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Map domain onto unit disc

0

$begingroup$

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:

1. I rotate $D$ via $varphi_1(z)=iz$ to get $mathbbC^-$.


2. I use the principal branch of the logarithm $varphi_2(z)=log(z)$ to map it further to the horizontal stripe between $-ipi$ and $ipi$.


3. I scale and shift the stripe using $varphi_3(z)=frac12z+ifracpi2$ to get the horizontal stripe between $0$ and $ipi$.


4. Then via the exponentialfunction $varphi_4(z)=exp(z)$ I map this stripe onto the upper half-plane $mathbb H^+$.


5. 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

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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 | 

0

$begingroup$

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:

1. I rotate $D$ via $varphi_1(z)=iz$ to get $mathbbC^-$.


2. I use the principal branch of the logarithm $varphi_2(z)=log(z)$ to map it further to the horizontal stripe between $-ipi$ and $ipi$.


3. I scale and shift the stripe using $varphi_3(z)=frac12z+ifracpi2$ to get the horizontal stripe between $0$ and $ipi$.


4. Then via the exponentialfunction $varphi_4(z)=exp(z)$ I map this stripe onto the upper half-plane $mathbb H^+$.


5. 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

share|cite|improve this question

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:

1. I rotate $D$ via $varphi_1(z)=iz$ to get $mathbbC^-$.


2. I use the principal branch of the logarithm $varphi_2(z)=log(z)$ to map it further to the horizontal stripe between $-ipi$ and $ipi$.


3. I scale and shift the stripe using $varphi_3(z)=frac12z+ifracpi2$ to get the horizontal stripe between $0$ and $ipi$.


4. Then via the exponentialfunction $varphi_4(z)=exp(z)$ I map this stripe onto the upper half-plane $mathbb H^+$.


5. 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

share|cite|improve this question

asked yesterday

RedLanternRedLantern

516

$endgroup$

share|cite|improve this question

asked yesterday

RedLanternRedLantern

516

share|cite|improve this question

asked yesterday

RedLanternRedLantern

516

asked yesterday

RedLanternRedLantern

516

asked yesterday

RedLanternRedLantern

516