# Contour Integral **The [[integral]] of a [[Complex Number|complex-valued]] function along contours in the complex plane.** ![[ContourIntegral1.svg|600]] ![[ContourIntegral2.svg|500]] *A contour integral; the contour of integration in the complex plane (above) and the modulus in $\mathbb{R}^{3}$ of the integrand function (below) is shown* A *contour integral* is the integral of a complex-valued function as evaluated along *contours* in the complex plane. They are a generalisation of [[Line Integral|line integrals]] to complex values. > [!NOTE] Contour > A *contour* is a directed curve in the complex plane consisting of a finite sequence of smooth arcs $\gamma_{1},\dots,\gamma_{n}$ such that the end point of some $\gamma_{i}$ is the start point of $\gamma_{i+1}$ for all $1\le i< n$ A *simple contour* is a contour which does not intersect itself. A *closed contour* is a contour in which the end point is equal to the start point. It is also possible for a contour to be simple and closed in which case it is also known as a *Jordan curve*. A Jordan curve is *positive* if the interior of the curve is on the left as it is traversed. ^39e92f Contours are almost always *parameterised* in terms of a single variable which converts the domain of integration into a closed interval on the real line. The value of a contour integral *does not depend* on the parameterisation. The notion of [[Antiderivative|antiderivatives]] and the [[fundamental theorem of calculus]] are both extendable to complex-valued functions, the latter of which dictates that the contour integral of a function *does not depend* on the path between any two endpoints. Contour integrals can be directly evaluated similarly to the definite integral of real-valued functions. However, there are additional applicable theorems that dictate the value of contour integrals in certain situations. ## Definition > [!NOTE] Contour integral > For some function $f(z)$, the *contour integral* along a smooth contour $\gamma$ is defined as > $\int_{a}^{b}f(\gamma(t))\gamma'(t)\;dt$ > where $\gamma(t):[a,b]\rightarrow\mathbb{C}$ is a *parameterisation* of the contour such that $\gamma(a)$ and $\gamma(b)$ are the endpoints of $\gamma$. The parameterisation of the contour can be expressed in terms of two functions $u:[a,b]\rightarrow\mathbb{R}$ and $v:[a,b]\rightarrow\mathbb{R}$ such that $\gamma(t)=u(t)+i\;v(t)$ where $a\le t\le b$. ## Notation The notation for a contour integral of the function $f(z)$ with respect to $z$ over the contour $\gamma$ is $\int_{\gamma}f(z)\;dz$ If the contour is closed, then the integral symbol may be notated with a circle. $\oint_{\gamma}f(z)\;dz$ ## Theorems ### Fundamental Theorem of Contour Integration > [!NOTE] Fundamental Theorem of Contour Integration > The *fundamental theorem of contour integration* states that if $f$ is a continuous function defined on a domain $D$ and $F$ is its antiderivative on $D$, then for any contour $\gamma$ with endpoints $z_{0}$ and $z_{1}$, > $\int_{\gamma}f(z)=F(z_{1})-F(z_{0})$ The fundamental theorem of contour integration is an extension of the [[Fundamental Theorem of Calculus#Second part|second part]] of the fundamental theorem of calculus and implies that as long as some function $f$ has an antiderivative on a domain, then the contour integral over the domain *does not depend* on the path between two given endpoints. ### Additional theorems - [[Cauchy's Integral Theorem]] - [[Cauchy's Integral Formula]] - [[Residue Theorem]] ## Examples > [!NOTE]- Direct evaluation > Suppose the integral > $\int_{\gamma}\bar{z}\;dz$ > is to be evaluated, where $\gamma=t+it^{2}$ between $z=0$ and $z=1+i$. > > The parameterisation of the contour is given as $\gamma(t)=t+i\;t^{2}$ with $a=0$ and $b=1$. The derivative of the contour with respect to the parameter is $\gamma'(t)=1+2i\;t$. > > Thus, > $\begin{align} > \int_{\gamma}\bar{z}\;dz&=\int_{0}^{1}(t-i\;t^{2})(1+2i\;t)\;dt\\ > &=\int_{0}^{1}2t^{3}+i\;t^{2}+t\\ \\ > &=\left[\frac{t^{4}}{2}+\frac{i\;t^{3}}{3}+\frac{t^{2}}{2}\right] \\ > &=1+\frac{1}{3}i > \end{align}$ #unfinished