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Problem: Find the image of domain \( D=\{\operatorname{Im} z>0\} \backslash\{z=i t, 0

10.5.8 Conformal mappings

6.37 $

Problem: Find the image of domain \( D \) for the mapping \( w=f(z) \), where \[ D:\left\{\begin{array}{l} 0<\operatorname{Re} z<\pi, \quad \operatorname{Im} z<0 \\ z \neq \frac{\pi}{2}+i t, \quad t \in[-1 ; 0) \end{array}, \quad f(z)=e^{i z} .\right. \]

10.5.9 Conformal mappings

5.1 $

Problem: Find the image of domain \( D \) for the mapping \( w=f(z) \), where \[ D:\left\{\begin{array}{c} -2<\operatorname{Re} z<0, \\ z \neq t, \quad t \in[-1 ; 0), \quad f(z)=\cos \frac{\pi z}{2} . \end{array}\right. \]

10.5.10 Conformal mappings

5.1 $

Problem: Find the image of domain \( D \) for the mapping \( w=f(z) \), where \[ D:\left\{\begin{array}{c} 0<\operatorname{Re} z<\frac{1}{2}, \quad f(z)=\tan \pi z . \\ \operatorname{Im} z>0 \end{array}\right. \]

10.5.11 Conformal mappings

6.37 $

Problem: Find the image of domain \( D \) for the linear-fractional mapping \( w=f(z) \), where \[ D:\{|z|<1\}, \quad f(z)=\frac{2 z}{z+2 i} . \]

10.5.12 Conformal mappings

5.1 $

Problem: Find the image of domain \[ D=\{|z|>1\} \cup\{|z| \leq 1 ; \operatorname{Im} z<0\} \backslash\{z=i t, 1

10.5.13 Conformal mappings

6.37 $

Problem: Find the image of domain \[ D=\{-\pi<\operatorname{Im} z<3 \pi\} \backslash\{z=i t, \pi \leq t<3 \pi)\} \] for the mapping \( w=f(z) \), where \[ f(z)=e^{-\frac{z}{2}} . \]

10.5.14 Conformal mappings

6.37 $

Problem: Find the image of domain \[ D=\{-\pi / 2<\operatorname{Re} z<0\} \backslash\{z=i t,-1 \leq t<0\} \] for the mapping \( w=f(z) \), where \( f(z)=\cos 2 z \).

10.5.15 Conformal mappings

5.1 $

Problem: Find the image of domain \( D \) for the mapping \( w=f(z) \), where \[ D:\left\{\begin{array}{l} \operatorname{Re} z<0 \\ |\operatorname{Im} z|<\frac{\pi}{4}, \quad f(z)=\operatorname{th} 2 z . \end{array}\right. \]

10.5.16 Conformal mappings

6.37 $

Problem: Find the image of domain \[ D=\{|z|>1\} \cup\{|z| \leq 1, \operatorname{Im} z>0\} \backslash\{z=i t,-2 \leq t<-1\} \] when mapped by the Joukowsky transform: \[ f(z)=\frac{1}{2}\left(z+\frac{1}{z}\right) \]

10.5.17 Conformal mappings

6.37 $

Problem: Find the image of domain \( D \) for the linear-fractional mapping \( w=f(z) \), where \( D: \operatorname{Re} z<-1, \quad f(z)=\frac{z}{z+1+i} \).

10.5.18 Conformal mappings

3.82 $

Problem: Find the image of domain \[ D=\{-\pi<\operatorname{Re} z<0\} \backslash\{z=t,-1 \leq t<0\} \] for the mapping \( f(z)=e^{-i z} \).

10.5.19 Conformal mappings

3.82 $

Problem: Find the image of domain \( D=\{|\operatorname{Re} z|<\pi\} \cup\{\operatorname{Im} z>0\} \backslash\{z=i t, t \in[1 ;+\infty)\} \) for the mapping \[ f(z)=\sin \frac{z}{2} \text {. } \]

10.5.20 Conformal mappings

6.37 $

Problem: Find the image of domain \( D \) for the mapping \( w=f(z) \), where \[ D:\left\{\begin{array}{c} \operatorname{Re} z>0 \\ 0<\operatorname{Im} z<\pi \end{array}, \quad f(z)=\operatorname{cth} z .\right. \]

10.5.21 Conformal mappings

5.1 $

Problem: Find the image of domain \( D \) for the linear-fractional mapping \( w=f(z) \), where \[ D:|z-1-i|<\sqrt{2}, \quad f(z)=\frac{2}{z} . \]

10.5.22 Conformal mappings

3.82 $

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