Math Problem Statement
How the constant u0 is evidently 0
Solution
From the provided image, the notes discuss extending a function to a disk, and its conjugate harmonic function , which is related to the reflection of a certain analytic function. Let's break it down to understand why the constant is evidently zero.
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Conjugate Harmonic Function: The function has a conjugate harmonic function , which together with forms a harmonic pair. That is, and satisfy the Cauchy-Riemann equations in their domain of definition.
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Reflection across the Real Axis: The function is symmetric with respect to the real axis in the sense that where is analytic. The real part of is harmonic, and therefore, and satisfy the harmonic conditions.
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Boundary Conditions and Cauchy-Riemann: It is given that . This reflects the fact that is symmetric with respect to the real axis. Moreover, we see from the equations that: on the real axis. These derivatives being zero on the real axis implies that vanishes on this axis.
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Conclusion that is Constant: Since vanishes on the real axis and satisfies the Cauchy-Riemann equations, it must be constant throughout the domain. The text concludes that this constant is zero, since a harmonic function that is constant and vanishes at some point must be zero everywhere.
Thus, the constant is evidently zero because:
- satisfies the Cauchy-Riemann equations.
- vanishes on the real axis.
- By the reflection principle and harmonicity, is constant across the entire disk, and since it vanishes at a point, it must be zero everywhere.
Let me know if you'd like further clarification on any point.
Here are five related questions you can explore:
- What are the Cauchy-Riemann equations and how do they relate to harmonic functions?
- What is the reflection principle in complex analysis?
- How does harmonicity constrain the behavior of solutions to boundary value problems?
- Why must a harmonic function that vanishes at a point be zero everywhere under certain conditions?
- What role does the real axis play in this construction for ?
Tip: The symmetry of harmonic functions with respect to the real axis is a common technique used in solving problems in complex analysis and potential theory.
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Math Problem Analysis
Mathematical Concepts
Harmonic Functions
Complex Analysis
Cauchy-Riemann Equations
Reflection Principle
Formulas
Cauchy-Riemann Equations: ∂u/∂x = ∂v/∂y, ∂u/∂y = -∂v/∂x
Definition of Harmonic Functions: Δu = 0
Theorems
Cauchy-Riemann Theorem
Reflection Principle
Uniqueness Theorem for Harmonic Functions
Suitable Grade Level
Advanced Undergraduate or Graduate
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