Isolated singular point

for an element of an analytic function

A point in the complex -plane satisfying the following properties: 1) the element of does not have an analytic continuation along any path to ; and 2) there exists a number such that analytic continuation of is possible along any path in the punctured neighbourhood of .

If a new element is obtained when is continued analytically along a closed path in encircling , for example along the circle , , then is called a branch point, or an isolated singular point of multi-valued character. Otherwise the element of defines a single-valued analytic function in and is called an isolated singular point of single-valued character. In a punctured neighbourhood of an isolated singular point of single-valued character, can be expanded in a Laurent series:

(1)

with regular part and principal part . The behaviour of an analytic function in a punctured neighbourhood of an isolated singular point of single-valued character is determined, in principle, by the principal part of its Laurent series. If all the coefficients of the principal part are zero, then on setting one gets a single-valued analytic function in a full neighbourhood of . This case of practical absence of a singularity is also characterized by the fact that is bounded in , or by the fact that the limit , , exists and is finite.

If among the coefficients of the principal part only finitely many are non-zero, and that with smallest index is , then is a pole of order (cf. Pole (of a function)). A pole is also characterized by the fact that

Finally, if there are infinitely many non-zero coefficients in the principal part, then is an essential singular point. In this case the following limit does not exist, neither finite nor infinite:

For an isolated singular point at infinity of the element , a punctured neighbourhood has the form , and the Laurent series is

Here the regular part is and the principal part is . With these conditions, the above descriptions of the classification and criteria for the type of an isolated singular point carry over to the case without further change (see also Residue of an analytic function). It should be noted that the elements of different branches of the complete analytic function at one and the same point may have singularities of completely-different types.

Holomorphic functions of several complex variables , , cannot have isolated singular points. For , the singular points form infinite sets of singularities.

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[2]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001

References

[a1]	E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) MR0593142 MR0197687 MR1523319 Zbl 0477.30001 Zbl 0336.30001 Zbl 0005.21004 Zbl 65.0302.01 Zbl 58.0297.01
[a2]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 MR0510197 MR1535085 MR0188405 MR1570643 MR1528598 MR0054016 Zbl 0395.30001