Contour integration

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Here we follow standard texts, such as Spiegel (1964)[1] or Levinson and Redheffer (1970). [2]

For a complex-valued function of a complex variable that is analytic in some region we may define the process of integration along a curve via the Riemann summation formalism

Figure 1: An integration path C in the complex z plane



where represents an increment along the curve


See Figure 1. Such a curve is called a contour, though this term does not refer to a level curve,

or a curve of equal elevation as it would in cartography, but to any curve in the complex plane that does not cross itself.

Formally we may write


Several important results allow us to make sense of, and to make use of the properties of complex integration.

Cauchy's Theorem

If is a closed contour, and the complex valued function is analytic inside the region bounded by, and on then

This is followed by a complementary theorem by Morerra

Morera's Theorem

If for every closed contour within a region of the complex and

then is analytic everywhere in .

Cauchy integral theorem

If is analytic on, and inside, a region bounded by a closed contour then for a point inside C,


Cauchy integral formulas

If is analytic on, and inside, a region bounded by a closed contour then for a point inside C, and for every integer

where is the -th derivative with respect to of .

Residue Theorem

Simple pole

If is analytic on, and inside, a region bounded by a closed contour then for a point except at a point inside C, such that is a simple pole, that is for analytic inside and on then

where, is called the residue of at

If has finite simple poles inside



Multiple pole

If for an integer where is analytic inside and on then is said to have an -th order pole at and


Here, the where is the -th derivative of

References

[3]
[4]

[5]

  1. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
  2. Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.
  3. Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
  4. Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
  5. Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.