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I am looking for a conformal mapping from the unit disc to itself $F:D\to D$ characterized by

  • it sends a point $a\in D$ to the origin;
  • it is the identity in $\partial D$.

I am aware of the exsitance of the Möbius transformations (see this post), that send $\partial D$ to $\partial D$: $F(\partial D)=\partial D$. However, the property I am looking for is pointwise along the boundary: $F(z)=z$ for all $z\in\partial D$.

The question arises from a physics context. I want to find a mapping that sends the potential lines of a point charge at $a\in D$ to the potential lines of a point charge at the origin, provided that the potential vanishes in $\partial D$.

Víctor
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    I don't think that can be done: If $F$ is such a function, then $g(z):=F(z)-z$ is zero on $\partial D$, and by the maximum principle, then $g=0$ in $D$. – Jose27 Mar 16 '23 at 17:29
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    @Jose27 thanks, but how would you apply the maximum principle for such function? Are conformal mappings harmonic? – Víctor Mar 16 '23 at 17:49
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    @Víctor If $f = u+iv$ is holomorphic, then $u,v$ are harmonic. – Nuke_Gunray Mar 16 '23 at 17:54
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    @Nuke_Gunray I see, thank you very much. – Víctor Mar 16 '23 at 17:55
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    Another way to think about it is every univalent map from $D\longrightarrow D$ is Möbius transformation (of a particular form) and if a Möbius transformation has 3 distinct fixed points on the extended complex plane, then it must be the identity. – user8675309 Mar 16 '23 at 22:09

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