Cauchy's integral formula states that if a function f(z) is analytic within and on a simple closed contour C enclosing a point z0, then the contour integral of f(z) around C equals 2πi times the value of the function at z0. The formula can be used to evaluate the value of an analytic function at a point inside the contour by integrating the function around the contour and dividing by the difference between the point and the integration variable. An example demonstrates applying the formula to evaluate the contour integral of the function z/(z2+1) around the unit circle.

1. PROJECT:-2
CAUCHY’S INTEGRAL FORMULA
PRESENTED BY :-
NAME – BISWAJEET BEHERA
REG.NO - 210101120045
SEC - A
BRANCH :-COMPUTER SCIENCE AND ENGINEERING
GUIDED BY :- Dr. BANITA MALLIK
CENTURION
UNIVERSITY

2. CAUCHY’S INTEGRAL FORMULA :-
Also known as CAUCHY’S SECOND THEOREM.
If f(z) is analytic in a simple connected domain D for any point 𝑧0 in D and for a
simple closed path C in D that encloses 𝑧0 , then
∫C f(z)/(z-z0) dz = 2πi X f(Zo)
Simple connected domain :-
A simple connected domain is a domain in which every simple path contains points
of D only.

3.  This means that the value of a function at a point inside a contour can be
calculated by integrating the function over the contour and dividing by the
difference between the point and the integration variable.
STEPS ARE AS FOLLOWS :-
 Find the analytic function f(z) that you want to integrate over the contour C.
 Identify a point z0 inside C where f(z) has a singularity (pole). This can be done
by finding the roots of the denominator of f(z).
 Calculate the residue of f(z) at z0 using the formula for residues.
 Apply Cauchy's Integral Formula to evaluate the integral ∫(C) f(z) dz by
substituting the value of Res(f, z0) into the formula.

4. Here is an example of using Cauchy's Integral Formula to evaluate a contour integral:
Example: Evaluate the integral ∫(C) z/(z^2 + 1) dz, where C is the unit circle |z| = 1.
Solution:
• We have f(z) = z/(z^2 + 1), which is analytic in the complex plane except at z = i
and z = -i.
• The singularities of f(z) inside the unit circle are at z = i and z = -i. We choose z0 =
i since it lies inside the unit circle.
• The residue of f(z) at z0 = i can be found using the formula:
• Res(f, i) = lim(z → i) (z - i) f(z) = lim(z → i) (z - i) z/(z^2 + 1) = 1/(2i)
• We apply Cauchy's Integral Formula to obtain:
∫(C) z/(z^2 + 1) dz = 2πi Res(f, i) = 2πi (1/(2i)) = πi
• Therefore, the value of the contour integral is πi.

5. THANK YOU 