Fundamental Principles of Conformal Mappings. Transformations of Polygons

Abstract

In this chapter, we provide the proofs of the main results on conformal mappings. The analyzed fundamental principles include the criterion for local univalence (Sect. 5.1), the principle of domain preservation (Sect. 5.2), the principle of boundary correspondence (Sect. 5.3), and the Riemann mapping theorem (Sect. 5.4). Unlike other parts of the text, the subjects considered here are almost exclusively theoretical. Even some more specific problems, such as a conformal mapping between two rings (Sect. 5.4) and construction of conformal mappings between half-planes, rectangles and polygons (Sects. 5.6 and 5.7), have rather theoretical significance than represent illustrative examples.

Exercises

• 1. Show that the following functions are not univalent in the given domains D, although they are univalent at every point of D:

  • (1) $\ f\left( z\right) =z^{2} ,\ \, D=\left\{ 1<\left| z\right| <2\right\} \, ;$

  • (2) $\ f\left( z\right) =z^{3} ,\ \, D=\left\{ \mathrm{Im\,}z>0\right\} \, ;$

  • (3) $\ f\left( z\right) =e^{z} ,\ \, D=\left\{ \left| z\right| <4\right\} \, .$

• 2. Prove that the function $f\left( z\right) =z^{2} +az$ is univalent in the upper half-plane $\mathrm{Im\,}z>0$ if and only if $\mathrm{Im\,}a\ge 0$.

• 3. Let $f\left( z\right) =z^{n} +ne^{i\alpha } z$, where $\forall \alpha \in {\mathbb {R}\,} ,\ n\in {\mathbb {N}},\ n\ge 2$. Show that $f\left( z\right)$ is univalent in the unit disk $\left| z\right| <1$.

• 4. Prove that the function $f\left( z\right) =z+e^{z}$ is univalent in the left half-plane $\mathrm{Re\,}z<0$.

• 5. Show that an analytic in a convex domain D function f(z), that satisfies the condition $\mathrm{Re} f'(z)>0$, is univalent in D. Give an example that shows that this result fails for non-convex domains.

• 6. Transform the following domains into the upper half-plane using the Riemann–Schwarz symmetry principle:

  • (1) the entire plane $z\, \, \left( z=x+iy\right)$ with the cuts $\left\{ \!\!\begin{array}{c} {\!-1\le x\le 1} \\ {y=0} \end{array}\right. \!\!$ and $\left\{ \!\!\begin{array}{c} {x=0} \\ {\!-b\le y\le a} \end{array}\right. \!\!\! ,\ a , b>0$;

  • (2*) the entire plane $z\, \, \left( z=x+iy\right)$ with the cuts $\left\{ \!\!\begin{array}{c} {-1\le x\le 1} \\ {y=0} \end{array}\right.$ and $\left\{ \!\!\begin{array}{c} {0\le \left| z\right| \le \sqrt{2} } \\ {\arg z=\pi /4 + k\pi /2 } \end{array}\right. \!\! ,\ k=0\, ,\, 1\, ,\, 2\, ,\, 3$;

  • (3) the strip $-2<y<2$ with the cuts $\left\{ \!\!\begin{array}{c} {x\ge 1} \\ {y=\pm 1} \end{array} ,\ \left\{ \!\!\begin{array}{c} {x\le -1} \\ {y=\pm 1} \end{array}\right. ,\ \left\{ \!\!\begin{array}{c} {x\ge 3} \\ {y=0} \end{array}\right. \right.$ and $\left\{ \!\!\begin{array}{c} {x\le 0} \\ {y=0} \end{array}\right.$;

  • (4) the entire plane z with the cuts $\left\{ \!\begin{array}{c} {\!\!-4\le x\le 2} \\ {y=0} \end{array}\right. , \, \left\{ \!\begin{array}{c} {\left| z\right| =2} \\ {\!\!-\frac{\pi }{3} \le \arg z\le \frac{\pi }{3} } \end{array}\right. \,$ and $\ \left\{ \!\begin{array}{c} {\left| z\right| =2} \\ {\!\! \frac{2\pi }{3} \le \arg z\le \frac{4\pi }{3} } \end{array}\right.$;

