Problem 10.6_2 Use Cauchy's integral formula to evaluate the following integral. $₁21-1 -dz sino (z) J|2|=1 (Z - π/6)3
Problem 10.6_2 Use Cauchy's integral formula to evaluate the following integral. $₁21-1 -dz sino (z) J|2|=1 (Z - π/6)3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 10.6_2**
Use Cauchy's integral formula to evaluate the following integral.
\[
\oint_{|z|=1} \frac{\sin^6(z)}{(z - \pi/6)^3} \, dz
\]
**Explanation:**
This problem involves evaluating a contour integral using Cauchy's integral formula. The integral is taken over a path where \(|z| = 1\). The integrand is \(\frac{\sin^6(z)}{(z - \pi/6)^3}\), which has a pole of order 3 at \(z = \pi/6\).
**Cauchy’s Integral Formula**:
Cauchy’s integral formula allows us to evaluate integrals of functions over closed contours in the complex plane. It states that for a function \(f(z)\) that is analytic inside and on some closed contour \(C\), and for a point \(a\) inside \(C\),
\[
f(a) = \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z - a} \, dz.
\]
The problem here requires the application of a generalized version of Cauchy’s formula for derivatives, due to the presence of a higher-order pole.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F780f9839-f737-4aba-91a2-6210989911b1%2Fdd51d67b-32e5-4e85-9c8a-da04c95d0f15%2Fooqdbre_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 10.6_2**
Use Cauchy's integral formula to evaluate the following integral.
\[
\oint_{|z|=1} \frac{\sin^6(z)}{(z - \pi/6)^3} \, dz
\]
**Explanation:**
This problem involves evaluating a contour integral using Cauchy's integral formula. The integral is taken over a path where \(|z| = 1\). The integrand is \(\frac{\sin^6(z)}{(z - \pi/6)^3}\), which has a pole of order 3 at \(z = \pi/6\).
**Cauchy’s Integral Formula**:
Cauchy’s integral formula allows us to evaluate integrals of functions over closed contours in the complex plane. It states that for a function \(f(z)\) that is analytic inside and on some closed contour \(C\), and for a point \(a\) inside \(C\),
\[
f(a) = \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z - a} \, dz.
\]
The problem here requires the application of a generalized version of Cauchy’s formula for derivatives, due to the presence of a higher-order pole.
Expert Solution
Step 1
By using Cauchy's integral formula we have to evaluate the following integral :
We know the Cauchy integral formula is given by :
Step by step
Solved in 2 steps
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