The mapping function for transforming the exterior of the unit circle with unequal cracks on both sides (in the $\zeta$-plane) to the exterior of the unit circle (in the $z$-plane) is as follows:
$${\omega}(\zeta ) = M(\zeta + \frac{1}{\zeta }) + N + Q(\zeta )$$
where $$\left\{ \begin{array}{l} Q(\zeta ) = \zeta \sqrt {\frac{1}{{{\zeta ^2}}}({{(M(\zeta + \frac{1}{\zeta }) + N)}^2} - 1)} \\ M = \frac{1}{8}(2 + {A_R} + {A_L} + \frac{1}{{1 + {A_R}}} + \frac{1}{{1 + {A_L}}})\\ N = \frac{1}{4}({A_R} - {A_L} + \frac{1}{{1 + {A_R}}} - \frac{1}{{1 + {A_L}}}) \end{array} \right.$$ Here, $A_R$ and $A_L$ are the length parameters.
When applying the mapping function, I observed incorrect mappings for certain $A_R$ and $A_L$ values. How to determine the parameter range in which the mapping function satisfies conformality? Or how should I evaluate the singularity of this function?
Thank you very much for your guidance.
