Finding certain Mobius tranformation

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Let $D$ denote the unit disc (|z| < 1). Let $a in mathbbC, B in mathbbR$ and $r > 0.$ I want to find a Mobius map $f$ such that



(1) $f(D) = z = x+iy : $



$textbfSol$ Let $g(z) = rz$ and $h(z) = z-a$. Then $f = h circ g.$



(2) $f(D) = > r$



$textbfSol$ Let $g(z) = rz$, $h(z) = z-a$ and $k(z) = z^-1.$ Then $f = k circ h circ g.$



(3) $f(D) = z = x+iy : x > B$



$textbfSol$ Let $g(z) = ifrac1-z1+z, h(z) = e^i(-pi/2)z$ and $k(z) = z+B.$ Then $f = k circ h circ g.$



Is it correct ?




complex-analysis holomorphic-functions mobius-transformation




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    Let $D$ denote the unit disc (|z| < 1). Let $a in mathbbC, B in mathbbR$ and $r > 0.$ I want to find a Mobius map $f$ such that



    (1) $f(D) = z = x+iy : $



    $textbfSol$ Let $g(z) = rz$ and $h(z) = z-a$. Then $f = h circ g.$



    (2) $f(D) = > r$



    $textbfSol$ Let $g(z) = rz$, $h(z) = z-a$ and $k(z) = z^-1.$ Then $f = k circ h circ g.$



    (3) $f(D) = z = x+iy : x > B$



    $textbfSol$ Let $g(z) = ifrac1-z1+z, h(z) = e^i(-pi/2)z$ and $k(z) = z+B.$ Then $f = k circ h circ g.$



    Is it correct ?




    complex-analysis holomorphic-functions mobius-transformation




    share|cite|improve this question






















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Let $D$ denote the unit disc (|z| < 1). Let $a in mathbbC, B in mathbbR$ and $r > 0.$ I want to find a Mobius map $f$ such that



      (1) $f(D) = z = x+iy : $



      $textbfSol$ Let $g(z) = rz$ and $h(z) = z-a$. Then $f = h circ g.$



      (2) $f(D) = > r$



      $textbfSol$ Let $g(z) = rz$, $h(z) = z-a$ and $k(z) = z^-1.$ Then $f = k circ h circ g.$



      (3) $f(D) = z = x+iy : x > B$



      $textbfSol$ Let $g(z) = ifrac1-z1+z, h(z) = e^i(-pi/2)z$ and $k(z) = z+B.$ Then $f = k circ h circ g.$



      Is it correct ?




      complex-analysis holomorphic-functions mobius-transformation




      share|cite|improve this question












      Let $D$ denote the unit disc (|z| < 1). Let $a in mathbbC, B in mathbbR$ and $r > 0.$ I want to find a Mobius map $f$ such that



      (1) $f(D) = z = x+iy : $



      $textbfSol$ Let $g(z) = rz$ and $h(z) = z-a$. Then $f = h circ g.$



      (2) $f(D) = > r$



      $textbfSol$ Let $g(z) = rz$, $h(z) = z-a$ and $k(z) = z^-1.$ Then $f = k circ h circ g.$



      (3) $f(D) = z = x+iy : x > B$



      $textbfSol$ Let $g(z) = ifrac1-z1+z, h(z) = e^i(-pi/2)z$ and $k(z) = z+B.$ Then $f = k circ h circ g.$



      Is it correct ?





      complex-analysis holomorphic-functions mobius-transformation





      share|cite|improve this question











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      asked Jun 4 at 19:52









      user117375

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          $z mapsto r z$ scales the unit disk by the factor $r$, then $z mapsto z - a$ translates the disk so that the center moves to $-a$. You want $(z mapsto z + a) circ (z mapsto r z)$ instead.



          $z mapsto 1/z$ performs the combination of plane inversion with conjugation. The center and the radius of the disk will change if $z mapsto 1/z$ is applied as the third step. You want $z mapsto r/z + a$.



          The third mapping is correct, $g$ maps the unit disk to the upper half-plane.






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            $z mapsto r z$ scales the unit disk by the factor $r$, then $z mapsto z - a$ translates the disk so that the center moves to $-a$. You want $(z mapsto z + a) circ (z mapsto r z)$ instead.



            $z mapsto 1/z$ performs the combination of plane inversion with conjugation. The center and the radius of the disk will change if $z mapsto 1/z$ is applied as the third step. You want $z mapsto r/z + a$.



            The third mapping is correct, $g$ maps the unit disk to the upper half-plane.






            share|cite|improve this answer
























              up vote
              1
              down vote













              $z mapsto r z$ scales the unit disk by the factor $r$, then $z mapsto z - a$ translates the disk so that the center moves to $-a$. You want $(z mapsto z + a) circ (z mapsto r z)$ instead.



              $z mapsto 1/z$ performs the combination of plane inversion with conjugation. The center and the radius of the disk will change if $z mapsto 1/z$ is applied as the third step. You want $z mapsto r/z + a$.



              The third mapping is correct, $g$ maps the unit disk to the upper half-plane.






              share|cite|improve this answer






















                up vote
                1
                down vote










                up vote
                1
                down vote









                $z mapsto r z$ scales the unit disk by the factor $r$, then $z mapsto z - a$ translates the disk so that the center moves to $-a$. You want $(z mapsto z + a) circ (z mapsto r z)$ instead.



                $z mapsto 1/z$ performs the combination of plane inversion with conjugation. The center and the radius of the disk will change if $z mapsto 1/z$ is applied as the third step. You want $z mapsto r/z + a$.



                The third mapping is correct, $g$ maps the unit disk to the upper half-plane.






                share|cite|improve this answer












                $z mapsto r z$ scales the unit disk by the factor $r$, then $z mapsto z - a$ translates the disk so that the center moves to $-a$. You want $(z mapsto z + a) circ (z mapsto r z)$ instead.



                $z mapsto 1/z$ performs the combination of plane inversion with conjugation. The center and the radius of the disk will change if $z mapsto 1/z$ is applied as the third step. You want $z mapsto r/z + a$.



                The third mapping is correct, $g$ maps the unit disk to the upper half-plane.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Aug 8 at 18:59









                Maxim

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