\textbf{Problem 3}

Consider a mapping $w = f ( z )$ of a domain $D$ on the complex $z$ plane to a domain $\Delta$ on the complex $w$ plane. Points on the complex $z$ and $w$ planes correspond to complex numbers $z = x + i y$ and $w = u + i v$, respectively. Here, $x , y$, $u$ and $v$ are real numbers, and $i$ is the imaginary unit.

I. Let $w = \sin z$.

\begin{enumerate}
  \item Express $u$ and $v$ as functions of $x$ and $y$, respectively.
  \item Suppose the domain $D _ { 1 } = \left\{ ( x , y ) \left\lvert \, 0 \leq x \leq \frac { \pi } { 2 } \right. , y \geq 0 \right\}$ on the $z$ plane is transformed to a domain on the $w$ plane. Show the transformed domain on the $w$ plane by drawing the transformed images corresponding to the three half-lines: $x = 0 , x = \frac { \pi } { 2 }$ and $x = c$ at $y \geq 0$ on the $z$ plane. Here, $c$ is a real constant on $0 < c < \frac { \pi } { 2 }$.
\end{enumerate}

II. If a real function $g ( x , y )$ has continuous first and second partial derivatives and satisfies Laplace's equation $\frac { \partial ^ { 2 } g } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } g } { \partial y ^ { 2 } } = 0$ in a domain $\Omega$ on a plane, $g ( x , y )$ is said to be harmonic in $\Omega$.

Suppose that a function $f ( z ) = u ( x , y ) + i v ( x , y )$ is holomorphic in $D$ on the $z$ plane:

\begin{enumerate}
  \item Show both $u ( x , y )$ and $v ( x , y )$ are harmonic in $D$ on the $z$ plane.
  \item Suppose a function $h ( u , v )$ is harmonic in $\Delta$ on the $w$ plane, show a function $H ( x , y ) = h ( u ( x , y ) , v ( x , y ) )$ is harmonic in $D$ on the $z$ plane.
\end{enumerate}

III. Suppose a function $h ( u , v )$ is harmonic in the domain $\Delta _ { 1 } = \{ ( u , v ) \mid u \geq 0 , v \geq 0 \}$ on the $w$ plane and satisfies the following boundary conditions:

$$\begin{aligned}
& h ( 0 , v ) = 0 \quad ( v \geq 0 ) \\
& h ( u , 0 ) = 1 \quad ( u \geq 1 ) \\
& \frac { \partial h } { \partial v } ( u , 0 ) = 0 \quad ( 0 \leq u \leq 1 )
\end{aligned}$$

\begin{enumerate}
  \item Let $z = \arcsin w$ and $H ( x , y ) = h ( u , v )$. Find the boundary conditions for $H ( x , y )$ corresponding to Equations (1), (2) and (3). Use the principal values of inverse trigonometric functions.
  \item Find the function $H ( x , y )$ which satisfies the boundary conditions obtained in Question III.1.
  \item Find $h ( u , 0 )$ on the interval $0 \leq u \leq 1$.
\end{enumerate}