In the mathematical field of complex analysis, contour integration is a
method of evaluating certain integrals along paths in the complex plane.
Contour integration is closely related to the calculus of residues, a method of
complex analysis. One use for contour integrals is the
evaluation of integrals along the real line that are not readily found by using
only real variable methods. Contour integration methods include
direct integration of a complex-valued function along a curve in the complex
plane application of the Cauchy integral
formula application of the residue theorem
One method can be used, or a combination of these methods, or various limiting
processes, for the purpose of finding these integrals or sums.
Curves in the complex plane In complex analysis a contour is a type
of curve in the complex plane. In contour integration, contours provide a
precise definition of the curves on which an integral may be suitably
defined. A curve in the complex plane is defined as a continuous function from a
closed interval of the real line to the complex plane: z : [a, b] → C.
This definition of a curve coincides with the intuitive notion of a curve,
but includes a parametrization by a continuous function from a closed
interval. This more precise definition allows us to consider what properties a
curve must have for it to be useful for integration. In the following
subsections we narrow down the set of curves that we can integrate to only
include ones that can be built up out of a finite number of continuous curves
that can be given a direction. Moreover, we will restrict the "pieces" from
crossing over themselves, and we require that each piece have a finite continuous
derivative. These requirements correspond to requiring that we consider
only curves that can be traced, such as by a pen, in a sequence of even, steady
strokes, which only stop to start a new piece of the curve, all without picking
up the pen. = Directed smooth curves=
Contours are often defined in terms of directed smooth curves. These provide a
precise definition of a "piece" of a smooth curve, of which a contour is
made. A smooth curve is a curve z : [a, b] → C
with a non-vanishing, continuous derivative such that each point is
traversed only once, with the possible exception of a curve such that the
endpoints match = z(b)). In the case where the endpoints match the curve is
called closed, and the function is required to be one-to-one everywhere
else and the derivative must be continuous at the identified point. A
smooth curve that is not closed is often referred to as a smooth arc.
The parametrization of a curve provides a natural ordering of points on the
curve: z(x) comes before z(y) if x 0 then
Therefore if t > 0 then A similar argument with an arc that
winds around −i rather than i shows that if t 1. But
where the integration is done over the Hankel contour, , is valid for all
complex . See also
Residue Cauchy principal value
Poisson integral Pochhammer contour
References and notes ^ John Stalker. Complex Analysis:
Fundamentals of the Classical Theory of Functions. Springer. p. 77. ISBN
0-8176-4038-X. ^ Joseph Bak & Donald J. Newman. Complex
Analysis. Springer. Chapters 11 & 12, pp. 130–156. ISBN 0-387-94756-6.
^ Steven George Krantz. "Chapter 2". Handbook of Complex Variables. Springer.
ISBN 0-8176-4011-8. ^ Dragoslav S. Mitrinovic & Jovan D.
Keckic. "Chapter 2". The Cauchy Method of Residues: Theory and Applications.
Springer. ISBN 90-277-1623-4. ^ Dragoslav S. Mitrinovic & Jovan D.
Keckic. Chapter 5. ISBN 90-277-1623-4. ^ a b c d e Edward B. Saff & Arthur
David Snider. Chapter 4. ISBN 01-390-7874-6.
Further reading Titchmarsh, E.C., The Theory of
Functions, Oxford University Press, ISBN 0-19-853349-7
Jean Jacquelin, Marko Riedel, Branche univalente, Les-Mathematiques.net, in
French. Marko Riedel et al., Problème
d'intégrale, Les-Mathematiques.net, in French.
Marko Riedel et al., Integral by residue, math.stackexchange.com.
Various authors, sin límites ni cotas, es.ciencia.matematicas, in Spanish.
W W L Chen, Introduction to Complex Analysis
External links Hazewinkel, Michiel, ed., "Complex
integration, method of", Encyclopedia of Mathematics, Springer, ISBN
978-1-55608-010-4 A collection of examples
Residue Calculus Module by John H. Mathews