Math Problem Statement

How the constant u0 is evidently 0

How the constant u0 is evidently 0

Solution

From the provided image, the notes discuss extending a function vv to a disk, and its conjugate harmonic function u0u_0, which is related to the reflection of a certain analytic function. Let's break it down to understand why the constant u0u_0 is evidently zero.

  1. Conjugate Harmonic Function: The function vv has a conjugate harmonic function u0u_0, which together with vv forms a harmonic pair. That is, u0u_0 and vv satisfy the Cauchy-Riemann equations in their domain of definition.

  2. Reflection across the Real Axis: The function u0u_0 is symmetric with respect to the real axis in the sense that u0(z)=Re(f(z))u_0(z) = \text{Re}(f(z)) where f(z)f(z) is analytic. The real part of f(z)f(z) is harmonic, and therefore, u0(x,y)u_0(x,y) and v(x,y)v(x,y) satisfy the harmonic conditions.

  3. Boundary Conditions and Cauchy-Riemann: It is given that u0(x,y)u0(x,y)=0u_0(x,y) - u_0(x,-y) = 0. This reflects the fact that u0u_0 is symmetric with respect to the real axis. Moreover, we see from the equations that: u0x=0andu0y=0\frac{\partial u_0}{\partial x} = 0 \quad \text{and} \quad \frac{\partial u_0}{\partial y} = 0 on the real axis. These derivatives being zero on the real axis implies that u0u_0 vanishes on this axis.

  4. Conclusion that u0u_0 is Constant: Since u0u_0 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 u0u_0 is evidently zero because:

  • u0u_0 satisfies the Cauchy-Riemann equations.
  • u0u_0 vanishes on the real axis.
  • By the reflection principle and harmonicity, u0u_0 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:

  1. What are the Cauchy-Riemann equations and how do they relate to harmonic functions?
  2. What is the reflection principle in complex analysis?
  3. How does harmonicity constrain the behavior of solutions to boundary value problems?
  4. Why must a harmonic function that vanishes at a point be zero everywhere under certain conditions?
  5. What role does the real axis play in this construction for u0u_0?

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