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