Conformal Mapping with Neuman Conditions

Question

I'm trying to solve the added question with Neumann conditions instead of Dirichlet. Also, I could not figure it out how to obtain $x^2$ at (4).

I would be pleased if anyone could help. Thank you.

Answer 1

1. The curve $xy=3$ is transformed to $v=3$ in u-v plane. Eq.(4) is obtained by solving algebra equation $x^4-u x^2-9=0$.
2. Suppose you have a Neumann conditions $\psi_y=30$ and $y \psi_x+x \psi_y=10 e^x$, then you will have $\psi_v=30$, and $ \psi_v=10 e^x$ with $x$ in Eq.(4).