On the complex plane we deal with functions where their argument is some complex number, $z = x(t) + iy(t)$, where $x,y \in \mathbb{R}$. Here we see that it's useful to parameterize with $t$ in order to establish a [continuous](Continuity.md) curve (or _contour_) over which we can integrate. This is the same approach to working entirely in real space where we consider $x(t)$ and $y(t)$ where these are components of _[vector fields](Vector%20fields.md)_, $f(\mathbf{v})$. On a complex plane, holomorphic functions, $f(z(t))$ take the place of vector fields and thus contour integrals are reminiscent of real [path integrarls](Path%20integrals%20of%20vector%20fields.md) in that sense. If we consider a curve on a complex plane, $\gamma:I\rightarrow \mathbb{C}$, we can define the contour integral within the domain $U$ of a function $f(z(t))$ for which $f$ is a [holomorphic function](Holomorphic%20functions.md) such that: $\int_{\gamma}dz f(z)=\int_{I}dt \frac{dz(t)}{dt}f(z(t))$  (Image adapted from Altland, A. von Delft, J. _Mathematics for Physicists_) This definition comes about from application of the [Riemann sum](Riemann%20integral.md#Riemann%20Sum) from which elementary integrals and line integrals are also defined: $\sum_l f(z(t_l))(z(t_{l+1})-z(t_l))\simeq \delta_t \sum_l f(z(t_l))\frac{d}{dt}z(t_l)$ where $\delta_t = t_{l+1}-t_l$ and as $\delta_t \rightarrow 0$, $\delta_t \sum_l \rightarrow \int_I$. We usually care most about [[Closed contour integral]]s since there are many important results that arise when we take the contour to be a closed path. For the most general theorems, unless stated otherwise, we're going to integrate along contours in the counter-clockwise direction (i.e. _along positively oriented contours_). Often we may need to split a closed contour into several open components and integrate directly by parameterizing $z(t)$. # Integrating over a circular arc Very often we deal with contours that form arcs where we parameterize with $z(\phi)=Re^{i\phi}$. $\int_{\gamma}dz f(z)=\int_{0}^{\phi}d\phi\,iRe^{i\phi}f(Re^{i\phi})$ We can show that as $R \rightarrow \infty$ that the contour integral over the arc may disappear by [[Jordan's lemma]]. --- # Proofs and examples ## Examples of contour integrals   #MathematicalFoundations/Analysis/ComplexAnalysis/Integrals