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.