View source for Logarithmic branch point

''branch point of infinite order''

A special form of a [[Branch point|branch point]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605601.png" /> of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605602.png" /> of one complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605603.png" />, when for no finite number of successive circuits in the same direction about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605604.png" /> the [[Analytic continuation|analytic continuation]] of some element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605605.png" /> returns to the original element. More precisely, an isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605606.png" /> is called a logarithmic branch point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605607.png" /> if there exist: 1) an annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605608.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l0605609.png" /> can be analytically continued along any path; and 2) a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056010.png" /> and an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056011.png" /> in the form of a power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056012.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056013.png" /> and radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056014.png" />, the analytic continuation of which along the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056015.png" />, taken arbitrarily many times in the same direction, never returns to the original element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056016.png" />. In the case of a logarithmic branch point at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056017.png" />, instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056018.png" /> one must consider a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056019.png" />. Logarithmic branch points belong to the class of transcendental branch points (cf. [[Transcendental branch point|Transcendental branch point]]). The behaviour of the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056020.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056021.png" /> in the presence of a logarithmic branch point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056022.png" /> is characterized by the fact that infinitely many sheets of the same branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056023.png" /> are joined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056024.png" />; this branch is defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056025.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056026.png" /> by the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056027.png" />.

See also [[Singular point|Singular point]] of an analytic function.

====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>



====Comments====
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056028.png" /> has a logarithmic branch point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060560/l06056030.png" /> is the (multiple-valued) [[Logarithmic function|logarithmic function]] of a complex variable.

====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>