  • (5*) the entire plane z with the cuts $\left\{ \!\!\begin{array}{c} {x\le 1} \\ {y=0} \end{array}\right.$ and $\left\{ \!\!\begin{array}{c} {0\le \left| z\right| \le 1} \\ {\arg z=\pm \pi /3} \end{array}\right.$;

  • (6*) the upper half-plane with the cuts $\left\{ \!\!\begin{array}{c} {x=k} \\ {0\le y\le 1} \end{array}\right. ,\ \forall k\in {\mathbb {Z}}$;

  • (7) the exterior of the unit disk (centered at the origin) with the cuts $\left\{ \!\!\begin{array}{c} {1\le \left| z\right| \le 2} \\ {\arg z={2k\pi /n } } \end{array}\right. ,\, k=0\, , 1\, ,\ldots ,n-1\, , \forall n\in {\mathbb {N}}\, ;$

  • (8*) the entire plane z with the cuts $\left\{ \!\begin{array}{c} {\!k\!-\!a\le x\le k\!+\!a} \\ {y=0} \end{array}\right.$ and $\, \left\{ \!\begin{array}{c} {\!x=k} \\ {\!y\le b} \end{array}\right. ,\, \forall k\in {\mathbb {Z}},\, \, b>0\, ,\, 0<2a<1$;

  • (9) the disk $\left| z\right| <2$ with the cuts $\left\{ \!\!\begin{array}{c} {-1\le x\le 2} \\ {y=0} \end{array}\right.$ and $\left\{ \!\!\begin{array}{c} {x=0} \\ {-1\le y\le 1} \end{array}\right.$;

  • (10) the upper half-plane with the cuts $\left\{ \!\!\begin{array}{c} {x=0} \\ {1/2 \le y \le 2 } \end{array}\right.$ and $\left\{ \!\!\begin{array}{c} {\left| z\right| =1} \\ {\pi /2 \le \arg z\le \pi } \end{array}\right.$.

• 7. Show that using an appropriate change of variable the integral

  $$\begin{aligned} K_{1} \left( k\right) =\int _{1}^{1/k }\frac{dt}{\sqrt{\left( t^{2} -1\right) \left( 1-k^{2} t^{2} \right) } } \ ,\ 0<k<1\, , \end{aligned}$$

  considered in Sect. 5.6, can be transformed to the form

  $$\begin{aligned} K_{1} \left( k\right) =K\left( k_{1} \right) =\int _{0}^{1}\frac{dt}{\sqrt{\left( 1-t^{2} \right) \left( 1-k_{1}^{2} t^{2} \right) } } \ , \ 0<k_{1} <1\, , \end{aligned}$$

  where $k_{1}$ is an additional parameter such that $k^{2} +k_{1}^{2} =1$.

• 8. Let T be an infinite triangle with the vertices $A=0\, ,\, \, B=1\, ,\, \, C=\infty$ and the corresponding angles $\pi \alpha ,\, \pi \beta , \, \pi \gamma$, where $0\!<\!\alpha \le 2\, ,\, 0\!<\!\beta \le 2\, ,\, -2\le \gamma \le 0\, ,\, \alpha +\beta +\gamma \!=\!1$. Find the function that transforms conformally the upper half-plane $\mathrm{Im\,}z>0$ into this triangle in the following cases:

  • (1) $\, \, \, \alpha =\frac{3}{4} \, ,\, \, \, \beta =\frac{1}{2} \, ,\, \, \, \gamma =-\frac{1}{4} \, ;$

  • (2) $\, \, \, \alpha =2\, ,\, \, \, \beta =\frac{1}{2} \, ,\, \, \, \gamma =-\frac{3}{2} \, ;$

  • (3) $\, \, \, \alpha =\frac{3}{2} \, ,\, \, \, \beta =\frac{3}{2} \, ,\, \, \, \gamma =-2\, ;\,$

  • (4) $\, \, \, \alpha =\frac{2}{3} \, ,\, \, \, \beta =\frac{2}{3} \, ,\, \, \, \gamma =-\frac{1}{3} \, .$

• 9. Construct conformal mappings of the upper half-plane $\mathrm{Im\,}z>0$ onto the domains in the plane w shown in Figs. 5.32, 5.33, 5.34, 5.35, 5.36, 5.37, 5.38, and 5.39, using the indicated correspondence between the points. Find the values of a and b for these mappings:

