Finding a conformal map from the intersection of two disks to the unit disk.












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I'm trying to solve a problem which asks me to find a conformal mapping from ${zin mathbb{C}: |z-i|< sqrt2$ and $|z+i|<sqrt2}$ onto the open unit disk.



I'm really new to these and I'm a bit lost because this doesn't resemble the examples I've seen so far.



Obviously the two disks intersect at $±1$, but I don't know where I should be looking to map them to, and while I know usually you'd try to map the region to something familiar and then compose this with another map to the unit disk from there, but I can't see how to map this to something simpler.



As I said, I haven't really done many examples of conformal maps so I'd really appreciate if you could help walk me through this example.



Thanks in advance!






complex-analysis complex-numbers conformal-geometry mobius-transformation







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    I'm trying to solve a problem which asks me to find a conformal mapping from ${zin mathbb{C}: |z-i|< sqrt2$ and $|z+i|<sqrt2}$ onto the open unit disk.



    I'm really new to these and I'm a bit lost because this doesn't resemble the examples I've seen so far.



    Obviously the two disks intersect at $±1$, but I don't know where I should be looking to map them to, and while I know usually you'd try to map the region to something familiar and then compose this with another map to the unit disk from there, but I can't see how to map this to something simpler.



    As I said, I haven't really done many examples of conformal maps so I'd really appreciate if you could help walk me through this example.



    Thanks in advance!






    complex-analysis complex-numbers conformal-geometry mobius-transformation







    share|cite|improve this question







    New contributor




    oxsam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      0












      0








      0







      I'm trying to solve a problem which asks me to find a conformal mapping from ${zin mathbb{C}: |z-i|< sqrt2$ and $|z+i|<sqrt2}$ onto the open unit disk.



      I'm really new to these and I'm a bit lost because this doesn't resemble the examples I've seen so far.



      Obviously the two disks intersect at $±1$, but I don't know where I should be looking to map them to, and while I know usually you'd try to map the region to something familiar and then compose this with another map to the unit disk from there, but I can't see how to map this to something simpler.



      As I said, I haven't really done many examples of conformal maps so I'd really appreciate if you could help walk me through this example.



      Thanks in advance!






      complex-analysis complex-numbers conformal-geometry mobius-transformation







      share|cite|improve this question







      New contributor




      oxsam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      I'm trying to solve a problem which asks me to find a conformal mapping from ${zin mathbb{C}: |z-i|< sqrt2$ and $|z+i|<sqrt2}$ onto the open unit disk.



      I'm really new to these and I'm a bit lost because this doesn't resemble the examples I've seen so far.



      Obviously the two disks intersect at $±1$, but I don't know where I should be looking to map them to, and while I know usually you'd try to map the region to something familiar and then compose this with another map to the unit disk from there, but I can't see how to map this to something simpler.



      As I said, I haven't really done many examples of conformal maps so I'd really appreciate if you could help walk me through this example.



      Thanks in advance!






      complex-analysis complex-numbers conformal-geometry mobius-transformation




      complex-analysis complex-numbers conformal-geometry mobius-transformation






      share|cite|improve this question







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      oxsam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question







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      Check out our Code of Conduct.






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          Start with the Möbius transformation $T(z) = frac{z-1}{z+1}$. $T(1) = 0$ and $T(-1) = infty$, therefore the two circles $|z pm i| = sqrt 2$ are mapped to lines through the origin. Use the preservation of angles to compute the directions of those lines. Conclude that $T$ maps the intersection of the two disks to a certain sector with opening angle $frac pi 2$. Then map this sector to a half-plane, and finally to the unit disk.






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            Start with the Möbius transformation $T(z) = frac{z-1}{z+1}$. $T(1) = 0$ and $T(-1) = infty$, therefore the two circles $|z pm i| = sqrt 2$ are mapped to lines through the origin. Use the preservation of angles to compute the directions of those lines. Conclude that $T$ maps the intersection of the two disks to a certain sector with opening angle $frac pi 2$. Then map this sector to a half-plane, and finally to the unit disk.






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              Start with the Möbius transformation $T(z) = frac{z-1}{z+1}$. $T(1) = 0$ and $T(-1) = infty$, therefore the two circles $|z pm i| = sqrt 2$ are mapped to lines through the origin. Use the preservation of angles to compute the directions of those lines. Conclude that $T$ maps the intersection of the two disks to a certain sector with opening angle $frac pi 2$. Then map this sector to a half-plane, and finally to the unit disk.






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                Start with the Möbius transformation $T(z) = frac{z-1}{z+1}$. $T(1) = 0$ and $T(-1) = infty$, therefore the two circles $|z pm i| = sqrt 2$ are mapped to lines through the origin. Use the preservation of angles to compute the directions of those lines. Conclude that $T$ maps the intersection of the two disks to a certain sector with opening angle $frac pi 2$. Then map this sector to a half-plane, and finally to the unit disk.






                share|cite|improve this answer














                Start with the Möbius transformation $T(z) = frac{z-1}{z+1}$. $T(1) = 0$ and $T(-1) = infty$, therefore the two circles $|z pm i| = sqrt 2$ are mapped to lines through the origin. Use the preservation of angles to compute the directions of those lines. Conclude that $T$ maps the intersection of the two disks to a certain sector with opening angle $frac pi 2$. Then map this sector to a half-plane, and finally to the unit disk.







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                Martin R

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                27.1k33253






















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