Isolated singular point

for an element of an analytic function $f ( z)$

A point $a$ in the complex $z$- plane satisfying the following properties: 1) the element of $f ( z)$ does not have an analytic continuation along any path to $a$; and 2) there exists a number $R > 0$ such that analytic continuation of $f ( z)$ is possible along any path in the punctured neighbourhood $U = \{ {z \in \mathbf C } : {0 < | z - a | < R } \}$ of $a$.

If a new element is obtained when $f ( z)$ is continued analytically along a closed path in $U$ encircling $a$, for example along the circle $| z - a | = \rho$, $0 < \rho < R$, then $a$ is called a branch point, or an isolated singular point of multi-valued character. Otherwise the element of $f ( z)$ defines a single-valued analytic function in $U$ and $a$ is called an isolated singular point of single-valued character. In a punctured neighbourhood $U$ of an isolated singular point $a$ of single-valued character, $f ( z)$ can be expanded in a Laurent series:

$$\tag{1 } f ( z) = \ \sum _ {k = - \infty } ^ { {+ } \infty } c _ {k} ( z - a) ^ {k}$$

with regular part $f _ {1} ( z) = \sum _ {k = 0 } ^ {+ \infty } c _ {k} ( z - a) ^ {k}$ and principal part $f _ {2} ( z) = \sum _ {k = - \infty } ^ {-} 1 c _ {k} ( z- a) ^ {k}$. The behaviour of an analytic function $f ( z)$ in a punctured neighbourhood $U$ 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 $f ( a) = c _ {0}$ one gets a single-valued analytic function in a full neighbourhood of $a$. This case of practical absence of a singularity is also characterized by the fact that $f ( z)$ is bounded in $U$, or by the fact that the limit $\lim\limits _ {z \rightarrow a } f ( z) = c _ {0}$, $z \in U$, exists and is finite.

If among the coefficients of the principal part only finitely many are non-zero, and that with smallest index is $c _ {-} m \neq 0$, then $a$ is a pole of order $m$( cf. Pole (of a function)). A pole $a$ is also characterized by the fact that

$$\lim\limits _ {z \rightarrow a } f ( z) = \infty ,\ \ z \in U.$$

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

$$\lim\limits _ {z \rightarrow a } f ( z),\ z \in U.$$

For an isolated singular point $a = \infty$ at infinity of the element $f( z)$, a punctured neighbourhood has the form $U = \{ {z \in \mathbf C } : {r < | z | < \infty } \}$, and the Laurent series is

$$f ( z) = \ \sum _ {k = - \infty } ^ { {+ } \infty } c _ {k} z ^ {k} .$$

Here the regular part is $f _ {1} = \sum _ {k = - \infty } ^ {0} c _ {k} z ^ {k}$ and the principal part is $f _ {2} ( z) = \sum _ {k = 1 } ^ {+ \infty } c _ {k} z ^ {k}$. With these conditions, the above descriptions of the classification and criteria for the type of an isolated singular point carry over to the case $a = \infty$ 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 $f ( z)$ at one and the same point $a \in \mathbf C$ may have singularities of completely-different types.

Holomorphic functions $f ( z)$ of several complex variables $z = ( z _ {1} \dots z _ {n} )$, $n \geq 2$, cannot have isolated singular points. For $n \geq 2$, 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