It states that the integral of an analytic function over a closed contour is zero, provided the function is analytic on and inside the curve.
Theorem says the integral around a closed loop is zero (if analytic). Formula gives the function’s value using a contour integral.
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Cauchy’s Integral Theorem and Formula
Cauchy’s Integral Theorem and Cauchy’s Integral Formula are fundamental results in complex analysis that link the values of analytic functions to contour integrals. The Integral Theorem states that if a function is analytic over a simply connected region, its integral over any closed contour is zero. The Integral Formula goes further, providing the exact value of a function at a point inside a contour using an integral. These tools are powerful for evaluating complex integrals and studying function behavior.
1.0What is Cauchy’s Integral Theorem?
Cauchy’s Integral Theorem states:
If a complex-valued function f(z) is analytic (holomorphic) on and inside a simple closed curve C, then the line integral of f(z) over C is zero.
Mathematically:∫cf(z)dz=0
Conditions:
f(z) must be analytic on and inside C
C must be a simple (non-intersecting), closed curve traversed in the positive (counter-clockwise) direction
2.0Cauchy Integral Formula
Cauchy's Integral Formula gives the value of an analytic function inside a closed curve in terms of an integral over that curve.
Statement:
Let f(z) be analytic inside and on a simple closed curve C, and let a be a point inside C. Then,
f(a)=2πi1∫cz−af(z)dz
This formula not only evaluates the function at a point but also is the foundation for deriving higher-order derivatives and solving complex integrals.
3.0What is Cauchy's Integral Formula for Derivatives?
Cauchy's formula can be extended to find derivatives of an analytic function:
General Form: fn(a)=2πin!∫c(z−a)n−1f(z)dz
This is known as the Cauchy's Integral Formula for Derivatives. It helps evaluate any order derivative of a function f(z) at a point a, using only values of f(z) around a contour.
Related Video:
4.0Application of Cauchy's Integral Theorem and Formula
Solving complex integrals quickly
Verifying analyticity of a function
Evaluating real definite integrals using contour integration
Deriving Taylor and Laurent series
Computing residues in advanced complex analysis
5.0What is Cauchy’s Integral Problem?
Cauchy's integral problem typically refers to problems involving evaluating contour integrals using Cauchy’s Theorem or Formula. These are common in exams like GATE, JEE Advanced, and undergraduate analysis.
Example: Evaluate ∫cz−2exdz, where C is a circle centered at the origin with radius 3.
Solution:
Since f(z)=ex is analytic everywhere, and z = 2 lies inside C, by Cauchy’s Integral Formula:
6.0Solved Examples on Cauchy’s Integral Theorem and Formula
Example 1: Evaluate ∫cz−1z2+3z+2dz, where C is a circle of radius 2 centered at origin.
Solution:
Since z = 1 is inside C, and f(z)=z2+3z+2 is analytic,
∫cz−1f(z)dz=2πi⋅f(1)=2πi⋅(1+3+2)=2πi⋅6=12πi
Example 2: Find f''(0) where f(z)=z2+11, and C is the circle |z| = 2.
Solution: Use the derivative form of Cauchy’s Integral Formula:
f"(0)=2πi2!∫cz3f(z)dz
f"(0)=2πi2∫c(z2+1)z31dz
Now use residue theorem or partial fractions to evaluate.
Example 3: Evaluate the integral ∫cz−1ezdz where C is the circle |z| = 2 traversed counterclockwise.
Solution: This is directly an application of Cauchy’s Integral Formula. The integrand is of the form z−af(z), where:
f(z)=ex, analytic in and on C
a = 1, and 1ϵInt(C)
By Cauchy’s Integral Formula:
∫cz−af(z)dz=2πif(a)
⇒∫cz−aezdz=2πie1=2πie
Answer:2πie
Example 4 : Let f(z)=z2+11. Evaluate ∫cf(z)dz where C is the circle |z| = 2.
Solution:
Factor the denominator:f(z)=(z−i)(z+i)1
Poles at z = i and z = -i, both inside the circle |z| = 2. Since f(z) is not analytic inside C, Cauchy's theorem does not apply directly. Instead, compute via Residue Theorem or decompose:
Let’s use Partial Fraction Decomposition:
f(z)=2i1(z−i1−z+i1)
Apply Cauchy’s Integral Formula for each:
∫cz−i1dz=2πi,∫cz+i1dz=2πi
So, ∫cf(z)dz=2i1(2πi−2πi)=0
Answer: 0
Example 5 : Evaluate the integral ∫czsinzdz where C is any simple closed contour enclosing the origin.
Solution:
f(z)=zsinz is analytic at all points including at z = 0, since:
x→0limzsinz=1
Thus, f(z) is entire.
By Cauchy’s Theorem, the integral over a closed contour is:
Answer: 0
Example 6 : Let f(z)=z3, and let C be the circle |z| = 2. Evaluate f"(0)=2πi2!∫cz3f(z)dz
Solution:
Cauchy’s Integral Formula for derivatives states:
f"(a)=2πin!∫c(z−a)n+1f(z)dz
Given:
f(z)=z3
a = 0
n = 2
Apply:
f"(0)=2πi2!∫cz3z3dz
⇒f"(0)=2πi2!∫c1dz
⇒f"(0)=2πi2.0=0
(Since the integral of a constant over a closed contour is 0.)
Answer: 0
Example 7: Evaluate: ∫cz1dzwhere C is the unit circle |z| = 1 oriented counterclockwise.
Solution:
This is a standard result:∫cz1dz=2πi
Answer: 2πi
7.0Practice Questions on Cauchy’s Integral Theorem and Formula
Evaluate ∫cz−iz3+2zdz, where C is a circle of radius 2 centered at origin.
Use Cauchy’s Integral Formula to evaluate ∫c(z−a)2ezdz.
Prove that the integral of any analytic function around a closed curve is zero using Cauchy's Integral Theorem.
State and prove the Cauchy's Integral Formula for Derivatives.
Find f(3)(0) for f(z)=z2sinz
8.0Sample Questions on Cauchy’s Integral Theorem and Formula
Q1: State Cauchy’s Integral Formula.
Ans:f(a)=2πi1∫cz−af(z)dz. It evaluates the function f(a) using an integral over the curve C.
Q2: What is Cauchy’s Integral for Derivatives?
Ans: f(n)(a)=2πin!∫c(z−a)n+1f(z)dz
Table of Contents
1.0What is Cauchy’s Integral Theorem?
2.0Cauchy Integral Formula
3.0What is Cauchy's Integral Formula for Derivatives?
4.0Application of Cauchy's Integral Theorem and Formula
5.0What is Cauchy’s Integral Problem?
6.0Solved Examples on Cauchy’s Integral Theorem and Formula
7.0Practice Questions on Cauchy’s Integral Theorem and Formula
8.0Sample Questions on Cauchy’s Integral Theorem and Formula