Conformal Mapping Between Two Domains (log)Does anyone have a recommendation as how to go about solving this problem?I want a conformal from G to H where G = { z ∈ C | | z | &lt; 1 , | z + i | &gt; 2 } , S = { z ∈ C | R e ( z ) ∈ ( − π , π ) } .The previous part of the question was to do with branches of the log. Any advice would be most appreciated!The sort of answer that I'm looking for is this:Use f(z) = α i log ⁡ ( z ) + β arg ⁡ ( z )then show how to find α and β using the boundaries.Thanks!

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Conformal Mapping Between Two Domains (log)Does anyone have a recommendation as how to go about solving this problem?I want a conformal from G to H where  G  =  {  z  ∈      C           |           |    z      |    &amp;lt;  1  ,      |    z  +  i      |    &amp;gt;      2    }  ,  S  =  {  z  ∈      C           |           R    e  (  z  )  ∈  (  −  π  ,  π  )  }  .The previous part of the question was to do with branches of the log. Any advice would be most appreciated!The sort of answer that I&#039;m looking for is this:Use f(z) =   α  i  log  ⁡  (  z  )  +  β  arg  ⁡  (  z  )then show how to find   α and   β using the boundaries.Thanks!

Brogan Navarro · Accepted Answer

Consider  f  (  z  )  =  −  5  π  +  8  i  Log  ⁡      (                  z        +        1                    z        −        1              )  The boundary of   G consists of two arcs :       γ    0   that has the parametrization   ϕ  (  t  )  =      e          i      t       for   t  ∈  [  0  ,  π  ], followed by       γ    1   that has the parametrization   ψ  (  t  )  =  −  (  i  +  (  1  −  i  )      e          −      i      t        ) for   t  ∈  [  0  ,  π      /    2  ]Now,   f  (  ϕ  (  t  )  )  =  −  π  +  8  i  log  ⁡  (  cot  ⁡  (  t      /    2  )  ) and as   t varies from   0 to   π, the point   f  (  ϕ  (  t  )  ) varies on the line   x  =  −  π from   i  (  +  ∞  ) to   i  (  −  ∞  )Similarly,   f  (  ψ  (  t  )  )  =  π  −  8  i  log  ⁡      (                            cot          ⁡          (          t                      /                    2          )          −          1                          2                      )   and as   t varies from   0 to   π      /    2, the point   f  (  ψ  (  t  )  ) varies on the line   x  =  π from   i  (  −  ∞  ) to   i  (  +  ∞  )Thus,   f  (  G  )  =  S as desired.

polynnxu · Accepted Answer

G is bounded by two circular arcs, thus it can be mapped by a Möbius transformation to an angular sector       A    α    =  {  z  :  0  &amp;lt;  arg  ⁡  z  &amp;lt;  α  }That angular sector can be mapped to a half-plane or a slit plane by a power function, and finally, the half-plane or slit plane can be mapped to a strip using a logarithm. Scalings and/or rotations at some steps may be helpful.

Mapping Between Two DomainsG=(z in bbb C||z|<1,|z+i|>sqrt(2)), S =(z in bbb C|bbb Re(z) in (-pi,pi)}

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