Methods of contour integration Guide, Meaning , Facts, Information and Description

In complex analysis, the evaluation of integrals of real-valued functions along intervals on the real line, is not readily found with certain integrands and methods involving only real variables. Complex analysis methods described below give means of calculating these real-valued integrals by means of contour integrals in the complex plane.

These methods include

• direct integration of a complex-valued function along a curve in the complex plane (a contour)
• application of the Cauchy integral formula
• application of the residue theorem

One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums.

Table of contents

1 Direct methods 1.1 Example 2 Applications of integral theorems 2.2 Example 3 See also

Direct methods

Direct methods involve the calculation of the integral by means of methods similar to those in calculating line integrals in several-variable calculus. This means that we use the following method:

• parametrizing the contour

The contour is parametrized by a differentiable complex-valued function of real variables, or the contour is broken up into pieces and parametrized seperately

• substitution of the parametrization into the integrand

Substituting the parametrization into the integrand transforms the integral into an integral of one real variable.

• direct evaluation

The integral is evaluated in a method akin to a real-variable integral.

Example

A fundamental result in complex analysis is that the integral around the contour C which is the unit circle (or any Jordan curve about 0) of z^-1 is 2πi. Let us evaluate the integral:

In evaluating this integral, we use the unit circle |z| = 1 as our contour, which we can parametrize by γ(t) = e^it, with t ∈ [0, 2π]. Observe that γ'(t) = ie^it. Now, substituting this for z, we have

which is the value of the integral.

Applications of integral theorems

Applications of integral theorems are also often used to evaluate the contour integral along a contour that means that the real-valued integral is calculated simultaneously along with calculating the contour integral.

Integral theorems such as the Cauchy integral formula or residue theorem are generally are used in the following method:

• a specific contour is chosen:

The contour is chosen so that the contour follows the part of the complex plane that describes the real-valued integral, and also encloses singularities of the integrand so application of the Cauchy integral formula or residue theorem is possible

• application of the Cauchy-Goursat theorem

The integral is reduced to only an integration around a small circle about each pole.

• application of the Cauchy integral formula or residue theorem

Application of these integral formula gives us a value for the integral around the whole of the contour.

• division of the contour into a contour along the real part and imaginary part

The whole of the contour can be divided into the contour that follows the part of the complex plane that describes the real-valued integral as chosen before (call it R), and the integral that crosses the complex plane (call it I). The integral over the whole of the contour is the sum of the integral over each of these contours.

• demonstration that the integral that crosses the complex plane plays no part in the sum

If the integral I can be shown to be zero, or if the real-valued integral that is sought is improper, then if we demonstrate that the integral I as described above tends to 0, the integral along R will tend to the integral around the contour R+I.

• conclusion

If we can show the above step, then we can directly calculate R, the real-valued integral.

Example

The integral

(which arises in probability theory as (a scalar multiple of) the characteristic function of the Cauchy distribution) resists the techniques of elementary calculus. We will evaluate it by expressing it as a limit of contour integrals along the contour C that goes along the real line from −a to a and then counterclockwise along a semicircle centered at 0 from a to −a. Take a to be greater than 1, so that the imaginary unit i is enclosed within the curve. The contour integral is

Since e^itz is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator z² + 1 is zero. Since z² + 1 = (z + i)(z − i), that happens only where z = i or z = −i. Only one of those points is in the region bounded by this contour. The residue of f(z) at z = i is

According to the residue theorem, then, we have

The contour C may be split into a "straight" part and a curved arc, so that

and thus

It can be shown that if t > 0 then

Therefore if t > 0 then

A similar argument with an arc that winds around −i rather than i shows that if t < 0 then

and finally we have

(If t = 0 then the integral yields immediately to calculus methods and its value is π.)

See also

• Cauchy integral formula for an example of the application of this integral theorem
• Residue (complex analysis)

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