Contour integral

A method of evaluating certain integrals along paths in the complex plane

In complex analysis, contour integration is a way to calculate an integral around a contour on the complex plane. In other words, it is a way of integrating along the complex plane.

More specifically, given a complex-valued function and a contour , the contour integral of along is written as or .[1][2]

Calculating contour integrals with the residue theoremEdit

For a standard contour integral, we can evaluate it by using the residue theorem. This theorem states that


where   is the residue of the function  ,   is the contour located on the complex plane. Here,   is the integrand of the function, or part of the integral to be integrated.

The following examples illustrate how contour integrals can be calculated using the residue theorem.

Example 1Edit


Example 2Edit


Multivariable contour integralsEdit

To solve multivariable contour integrals (contour integrals on functions of several variables), such as surface integrals, complex volume integrals and higher order integrals, we must use the divergence theorem. For right now, let   be interchangeable with  . These will both serve as the divergence of the vector field written as  . This theorem states that:


In addition, we also need to evaluate  , where   is an alternate notation of  . [1]The divergence of any dimension can be described as


The following examples illustrate the use of divergence theorem in the calculation of multivariate contour integrals.

Example 1Edit

Let the vector field   be bounded by the following conditions


The corresponding double contour integral would be set up as such:


We now evaluate   by setting up the corresponding triple integral:


From this, we can now evaluate the integral as follows:


Example 2Edit

Given the vector field   and   being the fourth dimension. Let this vector field be bounded by the following:


To evaluate this, we use the divergence theorem as stated before, and evaluate   afterwards. Let  , then:


From this, we now can evaluate the integral:


Thus, we can evaluate a contour integral of the fourth dimension.

Related pagesEdit


  1. 1.0 1.1 "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-09-18.
  2. "Contour Integration | Brilliant Math & Science Wiki". Retrieved 2020-09-18.