Can two conformal mappings be combined?

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Shengfan Bi 2024-4-30

A conformal mapping function

within a complex domain maps the unit circle in the ζ-plane to the

-plane. Similarly, there is another function

. Combining

and

, we can obtain

Can this function also satisfy conformal mapping, and are there any applicable conditions for it to do so?

The following figure is an attempt I made. If feasible, can any smooth and simply connected region be combined with

for conformal mapping?

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