View source for Transcendental branch point

''of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936001.png" />''

A [[Branch point|branch point]] that is not an [[Algebraic branch point|algebraic branch point]]. In other words, it is either a branch point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936002.png" /> of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936003.png" /> at which, however, there does not exist a finite or infinite limit

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936004.png" /></td> </tr></table>

or a [[Logarithmic branch point|logarithmic branch point]] of infinite order. For example, the first possibility is realized at the transcendental branch point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936005.png" /> for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936006.png" />, the second for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936007.png" />.

In the first case the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936008.png" /> can be expanded in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t0936009.png" /> in the form of a [[Puiseux series|Puiseux series]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t09360010.png" /></td> </tr></table>

with an infinite number of non-zero coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093600/t09360011.png" /> with negative indices.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>