Symmetry principle (Schwarz reflection)

Let $f(z)$ be analytic for $z \neq 0$, and suppose that $f(1/z) = f(z)$. Suppose also that $f(z)$ is real for all $z$ on the unit circle $\left|z\right| = 1$. Prove that $f(z)$ is real for all real $z \neq 0$.

Suppose $f$ is meromorphic in a neighborhood of $\overline{D}$ ($D = \left\{ \left|z\right| < 1\right\}$) whose only pole is a simple one at $z = a \in D$. If $f(\partial D) \subseteq \mathbb {R}$, show that there is a complex constant $A$ and a real constant $B$ such that

Let $\gamma$ be an arc of the unit circle. Suppose that $u$ and $v$ are harmonic in $D = \left\{ z : \left|z\right| < 1\right\}$ and continuously differentiable on $D \cup \gamma$. If the boundary values satisfy $u = v$ on $\gamma$ and the radial derivatives satisfy $\dfrac {\partial u}{\partial r} = \dfrac {\partial v}{\partial r}$ on $\gamma$, prove that $u = v$ in $D$.

Let $0 < \varepsilon < \pi$, let $\gamma_\varepsilon$ denote the arc $\left\{ e^{it} : \varepsilon \leq t \leq 2\pi -\varepsilon \right\}$ and let $\Omega_\varepsilon$ be the complement of $\gamma_\varepsilon$ in the Riemann sphere. If $f$ is the conformal map of the unit disk onto $\Omega_\varepsilon$, $f(0) = 0$, $f'(0) > 0$, describe the part of the unit disc that $f$ maps onto $\left\{ \left|z\right| > 1\right\}$.

Let $f$ be holomorphic in the unit disk $\left\{ \left|z\right| < 1\right\}$, continuous in $\left\{ \left|z\right| \leq 1\right\}$ and $\left|f(z)\right| = 1$ whenever $\left|z\right| = 1$. Prove that $f$ is a rational function.

Let $\mathfrak {R}$ be a region symmetric with respect to the real axis, $a$ real. Then the power series expansion of the normed mapping function associated with $a$ in powers of $z-a$ has only real coefficients. A similar result holds for the normed mapping function associated with the point at infinity as central point, if $\mathfrak {R}$ contains the point at infinity.

For which pairs of lines $L_{1}, L_{2}$ do there exist real functions, harmonic in the whole plane, that are 0 at all points of $L_{1} \cup L_{2}$ without vanishing identically?

In Theorem 11.14 the hypotheses were, in simplified form, that $\Omega \subset \Pi^{+}, L$ is on the real axis, and $\mathrm{Im}\left(f(z)\right) \rightarrow 0$ as $z \rightarrow L$. Use this theorem to establish analogous reflection theorems under the following hypotheses:

Here is yet another proof of Theorem 14.22. If $f$ is as in 14.22, repeated use of the reflection principle extends $f$ to an entire function such that $\left|f(z)\right| = 1$ whenever $\left|z\right| = 1$. This implies $f(z) = \alpha z^{n}$, where $\left|\alpha \right| = 1$ and $n$ is an integer. Complete the details.

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 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 $\infty$).