# contour integral

Let $f$ be a complex-valued function defined on the image of a curve (http://planetmath.org/Curve) $\alpha$: $[a,b]\rightarrow\mathbb{C}$, let $P=\{a_{0},...,a_{n}\}$ be a partition (http://planetmath.org/Partition3) of $[a,b]$. We will restrict our attention to contours, i.e. curves for which the parametric equations consist of a finite number of continuously differentiable arcs. If the sum

 $\sum_{i=1}^{n}f(z_{i})(\alpha(a_{i})-\alpha(a_{i-1})),$

where $z_{i}$ is some point $\alpha(t_{i})$ such that $a_{i-1}\leqslant t_{i}\leqslant a_{i}$, converges as $n$ tends to infinity and the greatest of the numbers $a_{i}-a_{i-1}$ tends to zero, then we define the contour integral of $f$ along $\alpha$ to be the integral

 $\int_{\alpha}f(z)dz:=\int_{a}^{b}f(\alpha(t))d\alpha(t)$

## Notes

(i) If $\operatorname{Im}(\alpha)$ is a segment of the real axis, then this definition reduces to that of the Riemann integral of $f(x)$ between $\alpha(a)$ and $\alpha(b)$.

(ii) An alternative definition, making use of the Riemann-Stieltjes integral, is based on the fact that the definition of this can be extended without any other changes in the wording to cover the cases where $f$ and $\alpha$ are complex-valued functions.

Now let $\alpha$ be any curve $[a,b]\rightarrow\mathbb{R}^{2}$. Then $\alpha$ can be expressed in terms of the components $(\alpha_{1},\alpha_{2})$ and can be associated with the complex-valued function

 $z(t)=\alpha_{1}(t)+i\alpha_{2}(t).$

Given any complex-valued function of a complex variable, $f$ say, defined on $\operatorname{Im}(\alpha)$ we define the contour integral of $f$ along $\alpha$, denoted by

 $\int_{\alpha}f(z)dz$

by

 $\int_{\alpha}f(z)dz=\int_{a}^{b}f(z(t))dz(t)$

whenever the complex Riemann-Stieltjes integral on the right exists.

(iii) Reversing the direction of the curve changes the sign of the integral.

(iv) The contour integral always exists if $\alpha$ is rectifiable and $f$ is continuous.

(v) If $\alpha$ is piecewise smooth and the contour integral of $f$ along $\alpha$ exists, then

 $\int_{\alpha}fdz=\int_{a}^{b}f(z(t))z^{\prime}(t)dt.$
 Title contour integral Canonical name ContourIntegral Date of creation 2013-03-22 12:51:44 Last modified on 2013-03-22 12:51:44 Owner Mathprof (13753) Last modified by Mathprof (13753) Numerical id 23 Author Mathprof (13753) Entry type Definition Classification msc 30A99 Classification msc 30E20 Synonym complex integral Synonym line integral Synonym curve integral Related topic CauchyIntegralFormula Related topic PathIntegral Related topic Integral Related topic IntegralTransform Related topic RealAndImaginaryPartsOfContourIntegral Defines contour