function holomorphic in 2 except p in an open neighborhood of w, the
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you solve this problem using a triangular keyhole contour as the method?
![### Exercise from Stein & Shakarchi, Chapter 2, Exercise #6
**Problem Statement:**
Let \( \Omega \) be an open subset of \( \mathbb{C} \), and let \( T \subset \Omega \) be a triangle whose interior is also contained in \( \Omega \). Suppose that \( f \) is a function holomorphic in \( \Omega \) except possibly at a point \( w \) inside \( T \). Prove that if \( f \) is bounded in an open neighborhood of \( w \), then
\[ \int_{T} f(z) \, dz = 0. \]
**Detailed Explanation:**
1. **Understanding the Problem:**
- \( \Omega \) is an open subset of the complex plane \( \mathbb{C} \).
- \( T \) is a triangle situated inside \( \Omega \), including its interior.
- The function \( f \) is holomorphic (complex differentiable) in \( \Omega \) except possibly at a single point \( w \) within the triangle \( T \).
- The function \( f \) is bounded in an open neighborhood around the point \( w \).
2. **Objective:**
- Show that the contour integral of the function \( f \) around the boundary of the triangle \( T \) is zero, i.e.,
\[ \int_{T} f(z) \, dz = 0. \]
**Key Concepts:**
- **Holomorphic Function:** A function that is complex differentiable in a neighborhood of every point in its domain.
- **Contour Integral:** An integral computed along a path (or contour) in the complex plane.
- **Bounding Neighborhood:** A region around a point where the function does not exceed a certain value.
**Contextual Explanation:**
- The exercise explores a fundamental concept in complex analysis related to the behavior of holomorphic functions and their integrals over contours.
- This result is related to the **Cauchy-Goursat Theorem**, which asserts that the integral of a holomorphic function around a closed contour is zero if it is holomorphic within the region enclosed by the contour.
**Note to Students:**
- This problem entails proving a specific behavior of a complex function under certain conditions (holomorphicity and boundedness).
- Familiarity with complex integration, properties of holomorphic functions, and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F6805c98e-7efa-4cef-bde3-3470abe27db2%2Fi826yvi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exercise from Stein & Shakarchi, Chapter 2, Exercise #6
**Problem Statement:**
Let \( \Omega \) be an open subset of \( \mathbb{C} \), and let \( T \subset \Omega \) be a triangle whose interior is also contained in \( \Omega \). Suppose that \( f \) is a function holomorphic in \( \Omega \) except possibly at a point \( w \) inside \( T \). Prove that if \( f \) is bounded in an open neighborhood of \( w \), then
\[ \int_{T} f(z) \, dz = 0. \]
**Detailed Explanation:**
1. **Understanding the Problem:**
- \( \Omega \) is an open subset of the complex plane \( \mathbb{C} \).
- \( T \) is a triangle situated inside \( \Omega \), including its interior.
- The function \( f \) is holomorphic (complex differentiable) in \( \Omega \) except possibly at a single point \( w \) within the triangle \( T \).
- The function \( f \) is bounded in an open neighborhood around the point \( w \).
2. **Objective:**
- Show that the contour integral of the function \( f \) around the boundary of the triangle \( T \) is zero, i.e.,
\[ \int_{T} f(z) \, dz = 0. \]
**Key Concepts:**
- **Holomorphic Function:** A function that is complex differentiable in a neighborhood of every point in its domain.
- **Contour Integral:** An integral computed along a path (or contour) in the complex plane.
- **Bounding Neighborhood:** A region around a point where the function does not exceed a certain value.
**Contextual Explanation:**
- The exercise explores a fundamental concept in complex analysis related to the behavior of holomorphic functions and their integrals over contours.
- This result is related to the **Cauchy-Goursat Theorem**, which asserts that the integral of a holomorphic function around a closed contour is zero if it is holomorphic within the region enclosed by the contour.
**Note to Students:**
- This problem entails proving a specific behavior of a complex function under certain conditions (holomorphicity and boundedness).
- Familiarity with complex integration, properties of holomorphic functions, and
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