# Contour integration

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

For a complex-valued function ${\displaystyle f(z)=u(x,y)+iv(x,y)}$ of a complex variable ${\displaystyle z=x+iy}$ that is analytic in some region ${\displaystyle {\mathcal {R}},}$ we may define the process of integration along a curve ${\displaystyle C}$ via the Riemann summation formalism

Figure 1: An integration path C in the complex z plane
${\displaystyle \int _{C}f(z)dz=\int _{a}^{b}f(z)dz=\lim _{\Delta z_{n}\rightarrow 0}\lim _{N\rightarrow \infty }\sum _{n=0}^{N}f(\zeta _{n})\Delta z_{n}}$

${\displaystyle \int _{C}f(z)dz=\lim _{\Delta z_{n}\rightarrow 0}\lim _{N\rightarrow \infty }\sum _{n=0}^{N}f(\zeta _{n})\Delta z_{n}}$

where ${\displaystyle \Delta z_{n}=z_{n+1}-z_{n}}$ represents an increment along the curve

${\displaystyle C=\left\{a=z_{0}<\zeta _{0}

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

${\displaystyle \int _{C}f(z)\;dz=\int _{C}(u(x,y)+iv(x,y))\;(dx+idy)=\left(\int _{C}u(x,y)\;dx-\int _{C}v(x,y)\;dy\right)+i\left(\int _{C}v(x,y)\;dx+\int _{C}u(x,y)\;dy\right).}$

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

## Cauchy's Theorem

If ${\displaystyle C}$ is a closed contour, and the complex valued function ${\displaystyle f(z)}$ is analytic inside the region bounded by, and on ${\displaystyle C}$ then

${\displaystyle \int _{C}f(z)\;dz=0.}$

This is followed by a complementary theorem by Morerra

## Morera's Theorem

If for every closed contour ${\displaystyle C}$ within a region ${\displaystyle {\mathcal {R}}}$ of the complex and

${\displaystyle \int _{C}f(z)\;dz=0}$

then ${\displaystyle f(z)}$ is analytic everywhere in ${\displaystyle {\mathcal {R}}}$.

## Cauchy integral theorem

If ${\displaystyle f(z)}$ is analytic on, and inside, a region bounded by a closed contour ${\displaystyle C}$ then for a point ${\displaystyle a}$ inside C,

${\displaystyle f(a)={\frac {1}{2\pi i}}\int _{C}{\frac {f(z)}{(z-a)}}\;dz.}$

## Cauchy integral formulas

If ${\displaystyle f(z)}$ is analytic on, and inside, a region bounded by a closed contour ${\displaystyle C}$ then for a point ${\displaystyle a}$ inside C, and for every integer ${\displaystyle k>0}$

${\displaystyle f^{(k)}(a)={\frac {k!}{2\pi i}}\int _{C}{\frac {f(z)}{(z-a)^{k+1}}}}$

where ${\displaystyle f^{(k)}(a)}$ is the ${\displaystyle k}$-th derivative with respect to ${\displaystyle a}$ of ${\displaystyle f(a)}$.

## Residue Theorem

### Simple pole

If ${\displaystyle f(z)}$ is analytic on, and inside, a region bounded by a closed contour ${\displaystyle C}$ then for a point except at a point ${\displaystyle a}$ inside C, such that ${\displaystyle a}$ is a simple pole, that is ${\displaystyle f(z)={\frac {g(z)}{(z-a)}}}$ for ${\displaystyle g(z)}$ analytic inside and on ${\displaystyle C}$ then

${\displaystyle \int _{C}f(z)\;dz=2\pi i\;Res(f(z);a)}$

where, ${\displaystyle Res(f(z);a)}$ is called the residue of ${\displaystyle f(z)}$ at ${\displaystyle a.}$

If ${\displaystyle f(z)}$ has ${\displaystyle M}$ finite simple poles inside ${\displaystyle C}$

${\displaystyle \int _{C}f(z)\;dz=2\pi i\;Res(f(z);a)=2\pi i\sum _{k=0}^{M}\;Res(f(z);a_{k})}$

### Multiple pole

If ${\displaystyle f(z)={\frac {g(z)}{(z-a)^{N+1}}}}$ for ${\displaystyle N>0}$ an integer where ${\displaystyle g(z)}$ is analytic inside and on ${\displaystyle C,}$ then ${\displaystyle f(z)}$ is said to have an ${\displaystyle N}$-th order pole at ${\displaystyle a}$ and

${\displaystyle \int _{C}f(z)\;dz=2\pi i\;Res(f(z);a).}$

Here, the ${\displaystyle Res(f(z);a)=\lim _{z\rightarrow a}N!(z-a)^{N+1}f^{(N)}(z),}$ where ${\displaystyle f^{(N)}}$ is the ${\displaystyle N}$-th derivative of ${\displaystyle f(z).}$

## References

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.