Finding a conformal map from the intersection of two disks to the unit disk.
$begingroup$
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
asked Jan 2 at 20:10
oxsamoxsam
41
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add a comment |
$begingroup$
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
asked Jan 2 at 20:10
oxsamoxsam
41
$endgroup$
add a comment |
$begingroup$
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
asked Jan 2 at 20:10
oxsamoxsam
41
$endgroup$
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
asked Jan 2 at 20:10
oxsamoxsam
41
asked Jan 2 at 20:10
oxsamoxsam
41
asked Jan 2 at 20:10
oxsamoxsam
41
asked Jan 2 at 20:10
oxsamoxsam
41
asked Jan 2 at 20:10
oxsamoxsam
41
41
add a comment |
add a comment |
1 Answer
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$begingroup$
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.
edited Jan 3 at 2:50
Xander Henderson
14.8k103555
answered Jan 2 at 20:38
Martin RMartin R
29.7k33558
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add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
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oldest
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active
oldest
votes
$begingroup$
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.
edited Jan 3 at 2:50
Xander Henderson
14.8k103555
answered Jan 2 at 20:38
Martin RMartin R
29.7k33558
$endgroup$
add a comment |
$begingroup$
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.
edited Jan 3 at 2:50
Xander Henderson
14.8k103555
answered Jan 2 at 20:38
Martin RMartin R
29.7k33558
$endgroup$
add a comment |
$begingroup$
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.
edited Jan 3 at 2:50
Xander Henderson
14.8k103555
answered Jan 2 at 20:38
Martin RMartin R
29.7k33558
$endgroup$
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.
edited Jan 3 at 2:50
Xander Henderson
14.8k103555
answered Jan 2 at 20:38
Martin RMartin R
29.7k33558
edited Jan 3 at 2:50
Xander Henderson
14.8k103555
edited Jan 3 at 2:50
Xander Henderson
14.8k103555
edited Jan 3 at 2:50
Xander Henderson
14.8k103555
14.8k103555
answered Jan 2 at 20:38
Martin RMartin R
29.7k33558
answered Jan 2 at 20:38
Martin RMartin R
29.7k33558
answered Jan 2 at 20:38
Martin RMartin R
29.7k33558
29.7k33558
add a comment |
add a comment |
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