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| Cauchy’s Integral Formula |
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| Cauchy’s Integral Theorem |
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| Generalisation of Cauchy’s Integral Formula |
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| Solved Numericals |
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Cauchy’s Integral Formula
Cauchy's integral formula is a fundamental result in complex analysis, a branch of mathematics. It asserts that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk. This formula provides integral representations for all derivatives of a holomorphic function. Named after Augustin-Louis Cauchy, this theorem is a cornerstone of complex analysis.Simply Connected Region
A region is termed as simply connected if, for any closed curve C within the region, all points inside C are also contained within the region. This implies that the region has no holes or isolated boundary points.
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.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.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 TheoremIf 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
Q1. 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:
f(z) is a polynomial of degree 101 so number of zeros of f(z) is 101
Hence using Cauchy's argument principle,
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 = 
∴ 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) = 
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 -
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FAQs on Cauchy’s integral theorem and integral formula - Engineering Mathematics for Mechanical Engineering
| 1. What is Cauchy’s Integral Formula? |  |
Ans. Cauchy’s Integral Formula states that if \( f(z) \) is a function that is continuous on and inside a simple closed curve \( C \), and has a derivative at every point inside \( C \), then for any point \( a \) inside \( C \), the value of \( f(a) \) can be expressed in terms of an integral around \( C \).
| 2. How is Cauchy’s Integral Formula different from Cauchy’s Integral Theorem? | |
Ans. Cauchy’s Integral Formula is used to calculate the value of a function at a point inside a closed curve \( C \), while Cauchy’s Integral Theorem states that if a function is analytic inside and on a simple closed curve \( C \), then the line integral of that function around \( C \) is equal to zero.
| 3. What is the significance of Cauchy’s Integral Formula in Mechanical Engineering? | |
Ans. In Mechanical Engineering, Cauchy’s Integral Formula is often used in the analysis and design of systems involving fluid flow, heat transfer, and stress distribution. It helps in calculating complex integrals to determine important parameters in engineering applications.
| 4. How is Cauchy’s Integral Formula applied in real-world engineering problems? | |
Ans. Cauchy’s Integral Formula is applied in real-world engineering problems to analyze the stress distribution in mechanical components, predict the flow of fluids in pipes and channels, and determine the temperature distribution in heat transfer systems. It helps engineers make informed decisions based on mathematical models.
| 5. What are some common misconceptions about Cauchy’s Integral Formula and Theorem? | |
Ans. One common misconception is that Cauchy’s Integral Formula and Theorem are only applicable in theoretical mathematics and have no practical engineering applications. In reality, these concepts play a crucial role in solving complex engineering problems and optimizing design processes.
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