2(z + 1)² (z + 3)4 2n (2-1)m {z |z| = 2}. What is the winding number of the curve f(C) around the origin? For positive integers m and n, let f(z) = = Let C denote the circle
2(z + 1)² (z + 3)4 2n (2-1)m {z |z| = 2}. What is the winding number of the curve f(C) around the origin? For positive integers m and n, let f(z) = = Let C denote the circle
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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---
### Understanding Complex Functions and Winding Numbers
**Problem Statement:**
For positive integers \( m \) and \( n \), let
\[
f(z) = \frac{2(z+1)^2 (z+3)^4}{z^n (z-1)^m}.
\]
Let \( C \) denote the circle
\[
\{ z : |z| = 2 \}.
\]
What is the winding number of the curve \( f(C) \) around the origin?
**Explanation:**
In this problem, we are analyzing a complex function \( f(z) \) given by the product and quotient of polynomial terms. The function is expressed as a ratio of two polynomials:
- **Numerator:** \( 2(z+1)^2 (z+3)^4 \)
- **Denominator:** \( z^n (z-1)^m \)
You'll need to consider the properties of \( f(z) \) as the variable \( z \) moves along the circle \( C \), which is defined by the modulus \( |z| = 2 \). This circle in the complex plane has a radius of 2 and is centered at the origin.
The goal is to determine the winding number of the curve \( f(C) \) around the origin. The winding number provides insight into how many times and in what direction the curve \( f(C) \) encircles the origin.
**Approach:**
1. **Understand the Components:** Analyze each component of the function \( f(z) \), especially the zeros and poles (roots of the numerator and denominator, respectively).
2. **Inclusive Points Toward the Origin:** Evaluate whether and how often \( f(C) \) encircles the origin, accounting for the poles and zeros that affect the winding direction and frequency.
3. **Calculate Winds:** The winding number calculation often involves contour integration or applying arguments about the symmetry and path traced by \( f(z) \).
This problem involves applying your knowledge in complex analysis, particularly concepts like holomorphic functions, residues, and winding numbers. Understanding these concepts will aid in solving not just the winding number but also grasping function behavior in the complex plane.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa62f7b35-7db6-46d8-92c3-a45ad2747ea7%2Fa65e8890-5cf5-4967-84d0-5b145b59bd72%2F34wsvs_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Below is the transcription and explanation tailored for an educational website:
---
### Understanding Complex Functions and Winding Numbers
**Problem Statement:**
For positive integers \( m \) and \( n \), let
\[
f(z) = \frac{2(z+1)^2 (z+3)^4}{z^n (z-1)^m}.
\]
Let \( C \) denote the circle
\[
\{ z : |z| = 2 \}.
\]
What is the winding number of the curve \( f(C) \) around the origin?
**Explanation:**
In this problem, we are analyzing a complex function \( f(z) \) given by the product and quotient of polynomial terms. The function is expressed as a ratio of two polynomials:
- **Numerator:** \( 2(z+1)^2 (z+3)^4 \)
- **Denominator:** \( z^n (z-1)^m \)
You'll need to consider the properties of \( f(z) \) as the variable \( z \) moves along the circle \( C \), which is defined by the modulus \( |z| = 2 \). This circle in the complex plane has a radius of 2 and is centered at the origin.
The goal is to determine the winding number of the curve \( f(C) \) around the origin. The winding number provides insight into how many times and in what direction the curve \( f(C) \) encircles the origin.
**Approach:**
1. **Understand the Components:** Analyze each component of the function \( f(z) \), especially the zeros and poles (roots of the numerator and denominator, respectively).
2. **Inclusive Points Toward the Origin:** Evaluate whether and how often \( f(C) \) encircles the origin, accounting for the poles and zeros that affect the winding direction and frequency.
3. **Calculate Winds:** The winding number calculation often involves contour integration or applying arguments about the symmetry and path traced by \( f(z) \).
This problem involves applying your knowledge in complex analysis, particularly concepts like holomorphic functions, residues, and winding numbers. Understanding these concepts will aid in solving not just the winding number but also grasping function behavior in the complex plane.
---
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