Abstract:
We construct a family of mappings $f=f(z,\tau)$, $\tau\in[0, \tau_0]$. When $\tau$ is fixed, the mapping $f$ translates the half-plane into a strip with a cut (the length of the cut depends on the parameter $\tau$) along the ray $\gamma$ going to infinity. The cut forms zero angles with the strip boundary. The decomposition of the governing function $\lambda(\tau)$ of the Loewner equation at the point $\tau = 0$, $\tau > 0$ generating such a family of regions is obtained. We formulate a hypothesis about the behavior of the control function generating a cut emerging from the zero corner of some single-connected region along the arc of a circle. The hypothesis is tested on one particular case.
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