayer & Müller (1895) pp. 77–124
[4]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[5]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) pp. Chapt. 2 (In Russian)

Comments

In the West, this theorem is known universally as the Casorati–Weierstrass theorem. It was, however, proved earlier by C. Briot and C. Bouquet and appears in the first edition [a1] of their book on elliptic functions (1859), though it is missing from the second edition of this work; cf. the discussion in [a2], pp. 4–5.

References

[a1]	C. Briot, C. Bouquet, "Théorie des fonctions doublement périodiques et, en particulier, des fonctions elliptiques" , Mallet–Bachelier (1859)
[a2]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9