  • (1)   $\, \, \, z=-1\, ;\, \, b\, ;\, \, 1\, ;\, \, \infty \, \, \leftrightarrow \, \, w=A\, ;\, \, B;\, \, C;\, \, D;$

  • (2)   $\, \, \, z=-1\, ;\, \, -b\, ;\, \, 0\, ;\, \, b\, ;\, \, 1\, ;\, \, \infty \, \, \leftrightarrow \, \, w=A\, ;\, \, B\, ;\, \, C;\, \, D;\, \, E;\, \, K;$

  • (3*) $\, \, z=-1\, ;\, \, 1\, ;\, \, \infty \, \, \leftrightarrow \, \, w=E;\, \, O;\, \, D;$

  • (4) $\, \, z=\infty \, ;\, \, 0\, ;\, \, 1\, \, \leftrightarrow \, \, w=C;\, \, B;\, \, A;$

  • (5) $\, \, z=0\, ;\, \, 1\, ;\, \, \infty \, \, \leftrightarrow \, \, w=O;\, \, C;\, \, A;$

  • (6) $\, \, z=-1\, ;\, \, 0\, ;\, \, 1\, \, \leftrightarrow \, \, w=E;\, \, O;\ C;$

  • (7) $\, \, z=\infty \, ;\, \, -\left( 1+a\right) \, ;\, \, -1\, ;\, 0\, \, \leftrightarrow \, \, w=A\, ;\, \, B;\, C;\, D; \, p=\mathrm{Re\,}\left( D-B\right) ;$

  • (8) $\, \, z=\infty \, ;\, \, -1\, ;\, \, a\, ;\, \, 1\, \, \leftrightarrow \, \, w=A\, ;\, \, B\, ;\, \, C;\, \, D;\, \, \, \, \, p{=}\mathrm{Re\,}\left( D-B\right) \, , h=\mathrm{Im\,}\left( B-D\right) .$

• 10*. Find a conformal transformation of the unit disk $|z|<1$ into a polygon of sixteen sides (regular star-like polygon with eight points), inscribed in the circle of the radius R, with the vertices at the points $A_{k} ,\, \, B_{k} ,\, \, \, k=1,\ldots ,8$, and the corresponding angles $\alpha _{k} \pi =\alpha \pi =\frac{\pi }{2} \, ,\, \, \, \beta _{k} \pi =\beta \pi =\frac{5}{4} \pi ,\, \, \, k=1,\ldots ,8$.

• 11*. Find a conformal transformation of the unit disk $|z|<1$ into a dodecagon (regular star-like polygon with six points), inscribed in the circle of the radius R, with the vertices $A_{k} ,\, \, B_{k} ,\, \, \, k=1,\ldots ,6$, and the corresponding angles $\alpha _{k} \pi =\alpha \pi =\frac{\pi }{3} \, ,\, \, \, \beta _{k} \pi =\beta \pi =\frac{4}{3} \pi ,\, \, \, k=1,\ldots ,6$.

• 12. Using the Legendre elliptic integral of the first type, construct conformal mappings of the following doubly-connected domains onto the ring $1<\left| w\right| <R$:

  • (1) the entire plane z with the cuts along the two intervals of the real axis $\left[ -{1 /k} ,-1\right]$ and $\left[ 1\, ,\, {1 /k} \right] \, ,\,$ $0<k<1$;

  • (2) the right half-plane $\mathrm{Re\,}z>0$ with the cut along the interval $\left[ 1\, ,\, {1 /k} \right] \, ,\, \, \, 0<k<1$;

  • (3*) the entire plane z with the cuts along the intervals $\left[ -4\, ,\, 0\right]$ and $\left[ 1\, ,\, 2\right]$ on the real axis;

  • (4) the entire plane z with the cuts along the intervals $\left( -\infty \, ,\, -{1 /k} \right] \, ,\, \, \left[ -1\, ,\, 1\right] \,$ and $\, \left[ {1 /k} \, ,\, +\infty \right) \, ,\,$ $0<k<1$ on the real axis;

  • (5*) the entire plane z with the cut along the interval $\left[ 1\, ,\, 1+d\right] \, ,\, \, d>0$ of the real axis and the cut along the interval $\left[ -ih\, ,\, ih\right]$, $h>0$ of the imaginary axis;

