Contour Integral

Contour Integral definition: It is the method of calculating an integral along the complex numbers plane. A contour integration method is used to determine the contour integral. It is used to evaluate the integrals along the line or plane of real numbers. Contours are basically the set of smooth curves which gives single direction endpoints. Contour Integral works even for a small sized shape of the contour and expresses the integral of real value.

Line integrals definition: These are also known as curvilinear or path integrals and are used to find out the surface areas of a three dimensional curve. It uses the contour integral for its determination.

Line Integrals formula:

It can be used for both vector fields as well as scalar fields.

For scalar field;

For a<b, C is the bijective parameter of the smooth curve.

∫C f(r) ds = ∫[a,b] f[r(t)] |r’(t)| dt

Here, the end points of the curve will be given by r(a) and r(b).

For vector field; 

A line integral in the direction of “r” along a smooth curve is represented as;

∫C F(r). dr = ∫[a,b] F[r(t)] . r’(t)dt

Fundamental concepts of line integral:

A line integral is the definite integral where the function f(x,y,z) has been integrated along the path of a smooth curve.

Applications of line integral:

The applications of line integral are;

• It is used in electromagnetics by calculating the work done by some charged particle forming curve under the influence of an electrical field.

• In mathematics, it is used to determine the surface of smooth curves.

• It is also used in classical mechanics by calculating the work done of the particle of m mass under the influence of gravitational field.

• It is used in Faraday’s law of magnetic induction.

• It is used in Ampere’s law.

• It is used in physics to calculate the wire mass and moment of inertia.

Contour integration:

Contour integration is the method of finding the contour integrals along the plane. This method includes following applications;

• In the Residue theorem

• In the Cauchy integral formula

• Along the curve in a contour plane, to find out the direct integration of complex functions.

The methods for calculating the contour integrals are same as that of the line integrals i.e. by either direct evaluation of real integral, by substitution of parameters or by parameterizing the contour.

Residue theorem

The residue theorem states that;

where Res represents the residue of f(z).

By using this theorem, we can determine;

Applications of integral theorems:

The applications of integral theorems are as follows;

• It is used to describe the real value integrals.

• To divide the contour into the imaginary part and the real part.

• To find the whole integral of the contour(Cauchy integral formula).

• In the integration around a small circle.

• To determine the integral of the complex planes.

Integral interpretation:

An integral interpretation are the special functions which involve the contour integrals which are important for numerical evaluations, complex equations of functions f(x,y,z) or analytical continuation.

Multivariable contour integrals:

Multivariable contour integrals include high order integrals, complex volume integrals and surface integrals. Divergence theorem is used for this integral. Divergence theorem is also known as Gauss’s theorem or Ostrogradsky’s theorem. This theorem states;

Conclusion:

Contour integrals is the integral by which we can find the surface area of the complex functions along the plane. It helps in Cauchy integral formula, Faraday’s law or residue theorem. It is also used to find the complex equations, in parameterizing the contour. It describes the integrals of real value and divides the contour into the imaginary part and real part of the integer. It is also used to determine the function f(x,y,z) of the complex integral along the smooth plane. It is applicable in classical mathematics, in determining the works done under the influence of gravitational or electrical field.