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Contour integral

From Simple English Wikipedia, the free encyclopedia

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 theorem[change | change source]

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 1[change | change source]

Example 2[change | change source]

Multivariable contour integrals[change | change source]

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 1[change | change source]

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 2[change | change source]

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 pages[change | change source]

References[change | change source]

  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". brilliant.org. Retrieved 2020-09-18.