Conformal maps by elementary functions
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1 The power function
Let \(t \in \mathbb {R}\). Consider on the domain \(G=\mathbb {C} \backslash [0,+\infty )\) the function
\[ w=|z|^{t} e^{i t \arg z}, \quad \text{ where } \quad \arg z \in (0,2 \pi ) \text{. } \tag {1} \]
This function is regular on \(G\). Moreover, for \(t \neq 0\) the function (1) is univalent on a domain \(D \subset G\) if \(D\) does not contain two distinct points \(z_{1}, z_{2}\) such that \(z_{2}=z_{1} \cdot e^{2 \pi i k / t}, k \in \mathbb {Z}\).
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.001 of the problem book, code: SHPK images_recreated/28/001.tex): a sector mapped to a sector under w = z^t. Interactive: sliders for t, phi_0, phi_1 with a live deformation of the angles.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/010_power_sector_map_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of figs. 28.002-28.004 of the problem book, code: SHPK images_recreated/28/): three tabulated maps w = z^2. Interactive: dragging the level a of the half-plane Im z > a, an animation unfolding the half-plane into a slit plane.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/020_square_map_fig.tex
2 The exponential
The function \(w=e^{z}\) is a conformal map on the domain \(D \subset \mathbb {C}\) if and only if \(D\) does not contain two distinct points \(z_{1}, z_{2}\) such that \(z_{2}=z_{1}+2 \pi k i\), where \(k \in \mathbb {Z}\).
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of figs. 28.005-28.007 of the problem book, code: SHPK images_recreated/28/): three tabulated maps w = e^z. Interactive: sliders for the width and position of the strip/half-strip, a live image.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/030_exp_map_fig.tex
3 The logarithm
The multivalued function \(w=\operatorname {Ln} z\) splits into regular branches on every simply connected domain \(G \subset \overline{\mathbb {C}}\) not containing the points \(0\) and \(\infty\). Every regular branch \(f(z) \in \operatorname {Ln} z\) on such a domain \(G\) is univalent (since its inverse function \(e^{z}\) is single-valued), so this branch \(f(z)\) is a conformal map from the domain \(G\) onto the domain \(f(G)\), which is the inverse of the map from \(f(G)\) onto \(G\) given by the function \(w=e^{z}\).
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of figs. 28.008-28.009 of the problem book, code: SHPK images_recreated/28/; in 009 the axis labels u/x, which were swapped in the source, have been corrected): maps of a regular branch of Ln z. Interactive: choosing a branch (the value of f(-1) or f(2+i0)), highlighting the correspondence of the boundaries.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/040_log_branch_map_fig.tex
4 The Zhukovsky function
The function:
\[ \operatorname {Zh}(z) := \frac{1}{2}\left(z + \frac{1}{z}\right) \]
4.1 Conformality
On which domains is it conformal? Injectivity: for \(z_1 \neq z_2\) we have \(\operatorname {Zh}(z_1) = \operatorname {Zh}(z_2)\) \(\iff\) \(z_1z_2 = 1\). A domain of conformality cannot contain \(\pm 1\), since a neighborhood of either point will always contain points that break injectivity. Also, \(0\) and \(\infty\) cannot both be present at once.
Regularity? \(\operatorname {Zh}'(z)\neq 0\), so \(z \neq \pm 1\).
It suffices for the function to be univalent and regular at every point. On the whole sphere, the Zhukovsky function has only 2 poles of the first order.
In total: \(\operatorname {Zh}\) is conformal on \(D \subset \overline{\mathbb {C}}\) \(\iff\) \(\forall z \in D\) \(\frac{1}{z}\not\in D\).
