Contour integration

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

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

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

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

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

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

$C=\left\{a=z_{0}<\zeta _{0}<z_{1}<\zeta _{1}<z_{2}\;\ldots \;z_{N-2}<\zeta _{N-1}<z_{N-1}<\zeta _{N}<z_{N}=b\right\}$

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

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

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

Cauchy's Theorem

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

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

This is followed by a complementary theorem by Morerra

Morera's Theorem

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

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

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

Cauchy integral theorem

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

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

Cauchy integral formulas

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

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

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

Residue Theorem

Simple pole

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

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

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

If $f(z)$ has $M$ finite simple poles inside $C$

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

Multiple pole

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

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

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