Map domain onto unit disc The Next CEO of Stack OverflowConformal map from a lune to the unit disc in $mathbbC$Conformal map from punctured disc to discMapping unit disc onto upper half planeCreating surjective holomorphic map from unit disc to $mathbbC$?Simple bijective map from upper half plane to whole planeMapping the upper half plane to unit discConformal map from $U_r = mathbbC backslash (-infty,r]$ to the unit discFinding a conformal map from this domain into the unit discFind biholomorphic function with certain propertyFind Conformal Mapping Between Circles
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Map domain onto unit disc
The Next CEO of Stack OverflowConformal map from a lune to the unit disc in $mathbbC$Conformal map from punctured disc to discMapping unit disc onto upper half planeCreating surjective holomorphic map from unit disc to $mathbbC$?Simple bijective map from upper half plane to whole planeMapping the upper half plane to unit discConformal map from $U_r = mathbbC backslash (-infty,r]$ to the unit discFinding a conformal map from this domain into the unit discFind biholomorphic function with certain propertyFind Conformal Mapping Between Circles
$begingroup$
I want to map the domain $D=mathbbCbackslashiy: ygeq 0$ onto the unit disc. I know that there are fast ways to do so but I am specifically interested in whether the following is correct:
- I rotate $D$ via $varphi_1(z)=iz$ to get $mathbbC^-$.
- I use the principal branch of the logarithm $varphi_2(z)=log(z)$ to map it further to the horizontal stripe between $-ipi$ and $ipi$.
- I scale and shift the stripe using $varphi_3(z)=frac12z+ifracpi2$ to get the horizontal stripe between $0$ and $ipi$.
- Then via the exponentialfunction $varphi_4(z)=exp(z)$ I map this stripe onto the upper half-plane $mathbb H^+$.
- Lastly via the Cayley-Transform $varphi_5(z)=fracz-iz+i$ we get from the upper half-plane to the unit-disc.
In total the function $phi:=varphi_5circvarphi_4circvarphi_3circvarphi_2circvarphi_1$ should get me from $D$ to the unit disc. Is this correct?
complex-analysis
asked yesterday
RedLanternRedLantern
516
$endgroup$
add a comment |
$begingroup$
I want to map the domain $D=mathbbCbackslashiy: ygeq 0$ onto the unit disc. I know that there are fast ways to do so but I am specifically interested in whether the following is correct:
- I rotate $D$ via $varphi_1(z)=iz$ to get $mathbbC^-$.
- I use the principal branch of the logarithm $varphi_2(z)=log(z)$ to map it further to the horizontal stripe between $-ipi$ and $ipi$.
- I scale and shift the stripe using $varphi_3(z)=frac12z+ifracpi2$ to get the horizontal stripe between $0$ and $ipi$.
- Then via the exponentialfunction $varphi_4(z)=exp(z)$ I map this stripe onto the upper half-plane $mathbb H^+$.
- Lastly via the Cayley-Transform $varphi_5(z)=fracz-iz+i$ we get from the upper half-plane to the unit-disc.
In total the function $phi:=varphi_5circvarphi_4circvarphi_3circvarphi_2circvarphi_1$ should get me from $D$ to the unit disc. Is this correct?
complex-analysis
asked yesterday
RedLanternRedLantern
516
$endgroup$
$begingroup$
your domain is a slit plane, i.e. the plane minus a half-line. How can this be rotated to a half plane via $varphi_1$?
$endgroup$
– JustDroppedIn
8 hours ago
add a comment |
$begingroup$
I want to map the domain $D=mathbbCbackslashiy: ygeq 0$ onto the unit disc. I know that there are fast ways to do so but I am specifically interested in whether the following is correct:
- I rotate $D$ via $varphi_1(z)=iz$ to get $mathbbC^-$.
- I use the principal branch of the logarithm $varphi_2(z)=log(z)$ to map it further to the horizontal stripe between $-ipi$ and $ipi$.
- I scale and shift the stripe using $varphi_3(z)=frac12z+ifracpi2$ to get the horizontal stripe between $0$ and $ipi$.
- Then via the exponentialfunction $varphi_4(z)=exp(z)$ I map this stripe onto the upper half-plane $mathbb H^+$.
- Lastly via the Cayley-Transform $varphi_5(z)=fracz-iz+i$ we get from the upper half-plane to the unit-disc.
In total the function $phi:=varphi_5circvarphi_4circvarphi_3circvarphi_2circvarphi_1$ should get me from $D$ to the unit disc. Is this correct?
complex-analysis
asked yesterday
RedLanternRedLantern
516
$endgroup$
I want to map the domain $D=mathbbCbackslashiy: ygeq 0$ onto the unit disc. I know that there are fast ways to do so but I am specifically interested in whether the following is correct:
- I rotate $D$ via $varphi_1(z)=iz$ to get $mathbbC^-$.
- I use the principal branch of the logarithm $varphi_2(z)=log(z)$ to map it further to the horizontal stripe between $-ipi$ and $ipi$.
- I scale and shift the stripe using $varphi_3(z)=frac12z+ifracpi2$ to get the horizontal stripe between $0$ and $ipi$.
- Then via the exponentialfunction $varphi_4(z)=exp(z)$ I map this stripe onto the upper half-plane $mathbb H^+$.
- Lastly via the Cayley-Transform $varphi_5(z)=fracz-iz+i$ we get from the upper half-plane to the unit-disc.
In total the function $phi:=varphi_5circvarphi_4circvarphi_3circvarphi_2circvarphi_1$ should get me from $D$ to the unit disc. Is this correct?
complex-analysis
complex-analysis
asked yesterday
RedLanternRedLantern
516
asked yesterday
RedLanternRedLantern
516
asked yesterday
RedLanternRedLantern
516
asked yesterday
RedLanternRedLantern
516
asked yesterday
RedLanternRedLantern
516
516
$begingroup$
your domain is a slit plane, i.e. the plane minus a half-line. How can this be rotated to a half plane via $varphi_1$?
$endgroup$
– JustDroppedIn
8 hours ago
add a comment |
$begingroup$
your domain is a slit plane, i.e. the plane minus a half-line. How can this be rotated to a half plane via $varphi_1$?
$endgroup$
– JustDroppedIn
8 hours ago
$begingroup$
your domain is a slit plane, i.e. the plane minus a half-line. How can this be rotated to a half plane via $varphi_1$?
$endgroup$
– JustDroppedIn
8 hours ago
$begingroup$
your domain is a slit plane, i.e. the plane minus a half-line. How can this be rotated to a half plane via $varphi_1$?
$endgroup$
– JustDroppedIn
8 hours ago
add a comment |
0
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$begingroup$
your domain is a slit plane, i.e. the plane minus a half-line. How can this be rotated to a half plane via $varphi_1$?
$endgroup$
– JustDroppedIn
8 hours ago