Question #125900
There exists an analytic univalent function f that maps the infinite strip {z : 0 < Im z < 1} onto the unit disk.
1
Expert's answer
2020-07-13T18:57:01-0400

z1=πzz_1=\pi z maps the horizontal strip {z: 0<Im z<1}\{z: \ 0<\text{Im } z<1\} onto the horizontal strip {z1: 0<Im z1<π}\{z_1: \ 0<\text{Im } z_1<\pi\} .

The exponential function z2=ez1z_2=e^{z_1} maps the horizontal strip {z1: 0<Im z1<π}\{z_1: \ 0<\text{Im } z_1<\pi\} to the half plane {z2: Im z2>0}\{z_2: \ \text{Im} \ z_2>0\} .

z3=eiπ/2z2z_3=e^{-i \pi/2}z_2 rotates the complex plane by 90-90^\circ, {z3: Re z3>0}\{z_3:\ \text{Re}\ z_3>0\} .

The linear fractional transformation w=1z31+z3w=\frac{1-z_3}{1+z_3} maps the right half plane {z3: Re z3>0}\{z_3:\ \text{Re} \ z_3>0\} onto the unit disk.

So, we have function f=1z31+z3=1eiπ/2z21+eiπ/2z2=1+iz21iz2=1+iez11iez1=1+ieπz1ieπzf=\frac{1-z_3}{1+z_3}= \frac{1-e^{-i \pi/2}z_2}{1+ e^{-i \pi/2}z_2}=\frac{1+i z_2}{1-i z_2}= \frac{1+i e^{z_1}}{1-i e^{z_1}}= \frac{1+i e^{\pi z}}{1-i e^{\pi z}} .

Answer: f(z)=1+ieπz1ieπz.f(z)= \frac{1+i e^{\pi z}}{1-i e^{\pi z}}.


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