# Difference between revisions of "Isolated singular point"

Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||

Line 1: | Line 1: | ||

− | + | <!-- | |

+ | i0527801.png | ||

+ | $#A+1 = 52 n = 0 | ||

+ | $#C+1 = 52 : ~/encyclopedia/old_files/data/I052/I.0502780 Isolated singular point | ||

+ | Automatically converted into TeX, above some diagnostics. | ||

+ | Please remove this comment and the {{TEX|auto}} line below, | ||

+ | if TeX found to be correct. | ||

+ | --> | ||

− | + | {{TEX|auto}} | |

+ | {{TEX|done}} | ||

− | + | ''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|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|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|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)|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|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|Residue of an analytic function]]). It should be noted that the elements of different branches of the [[Complete analytic function|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==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR></table> | ||

− | |||

− | |||

====Comments==== | ====Comments==== | ||

− | |||

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) {{MR|0593142}} {{MR|0197687}} {{MR|1523319}} {{ZBL|0477.30001}} {{ZBL|0336.30001}} {{ZBL|0005.21004}} {{ZBL|65.0302.01}} {{ZBL|58.0297.01}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 {{MR|0510197}} {{MR|1535085}} {{MR|0188405}} {{MR|1570643}} {{MR|1528598}} {{MR|0054016}} {{ZBL|0395.30001}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) {{MR|0593142}} {{MR|0197687}} {{MR|1523319}} {{ZBL|0477.30001}} {{ZBL|0336.30001}} {{ZBL|0005.21004}} {{ZBL|65.0302.01}} {{ZBL|58.0297.01}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 {{MR|0510197}} {{MR|1535085}} {{MR|0188405}} {{MR|1570643}} {{MR|1528598}} {{MR|0054016}} {{ZBL|0395.30001}} </TD></TR></table> |

## Latest revision as of 22:13, 5 June 2020

*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 |

#### Comments

#### 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 |

**How to Cite This Entry:**

Isolated singular point.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Isolated_singular_point&oldid=24479