Cauchy’s integral theorem and integral formula | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

Cauchy’s Integral Formula

Simply Connected Region
A region is termed as simply connected if, for any closed curve C within the region, all points inside $C$C are also contained within the region. This implies that the region has no holes or isolated boundary points.

Cauchy’s Integral Theorem

If f(z) is an analytic function and its derivative f'(z) is continuous at all points within and on a simple closed curve C, then ∫_c f(z) dz = 0.

Cauchy’s Integral Formula

Generalisation of Cauchy’s Integral Formula

Converse of Cauchy’s Integral Theorem

If a complex function f(z) is continuous throughout the simple connected domain D and if ∫_c f(z) dz = 0 for every closed contour c in D, then f(z) will be an analytic function in D.

This theorem is also known as Morera’s theorem.

Solved Numericals

 The value of the integral where C is the circle of radius 2 centred at the origin taken in the anti-clockwise direction is:
Solution:

Q2. What is the value of , where C is the circle |z| = 1 is with positive orientation.
Solution: Given,

C is the circle |z| = 1 is with positive orientation

Let f(z) =  

Putting 2z² - 5z + 2 = 0

⇒ 2z2 - 4z - z + 2 = 0

⇒ 2z(z - 2) - 1(z - 2) = 0

⇒ (z - 2)(2z - 1) = 0

Now z = $\frac{1}{2}$ 1/2 lies inside C 

∴ The value of the given integral is 2π/3.

Q3. The value of  is where C is a circle |z| = 1,
Solution: Given,

C is a circle, lzl=1
Let f(z) = $\frac{𝑧}{2}$
Putting z(z² + 9) = 0
⇒ z = 0, 3i, - 3i
Now, z = 0 lies inside C.

∴ The value of given integral is

Q4. Let γ be the positively oriented circle in the complex plane given by {z ∈ ℂ: |z – 1| = 1}.  Then  equals:
Solution: Given,
Let γ be the positively oriented circle in the complex plane given by {z ∈ ℂ: |z – 1| = 1}.  Then

Cauchy Integral Formula - 
Let f(z) be analytic in a region D and let C be a closed curve in D. If a is any point in D, then 

here  is a winding number. The winding number measures the number of times a path (counter-clockwise) winds around a point. 
We have γ = {z ∈ ℂ: |z – 1| = 1}.

Now using Cauchy Integral formula we get -