Integrals Involving Branch Points || Keyhole Contour || Contour Integration || Math Contributor

In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis[citation needed]) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point.[1][dubious – discuss] Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept.

Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation w2 = z for w as a function of z. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. Despite the algebraic branch point, the function w is well-defined as a multiple-valued function and, in an appropriate sense, is continuous at the origin. This is in contrast to transcendental and logarithmic branch points, that is, points at which a multiple-valued function has nontrivial monodromy and an essential singularity. In geometric function theory, unqualified use of the term branch point typically means the former more restrictive kind: the algebraic branch points.[2] In other areas of complex analysis, the unqualified term may also refer to the more general branch points of transcendental type.

Chebfun is able to represent complex functions of a real variable, which lends itself very well to computing paths and path integrals in the complex plane. In this brief example we demonstrate this by integrating the function

f = @(x) log(x).*tanh(x);
around a 'keyhole' contour which avoids the branch cut on the negative real axis.

We'll first define our keyhole. Let r, R, and e be the inner and outer radii and the width of the key respectively:
Integrals Involving Branch Points || Keyhole Contour || Contour Integration || Math Contributor