page_title: Geometric function theory
section_title: Analytic continuation
hierarchical_section_title: Geometric function theory / Topics in geometric function theory / Analytic continuation
caption_attribution_description: analytic continuation of the imaginary part of logarithmic function on the complex plane.This depicts a branch that vanishes at the x-axis, and indicates the possibility of continuation on the complement of the negative closed half line.
context_page_description: Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.
context_section_description: Analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
  The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of mathematical singularities. The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology.
caption: Analytic continuation of natural logarithm (imaginary part)
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