# contour integral

Let $f$ be a complex-valued function defined on the image of a curve (http://planetmath.org/Curve) $\alpha $: $[a,b]\to \u2102$, let $P=\{{a}_{0},\mathrm{\dots},{a}_{n}\}$ 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 ${a}_{i-1}\u2a7d{t}_{i}\u2a7d{a}_{i}$, 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

$${\int}_{\alpha}f(z)\mathit{d}z:={\int}_{a}^{b}f(\alpha (t))\mathit{d}\alpha (t)$$ |

## Notes

(i) If $\mathrm{Im}(\alpha )$ 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 $[a,b]\to {\mathbb{R}}^{2}$. 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 $\mathrm{Im}(\alpha )$ we define the contour integral of $f$ along $\alpha $, denoted by

$${\int}_{\alpha}f(z)\mathit{d}z$$ |

by

$${\int}_{\alpha}f(z)\mathit{d}z={\int}_{a}^{b}f(z(t))\mathit{d}z(t)$$ |

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

$${\int}_{\alpha}f\mathit{d}z={\int}_{a}^{b}f(z(t)){z}^{\prime}(t)\mathit{d}t.$$ |

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 |