Conformal mappings

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Problem: Find the image of domain \[ D=\{\operatorname{Im} z<0\} \backslash\left\{z=e^{i t}, \pi

10.5.31 Conformal mappings

6.35 $

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

10.5.32 Conformal mappings

5.08 $

Problem: Find the image of domain \[ D=\{\operatorname{Im} z<0\} \backslash\{z=i t,-\infty

10.5.33 Conformal mappings

6.35 $

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

10.5.34 Conformal mappings

5.08 $

Problem: Find the image of domain \[ D=\{\operatorname{Re} z<0\} \cup\left\{|\operatorname{Im} z|<\frac{1}{4}\right\} \backslash\{z=t,-1 \leq t<0\} \] for the mapping \( f(z)=\operatorname{sh} z \).

10.5.35 Conformal mappings

5.08 $

Problem: Find the image of domain \( D=\left\{\left|\operatorname{Im} z<\frac{\pi}{2}\right|\right\} \) for the mapping \( f(z)=\operatorname{cth} \frac{z}{2} \).

10.5.36 Conformal mappings

6.35 $

Problem: Find the image of domain \( D \) with a linear-fractional mapping \( w=f(z) \), where \[ D:\{z \in \mathbb{C} \mid \operatorname{Im} z>1\}, \quad f(z)=\frac{z+2 i}{z+i} . \]

10.5.37 Conformal mappings

5.08 $

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

10.5.38 Conformal mappings

6.35 $

Problem: Find the image of domain \[ D=\left\{|\operatorname{Re} z|<\frac{\pi}{2}\right\} \backslash\left\{z=t, \quad t \in\left[0 ; \frac{\pi}{2}\right)\right\} \] for the mapping \( f(z)=e^{i z} \).

10.5.39 Conformal mappings

5.08 $

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

10.5.40 Conformal mappings

6.35 $

Problem: Find the image of domain \( D=\{0<\operatorname{Im} z<\pi\} \) for the mapping \( f(z)=\operatorname{cth} z \).

10.5.41 Conformal mappings

6.35 $

Problem: Find the image of domain \[ D=\{0<\operatorname{Re} z<\pi\} \backslash\left\{z=t, t \in\left(0 ; \frac{\pi}{2}\right)\right\} \] for the mapping \( f(z)=\operatorname{ch} i z \).

10.5.42 Conformal mappings

6.35 $

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

10.5.43 Conformal mappings

6.35 $

Problem: Find the image of domain \( D \) when mapped by the Joukowsky transform: \[ D:\left\{\begin{array}{c} \operatorname{Re} z>0, \quad \operatorname{Im} z>0 \\ |z|>1 \end{array}, \quad f(z)=\frac{1}{2}\left(z+\frac{1}{z}\right)\right. \text {. } \]

10.5.44 Conformal mappings

4.32 $

Problem: Find the image of line \( L \) for the mapping \( w=f(z) . L \) is the segment, connecting the points \( 1, i \), where \( w=(1+i) z+1-i \).

10.5.45 Conformal mappings

0 $

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