Contour integral

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Introduction[change | change source]

Contour integration is a way to calculate an integral on the complex plane. In other words, we're just integrating along the complex plane.

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

Where the "Res" of

Where is the residue of the function is also the integrand, or part of the integral to be integrated, and is the contour located on the complex plane.

Now, we can evaluate any contour integral! For example, let's take these:

Example 1[change | change source]

See what we did there? Using the residue theorem, we have evaluated the . Let's try another example:

Example 2[change | change source]

Multivariable Contour Integrals[change | change source]

To solve multivariable (contour integrals with more than one variable to integrate) contour integrals (i.e. 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 denoted as . This theorem states:

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

Let's try an example:

Example 1[change | change source]

Let the vector field and be bounded by the following

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


We now evaluate . While we're at it, let's set up the corresponding triple integral:

From knowing this, we can now evaluate the integral.

Now that we know this, let's try another!

Example 2[change | change source]

For example, let the vector field , and is the fourth dimension. Let this vector field be bounded by the following:

To evaluate this, we must utilize the divergence theorem as stated before, and we must evaluate . For right now, let


From this, we now can evaluate the integral.

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