z1=πz maps the horizontal strip {z: 0<Im z<1} onto the horizontal strip {z1: 0<Im z1<π} .
The exponential function z2=ez1 maps the horizontal strip {z1: 0<Im z1<π} to the half plane {z2: Im z2>0} .
z3=e−iπ/2z2 rotates the complex plane by −90∘, {z3: Re z3>0} .
The linear fractional transformation w=1+z31−z3 maps the right half plane {z3: Re z3>0} onto the unit disk.
So, we have function f=1+z31−z3=1+e−iπ/2z21−e−iπ/2z2=1−iz21+iz2=1−iez11+iez1=1−ieπz1+ieπz .
Answer: f(z)=1−ieπz1+ieπz.
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