  • (6) the disk of the radius ${1 / \sqrt{k} } \, ,\, \, 0<k<1$ (centered at the origin) with the cut along the interval $[-1,1]$;

  • (7) the exterior of the disk of the radius ${1 / \sqrt{k} } ,\, \, 0<k<1$ (centered at the origin) with the cuts $\left( -\infty \, ,\, -{1 /k} \right]$ and $\left[ {1 /k} ,\, +\infty \right)$ on the real axis;

  • (8) the entire plane with the cuts $\left[ 0\, ,\, k\right]$ and $\left[ {1 /k} ,\, +\infty \right) \, ,\, \, \, 0<k<1$ on the real axis;

  • (9) the unit disk $|z|<1$ with the cut along the interval $\left[ 0\, ,\, k\right] \, ,\, \, \, 0<k<1$ of the real axis;

  • (10) the exterior of the unit disk $|z|>1$ with the cut along the interval $\left[ {1 /k } ,+\infty \right) , 0<k<1$ of the real axis;

  • (11) the entire plane with the cuts $\left[ -1\, ,\, 0\right]$ and $\left[ p\, ,\, +\infty \right) \, ,\, \, \, p>0$ on the real axis;

  • (12) the entire plane with one cut along the interval $\left( -\infty \, ,\, 0\right]$ on the real axis and another cut along the arc of the unit circle: $\left| z\right| =1\, ,\, \, \left| \arg z\right| \le \alpha ,\, \, \, 0<\alpha <{\pi /2}$;

  • (13) the entire plane with the cuts $\left( -\infty \, ,\, -\sin \alpha \right] \, ,\, \, \, \left[ \sin \alpha ,\, +\infty \right)$ on the real axis and the cut $\left[ -i\cos \alpha ,\, i\cos \alpha \right] \$ on the imaginary axis, where $\ \ 0<\alpha <{\pi /2}$;

  • (14) the strip $-\frac{\pi }{2}\!<\!\mathrm{Re\,}z\!<\!\frac{\pi }{2}$ with the cut along the interval $\left[ -iH, iH\right] , H\!>\!0$ on the imaginary axis;

  • (15*) the strip $-\frac{\pi }{2} \!<\!\mathrm{Re\,}z<\frac{\pi }{2}$ with the cut along the interval $\left[ \alpha ,\, \beta \right] \, ,\, \, -\frac{\pi }{2}<\alpha<\beta <\frac{\pi }{2}$ on the real axis;

  • (16) the half-strip $\left\{ \!\!\begin{array}{c} {-{\pi /2} {<}\mathrm{Re\,}z{<}{\pi /2} } \\ {\mathrm{Im\,}z>0} \end{array}\right.$ with the cut along the interval $\left[ iH_{1} ,\, iH_{2} \right]$, $0<H_{1}<H_{2} <+\infty$ on the imaginary axis.

• 13. Suppose f(z) is analytic in the rectangle $D=\{z=x+iy: 0<x<1, |y|<h \}$, continuous in $\overline{D}$ and satisfies the conditions $\mathrm{Im} f(z)=0$, for $\forall z: x=0, |y|<h$ and $\mathrm{Im} f(z)=1$, for $\forall z: x=1, |y|<h$. Demonstrate that f(z) can be analytically extended to the strip $|y|<h$ ($\forall x$) and the corresponding function F(z) has the form $F(z)=iz+F_1(z)$, where $F_1(z)$ is analytic in the strip $|y|<h$ and has the period 2.

  Hint: use the Riemann–Schwarz symmetry principle.

• 14 . Suppose that $f(z)\ne constant$ is an analytic function in a domain D and ${\varGamma }=\{\forall z\in D: |f(z)|=c\}$, $c=constant$, is a simple closed curve such that ${\varGamma }=\partial G$, $G\cup {\varGamma }=\overline{G} \subset D$. Prove that there exists at least one zero of f(z) in G.

• 15. Suppose that f(z) is analytic in the disk $|z|<R$, $f(a)=0$, $|a|<R$, and $|f(z)|\le M$, $\forall z\in \{|z|<R \}$. Demonstrate that $|f(z)|\le MR \left| \frac{z-a}{R^2-z\bar{a}}\right|$, $\forall z\in \{|z|<R \}$, and $|f'(a)|\le \frac{MR}{R^2-|a|^2}$. For what functions the equality sign in these evaluations is achieved.

