The Symmetry Principle

Denote by $w(z)$ the function conformally mapping the angle $\left\{ z: 0 < \arg z < \frac{\pi }{2}\right\}$ onto the sector $\left\{ w:\left|w\right| > 1,0 < \arg w < \frac{\pi }{2}\right\}$ and satisfying the conditions $w(1) = 1, w(i) = i, w(\infty ) = \infty$. Having found the function $w(z)$, verify that it:

Denote by $w(z) = w\left(z ; z_{1}, z_{2}, z_{3}\right)$ the function conformally mapping the half-disk $\left\{ z:\left|z\right| < 1, \mathrm{Im}\left(z\right) > 0\right\}$ onto the same half-disk in the $w$-plane and satisfying the conditions

(the points $z_{1}, z_{2}, z_{3}$ lie on the boundary of the half-disk $\left\{ z:\left|z\right| < 1, \mathrm{Im}\left(z\right) > 0\right\}$). Determine what conditions the points $z_{1}, z_{2}, z_{3}$ must satisfy in order for the function $w(z)$ to conformally map the disk $\left\{ z:\left|z\right| < 1\right\}$ onto:

Find some functions $w(z)$ conformally mapping the domains shown in Figs. 29.18-29.29 onto the half-plane $\left\{ w: \mathrm{Im}\left(w\right) > 0\right\}$.

Find some function $w(z)$ conformally mapping the domain

($\alpha$ is a constant, $0 < \alpha < \pi / 2$) onto the half-plane $\left\{ w: \mathrm{Im}\left(w\right) > 0\right\}$.

Find some function $w(z)$ conformally mapping the domain

onto the half-plane $\left\{ w: \mathrm{Im}\left(w\right) > 0\right\}$.

Find some functions $w(z)$ that conformally map the regions depicted in Fig. 29.30-29.33 onto the disk $\left\{ w:\left|w\right| < 1\right\}$.

Find some function $w(z)$ that conformally maps the region

onto the half-plane $\left\{ w: \mathrm{Im}\left(w\right) > 0\right\}$.

Find some function $w(z)$ that conformally maps the region

onto the half-plane $\left\{ w: \mathrm{Im}\left(w\right) > 0\right\}$.

Find some function $w(z)$ that conformally maps the region $\left\{ z: \operatorname {Im} > 0, z \notin [2 k, 2 k+2 i], z \notin [2 k+1,2 k+1+i] \quad (k = 0, \pm 1, \ldots )\right\}$ onto the half-plane $\left\{ w: \mathrm{Im}\left(w\right) > 0\right\}$.

Find some functions $w(z)$ that conformally map the regions depicted in Fig. 29.34-29.36 onto the half-plane $\left\{ w: \mathrm{Im}\left(w\right) > 0\right\}$.

Let the function $f(z)$ be regular on $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$, continuous on $\left\{ z: \mathrm{Im}\left(z\right) \geqslant 0\right\}$, and take real values on the real axis. Prove that the function $f(z)$ can be regularly continued to the entire complex plane.

Let the function $f(z)$ be regular on $\left\{ z: 0 < \mathrm{Re}\left(z\right) < 1\right\}$, continuous on $\left\{ z: 0 \leqslant \mathrm{Re}\left(z\right) \leqslant 1\right\}$, and take real values on the lines $\left\{ z: \mathrm{Re}\left(z\right) = 0\right\}$ and $\left\{ z: \mathrm{Re}\left(z\right) = 1\right\}$. Prove that the function $f(z)$ can be regularly continued to the entire complex plane and that the function $F(z)$ effecting this analytic continuation satisfies the condition $F(z+2) = F(z)$.

Let the function $f(z)$ be regular on $\left\{ z: 0 < \mathrm{Im}\left(z\right) < 1\right\}$, continuous on $\left\{ z: 0 \leqslant \mathrm{Im}\left(z\right) \leqslant 1\right\}$, and satisfy the conditions

Prove that the function $f(z)$ can be regularly continued to the entire plane and that the function $F(z)$ effecting this analytic continuation satisfies the condition $F(z+2 i) = -F(z)$.

Let the function $f(z)$ be regular in the rectangle

continuous in the closure of this rectangle, and satisfy the conditions

Prove that the function $f(z)$ can be regularly continued to the strip $\left\{ z:\left|\mathrm{Im}\left(z\right)\right| < h\right\}$ and that the function $F(z)$ effecting this continuation has the form $i z+F_{1}(z)$, where the function $F_{1}(z)$ is regular in the strip $\left\{ z:\left|\mathrm{Im}\left(z\right)\right| < h\right\}$ and periodic with period 2.

Let the function $f(z)$ be regular in the half-annulus

continuous in its closure, and take real values on the segments $(-R,-\rho )$ and $(\rho , R)$. Prove that the function $f(z)$ can be regularly continued into the annulus $\left\{ z: \rho < \left|z\right| < R\right\}$ to a function regular in this annulus.

Let the function $f(z)$ conformally map the rectangle $\left\{ z:\left|\mathrm{Re}\left(z\right)\right| < h,\left|\mathrm{Im}\left(z\right)\right| < h\right\}$ onto some disk (or half-plane). Prove that the regular continuation of the function $f(z)$ is a function meromorphic in the entire plane and periodic with periods $4 h$ and $4 i h$.

Let the function $f(z)$ conformally map the sector

onto the triangle with the same vertices $0,1, e^{\pi i / n}$, in such a way that these vertices remain fixed. Prove that the function $f(z)$ can be regularly continued into the disk $\left\{ z:\left|z\right| < 1\right\}$ and that the function $F(z)$ effecting this analytic continuation conformally maps the disk $\left\{ z:\left|z\right| < 1\right\}$ onto a regular $2 n$-gon (with center at the point 0 and one of its vertices at the point 1).

Prove that any conformal mapping of a disk onto a disk (or onto a half-plane) is a fractional-linear mapping.

Prove that there is no function conformally mapping the annulus $\left\{ z: 1 < \left|z\right| < R_{1}\right\}$ onto the annulus $\left\{ z: 1 < \left|z\right| < R_{2}\right\}$ if $R_{1} \neq R_{2}$.

Let the function $f(z)$ conformally map the half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$ onto the region:

Prove that the regular continuation of the function $f(z)$ leads to a function $F(z)$ regular in the entire extended complex plane with three punctured points (the preimages of the points 0, 1, and $\infty$).