Overview- Cauchy’s integral theorem
The integration of a function of a complex variable along an open or close curve in the plane of the complex variables is known as complex integration. Cauchy’s integral theorem is the part of complex integration.
The study of complex integration is very useful in engineering physics and mathematics as well.
The concept we use to calculate the centre of mass, centre of gravity, mass moment of inertia of vehicles etc.
We use it in placing a satellite in its orbit to calculate the velocity and trajectory.
Definition of complex line integral
In case of a complex function f(z) the path of the definite integral 
In case the initial point and final point coincide so that c is a close curve, then this integral called contour integral and is denoted by-
If f(z) = u(x, y) + iv(x, y), then since dz = dx + i dy
We have-
It shows that the evaluation of the line integral of a complex function can be reduced to the evaluation of two line integrals of the real function.
Properties of line integral
Linearity-
Sense reversal-
Partitioning of path-
ML – inequality-
Example: Evaluate along the path y = x.
Sol.
Along the line y = x,
dy = dx that dz = dx + i dy
dz = dx + i dx = (1 + i) dx
On putting y = x and dz = (1 + i)dx
Cauchy integral theorem
A function f(z) is analytic and its derivative f’(z) continuous at all points inside and on a closed curve c, then .
Proof:
Suppose the region is R which is close by curve c and let-
By using Green’s theorem-
Replace
So that
Cauchy’s integral formula
Cauchy’s integral formula is-
Where f(z) is an analytic function within and on a closed curve C, a is any point within C.
Example-1: Evaluate by using Cauchy’s integral formula.
Here c is the circle |z – 2| = 1/2
Sol.
here
Find its poles by equating the denominator to zero.
There is one pole inside the circle, z = 2,
So that-
Now by using Cauchy’s integral formula, we get-
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