  Hint: use the fractional linear transformation of a disk into another disk.

• 16. Suppose that f(z) is analytic in the disk $|z|<R$, $|f(z)|\le M$, $\forall z\in \{|z|<R \}$ and $f(0)=w_0$, $|w_0|<M$. Prove the following inequalities:

  • (1) $\left| \frac{f(z)-w_0}{M^2-f(z)\overline{w}_0}\right| \le \frac{|z|}{MR}$, $\forall z\in \{|z|<R \}$;

  • (2) $|f'(0)|\le \frac{M^2-|w_0|^2}{MR}$.

    Investigate if there exist functions for which the equality in the above evaluation is achieved.

• 17. Suppose that f(z) is analytic in the right half-plane $\mathrm{Re} z>0$, continuous in $\mathrm{Re} z\ge 0$ and $|f(z)|<1$, $\forall z\in \{\mathrm{Re} z>0\}$. Suppose also that $f(z_k)=0$ at the points $z_k$, $k=1,\ldots , m$ located in the right half-plane. Show that $|f(z)|\le \frac{|z-z_1|\cdot |z-z_2|\cdot \ldots \cdot |z-z_m|}{|z+\bar{z}_1|\cdot |z+\bar{z}_2|\cdot \ldots \cdot |z+\bar{z}_m|}$ for any point z in the right half-plane.

• 18*. Let P(z) be a polynomial of degree n and $M(r)=\mathop {\max }\limits _{|z|=r} |P(z)|$. Demonstrate that the inequality $\frac{M(r_1)}{r_1^n}\ge \frac{M(r_2)}{r_2^n}$ holds for $\forall r_1, r_2$, $0<r_1<r_2$. Additionally, show that $P(z)=az^n$ if the equality in the last evaluation occurs at least for one pair of $r_1, r_2$.

  Hint: consider the auxiliary function $g(t)=\frac{1}{a_n}P(\frac{1}{t}) t^n$ and apply the maximum modulus principle.

• 19. Let $P(z)=z^n+a_{n-1}z^{n-1}+\ldots +a_0$. Prove that either $|P(z)|>1$ at least at one point of the circle $|z|=1$ or $P(z)=z^n$.

  Hint: make use of the result of Exercise 18.

• 20*. Suppose $g(z)=z+\sum _{n=1}^{+\infty } \frac{a_n}{z^n}$ is analytic in the domain $|z|>1$ except a simple pole at $z=\infty$, and univalent in $|z|>1$. Show that:

  • (1) $\sum _{n=1}^{+\infty } n |a_n|^2 \le 1$;

  • (2) $|a_n|\le \frac{1}{\sqrt{n}}$, $\forall n\in \mathbb {N}$; investigate for which function this inequality turns into equality $|a_m|= \frac{1}{\sqrt{m}}$ for some $m\in \mathbb {N}$.

    Find the functions for which $|a_1|=1$, and the image of the domain $|z|>1$ under transformation by these functions.

    Hint: use the fact that the area of a domain bounded by the image of the circle $|z|=\rho >1$ is positive and calculate this area using Green’s formula.

• 21*. Let function $f(z)=z+\sum _{n=2}^{+\infty } c_n z^n$ be analytic and univalent in the disk $|z|<1$. Prove that $|c_2|\le 2$. Investigate when the inequality turns into equality and find the image of the unit disk under such functions.

  Hint: apply the result of Exercise 20 to the function $g(z)=\frac{1}{f_2(\frac{1}{z})}$, where $f_2(z)=\sqrt{f(z^2)}$ is one of the functions considered in Exercise 28 in Chap. 4.

Fig. 5.32

Domain of exercise (9.1)

Fig. 5.33

Domain of exercise (9.2)

Fig. 5.34

Domain of exercise (9.3)

Fig. 5.35

Domain of exercise (9.4)

Fig. 5.36

Domain of exercise (9.5)

Fig. 5.37

Domain of exercise (9.6)

Fig. 5.38

Domain of exercise (9.7)

Fig. 5.39

Domain of exercise (9.8)