Harmonic Function Notes

Harmonic Function Composition

  • A harmonic function composed with a holomorphic function is also harmonic.

Applying the Concept

  • The goal is to convert back to x and y.
  • Given: v = 2xy and u = x^2 - y^2
  • Replace u and v with their respective definitions in terms of x and y.

The Challenge of Intuition

  • The process might not be immediately intuitive.
  • A key question is how to determine the appropriate function (e.g., z^2).

Boundary Conditions and Transformations

  • The goal is to find a transformation that converts the boundary.
  • In this specific case, an angle is transformed into a straight line.
  • The transformation isn't always straightforward (e.g., it's not always just xy).

Example: Handling x^2

  • If the boundary condition involves x^2, the mapping to a straight line may involve an unknown function instead of a simple z^2.
  • If a map is found such that something maps to a straight line, it's important to consider if the mapping function ([f]) is holomorphic.

Holomorphic Requirement

  • If f is not holomorphic, issues may arise.
  • Even if one boundary maps to a straight line, another boundary might map to a more complex shape.

Checking for Holomorphic Functions

  • It's crucial to ensure that the chosen function is holomorphic.
  • If a holomorphic function is found, a replacement can be made accordingly.
  • It's possible to construct a function that fits specific boundary conditions b.

Determining the Initial Function

  • It's necessary to determine the initial holomorphic function.
  • For example, if x^2y = c, it may not be holomorphic.
  • The function must be absolutely holomorphic for this method to work.

Practical Considerations

  • The process of figuring out the function should not be overly difficult.
  • Techniques may vary based on whether the problem is directly centered or involves movement.