Cauchy’s integral formula is a central statement in complex analysis in mathematics. It expresses that a holomorphic function defined on a disk is determined entirely by its values on the disk boundary. For all derivatives of a holomorphic function, it provides integration formulas. Also, this formula is named after Augustin-Louis Cauchy. In this article, you will learn Cauchy’s Integral theorem and the formula with the help of solved examples.
Before going to the theorem and formula of Cauchy’s integral, let’s understand what a simply connected region is.
Simply connected Region:
A connected region is called a simply connected region, if all the interior points of a closed curve C are illustrated in region D, are also the points of region D.
Cauchy’s Integral Theorem
Statement: If f(z) is an analytic function in a simply-connected region R, then ∫c f(z) dz = 0 for every closed contour c contained in R.
(or)
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
If a complex function f(z) is analytic within and on a closed contour c inside a simply-connected domain, and if z0 is any point in the middle of C, then
Here, the integral should be taken in the positive sense around c.
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Generalisation of Cauchy’s Integral Formula
If f(z) is an analytic function within and on a simple closed curve C and if z0 is any point within c, then
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 Examples
Question 1:
Solution:
f(z) = z2/(z – 5)
This function is not analytic at z = 5.
However, this point lies outside the circle defined by |z| = 2.
Therefore, f(z) is an analytic function at all points inside and on the closed curve c.
Thus, by Cauchy’s theorem, we can write ∫c f(z) dz = 0.
Question 2:
Solution:
Given,
1/(z2 + 4)2 = 1/(z2 + 22)2
As we know, i2 = -1.
So, we can write (2)2 as -(2i)2.
Thus,
1/(z2 + 4)2 = 1/[z2 – (2i)2]2
= 1/ [(z – 2i)2 (z + 2i)2]
Let us consider f(z) = 1/(z + 2i)2
So, we can say that f(z) is an analytic function inside C.
Thus, by Cauchy’s integral formula, we can write it as:
Using f’(z) = -2/(z + 2i)3, we get;
= 2πi [-2/ (2i + 2i)3]
= -4πi/ (4i)3
= -4πi/(-64i) {since i3 = -i}
= π/16
Practice Problems
- \(\begin{array}{l}\text{Evaluate }\int_{c}\ \frac{1}{(z^2+9)}dz \text{ where c is the circle |z + 3i| = 2 and |z| = 5.}\end{array} \)
- If “c” is any simple closed curve, evaluate the integral ∫c f(z) dz, if f(z) = cos 8z.
- Evaluate ∫c f(z) dz when f(z) = 8z3 + sin 4z for a simple closed curve c.
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