Difference between revisions of "Logarithmic branch point"
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''branch point of infinite order'' | ''branch point of infinite order'' | ||
| − | A special form of a [[Branch point|branch point]] | + | A special form of a [[Branch point|branch point]] $ a $ |
| + | of an analytic function $ f ( z) $ | ||
| + | of one complex variable $ z $, | ||
| + | when for no finite number of successive circuits in the same direction about $ a $ | ||
| + | the [[Analytic continuation|analytic continuation]] of some element of $ f ( z) $ | ||
| + | returns to the original element. More precisely, an isolated singular point $ a $ | ||
| + | is called a logarithmic branch point for $ f ( z) $ | ||
| + | if there exist: 1) an annulus $ V = \{ {z } : {0 < | z - a | < \rho } \} $ | ||
| + | in which $ f ( z) $ | ||
| + | can be analytically continued along any path; and 2) a point $ z _ {1} \in V $ | ||
| + | and an element of $ f ( z) $ | ||
| + | in the form of a power series $ \Pi ( z _ {1} ; r ) = \sum _ {\nu = 0 } ^ \infty c _ \nu ( z - z _ {1} ) ^ \nu $ | ||
| + | with centre $ z _ {1} $ | ||
| + | and radius of convergence $ r > 0 $, | ||
| + | the analytic continuation of which along the circle $ | z - a | = | z _ {1} - a | $, | ||
| + | taken arbitrarily many times in the same direction, never returns to the original element $ \Pi ( z _ {1} ; r ) $. | ||
| + | In the case of a logarithmic branch point at infinity, $ a = \infty $, | ||
| + | instead of $ V $ | ||
| + | one must consider a neighbourhood $ V ^ \prime = \{ {z } : {| z | > \rho } \} $. | ||
| + | Logarithmic branch points belong to the class of transcendental branch points (cf. [[Transcendental branch point|Transcendental branch point]]). The behaviour of the Riemann surface $ R $ | ||
| + | of a function $ f ( z) $ | ||
| + | in the presence of a logarithmic branch point $ a $ | ||
| + | is characterized by the fact that infinitely many sheets of the same branch of $ R $ | ||
| + | are joined over $ a $; | ||
| + | this branch is defined in $ V $ | ||
| + | or $ V ^ \prime $ | ||
| + | by the elements $ \Pi ( z _ {1} ; r ) $. | ||
See also [[Singular point|Singular point]] of an analytic function. | See also [[Singular point|Singular point]] of an analytic function. | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
| − | The function | + | The function $ \mathop{\rm Ln} ( z - z _ {0} ) $ |
| + | has a logarithmic branch point at $ z _ {0} $, | ||
| + | where $ \mathop{\rm Ln} $ | ||
| + | is the (multiple-valued) [[Logarithmic function|logarithmic function]] of a complex variable. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8</TD></TR></table> | ||
Latest revision as of 22:17, 5 June 2020
branch point of infinite order
A special form of a branch point $ a $ of an analytic function $ f ( z) $ of one complex variable $ z $, when for no finite number of successive circuits in the same direction about $ a $ the analytic continuation of some element of $ f ( z) $ returns to the original element. More precisely, an isolated singular point $ a $ is called a logarithmic branch point for $ f ( z) $ if there exist: 1) an annulus $ V = \{ {z } : {0 < | z - a | < \rho } \} $ in which $ f ( z) $ can be analytically continued along any path; and 2) a point $ z _ {1} \in V $ and an element of $ f ( z) $ in the form of a power series $ \Pi ( z _ {1} ; r ) = \sum _ {\nu = 0 } ^ \infty c _ \nu ( z - z _ {1} ) ^ \nu $ with centre $ z _ {1} $ and radius of convergence $ r > 0 $, the analytic continuation of which along the circle $ | z - a | = | z _ {1} - a | $, taken arbitrarily many times in the same direction, never returns to the original element $ \Pi ( z _ {1} ; r ) $. In the case of a logarithmic branch point at infinity, $ a = \infty $, instead of $ V $ one must consider a neighbourhood $ V ^ \prime = \{ {z } : {| z | > \rho } \} $. Logarithmic branch points belong to the class of transcendental branch points (cf. Transcendental branch point). The behaviour of the Riemann surface $ R $ of a function $ f ( z) $ in the presence of a logarithmic branch point $ a $ is characterized by the fact that infinitely many sheets of the same branch of $ R $ are joined over $ a $; this branch is defined in $ V $ or $ V ^ \prime $ by the elements $ \Pi ( z _ {1} ; r ) $.
See also Singular point of an analytic function.
References
| [1] | A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) pp. Chapt. 8 (Translated from Russian) |
Comments
The function $ \mathop{\rm Ln} ( z - z _ {0} ) $ has a logarithmic branch point at $ z _ {0} $, where $ \mathop{\rm Ln} $ is the (multiple-valued) logarithmic function of a complex variable.
References
| [a1] | L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8 |
How to Cite This Entry:
Logarithmic branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_branch_point&oldid=11411
Logarithmic branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_branch_point&oldid=11411
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article