The mapping function $z=\omega\left(t\right)$, which transforms the outside region of a unit circle (in the $\zeta$-plane) onto the outside region of a closed shape(in the $z$-plane), is expressed as follows (the form of series):
$$ \omega \left( \zeta \right) =a\left( \frac{1}{\zeta}+\zeta \right) +b\zeta \left( c+\sum_{i=1}^4{\sum_{k=1}^{14}{\frac{E_{k,i}}{1-\alpha _kr_i/\zeta}}} \right) $$
where $a, b, c, \alpha_{k}, r_{i}$ and $E_{k,i}$ are constants. The mapping function has one pole on $\zeta = 0$ and 48 poles on $\zeta={\alpha _k}{r_i}$ in the interior of the unit circle. We can calculate the expression:
$$ \frac{1}{2\pi i}\int_r{\frac{\omega \left( t \right)}{\overline{w'\left( t \right) }}}\frac{1}{t-\zeta}dt=-b\sum_{i=1}^4{\sum_{k=1}^{14}{\frac{E_{k,i}\left( \zeta _{k,i} \right) ^2}{\zeta -\zeta _{k,i}}\frac{1}{\overline{w'\left( \delta _{k,i} \right) }}}} $$
$$ {\zeta _{k,i}} = {\alpha _k}{r_i}, {\delta _{k.i}} = \frac{1}{{\overline {{\zeta _{k,i}}} }}\ $$
I encountered some issues in the derivation of the expression above:
- For this mapping function, how to determine the analyticity inside or outside the circle, and how to obtain the pole at $\zeta = 0$?
- Using the Cauchy integral theorem, when substituting the poles, why are we using $\delta_{k.i}$?
- The mapping function $\omega(\zeta)$ can be expanded into a series for approximation, while the $\overline {\omega '(t)} $ isn't expanded. Will this calculation method cause significant errors?
I look forward to your response regarding these questions or an explanation of the derivation process. Any additional information you may require, please let me know. Thank you very much!