Contour integral

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 formula_1 and a contour formula_2, the contour integral of formula_1 along formula_2 is written as formula_5 or formula_6.
Calculating contour integrals with the residue theorem.
For a standard contour integral, we can evaluate it by using the residue theorem. This theorem states that 
formula_7
where formula_8 is the residue of the function formula_9, formula_2 is the contour located on the complex plane. Here, formula_9 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.
Multivariable contour integrals.
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 formula_14 be interchangeable with formula_15. These will both serve as the divergence of the vector field written as formula_16. This theorem states that:
formula_17
In addition, we also need to evaluate formula_18, where formula_19 is an alternate notation of formula_20. 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.
Let the vector field formula_22 be bounded by the following conditions
The corresponding double contour integral would be set up as such:
We now evaluate formula_24 by setting up the corresponding triple integral:
From this, we can now evaluate the integral as follows:
formula_26
Example 2.
Given the vector field formula_27 and formula_28 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 formula_24 afterwards. Let formula_31, then:
From this, we now can evaluate the integral:
Thus, we can evaluate a contour integral of the fourth dimension.