4.2 Transformations of the basic figures
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ: a circle of radius R maps to an ellipse.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/050_circle_to_ellipse_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ: semicircles from the upper half-plane.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/060_upper_semicircles_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ: semicircles from the lower half-plane.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/070_lower_semicircles_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ: rays map to hyperbolas.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/080_rays_to_hyperbolas_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.010 of the problem book, code: SHPK images_recreated/28/010.tex): the upper half-plane to a plane with two slits under w = Zh(z). Interactive: dragging the points a, b along the boundary with live images aβ, bβ on the banks of the slits.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/090_halfplane_to_slit_plane_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.011 of the problem book, code: SHPK images_recreated/28/011.tex): the unit disk to a plane with a slit [-1,1] under w = Zh(z). Interactive: dragging a, b along the circle, highlighting the correspondence of the upper/lower semicircle to the banks of the slit.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/100_disk_to_slit_plane_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.012 of the problem book, code: SHPK images_recreated/28/012.tex): the exterior of the unit disk to a plane with a slit [-1,1] under w = Zh(z). Interactive: dragging a, b; comparing the traversal orientations with the previous figure.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/110_disk_exterior_to_slit_plane_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.013 of the problem book, code: SHPK images_recreated/28/013.tex): the region {Im z > 0, |z| > 1} to the upper half-plane under w = Zh(z). Interactive: a point running along the boundary with a synchronized image.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/120_upper_exterior_to_halfplane_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.014 of the problem book, code: SHPK images_recreated/28/014.tex): the lower half-disk to the upper half-plane under w = Zh(z). Interactive: a point running along the boundary with a synchronized image.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/130_lower_halfdisk_to_halfplane_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.015 of the problem book, code: SHPK images_recreated/28/015.tex): |z| > rho (and the disk |z| < 1/rho) to the exterior of an ellipse under w = Zh(z). Interactive: a slider for rho, with the ellipse degenerating into a segment as rho -> 1.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/140_circles_to_ellipse_exterior_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.016 of the problem book, code: SHPK images_recreated/28/016.tex): the sector alpha < arg z < pi - alpha to the exterior of a hyperbola under w = Zh(z). Interactive: a slider for alpha, deforming the hyperbola and its asymptotes.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/150_sector_to_hyperbola_exterior_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.017 of the problem book, code: SHPK images_recreated/28/017.tex): the sector 0 < arg z < alpha to the interior of the right branch of a hyperbola with a slit under w = Zh(z). Interactive: a slider for alpha; highlighting the image of the ray arg z = alpha (the branch) and of the ray [1, +inf) (the two-sheeted slit).
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/160_sector_to_hyperbola_branch_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.018 of the problem book, code: SHPK images_recreated/28/018.tex): the region {0 < arg z < alpha, |z| > 1} under w = Zh(z): the upper half-plane to the right of the upper half of the right branch of the hyperbola. Interactive: a slider for alpha. NOTE: the label has been corrected (the source had pi - alpha, which contradicts the figure and the computation).
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/170_sector_exterior_to_branch_halfplane_fig.tex
5 The inverse Zhukovsky function
The multivalued function \(w=z+\sqrt{z^{2}-1}\), which is the inverse of the Zhukovsky function, splits into two regular branches on every simply connected domain \(G \subset \overline{\mathbb {C}}\) that does not contain at least one curve joining the points \(z= \pm 1\). Every branch of the function \(w=z+\sqrt{z^{2}-1}\) regular on the domain \(G\) is univalent (since its inverse, the Zhukovsky function, is single-valued).
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.019 of the problem book, code: SHPK images_recreated/28/019.tex): branches of the inverse Zhukovsky function: a plane with a slit [-1,1] to the exterior (f_1) or the interior (f_2) of the unit disk. Interactive: a branch toggle, dragging a, b along the banks of the slit.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/180_inverse_zhukovsky_to_disk_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.020 of the problem book, code: SHPK images_recreated/28/020.tex): branches of the inverse Zhukovsky function: a plane with slits (-inf,-1], [1,+inf) to the upper (f_1) or lower (f_2) half-plane. Interactive: a branch toggle, dragging a, b.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/190_inverse_zhukovsky_to_halfplanes_fig.tex
Figure in development. Will be recreated as an interactive figure (Plotly/Three.js).
Plan: TikZ (redrawing of fig. 28.021 of the problem book, code: SHPK images_recreated/28/021.tex): branches of the inverse Zhukovsky function on the upper half-plane: {Im w > 0, |w| > 1} (f_1) or the lower half-disk (f_2). Interactive: a branch toggle, a point running along the boundary.
Source: courses/nedosh_courses/courses/01-analysis/CA/contents/specials/340/200_inverse_zhukovsky_halfplane_fig.tex