\def\arg{\mathop{arg}\nolimits}
\let\rule\hrule
\def\hrule{\goodbreak\medskip\rule\medskip\goodbreak}

%page 293
\noindent{\bf 2.}

The graph of the contour image encircles the origin three times, giving
$\Delta_C \arg f(z) = 2\pi\cdot 3 = 6\pi$ and, because it is analytic
inside and on $C,$ $P = 0 \to 3 = Z - P = Z,$ or there are three zeroes,
counting multiplicities, on $f(z)$ within $C.$

\noindent{\bf 7.}

\noindent{\it (a)}

On $|z|=2,$
$$36 = |9z^2| \geq 35 = 16 + 16 + 2 + 1 \geq |z^4 - 2z^3 + z - 1|,$$
meaning the circle contains the same number of zeroes on the original
and $9z^2,$ or 2, by Rouch\'e's theorem.

\noindent{\it (b)}

Similarly,
$$32 = |z^5| \geq 29 = 24 + 4 + 1 \geq |3z^3+z^2+1|,$$
giving 5 zeroes by the same method.

\hrule
%page 301
\noindent{\bf 2.}

$$w = iz + i = i(x+yi) + i = -y + (x+1)i = u+vi \to v = x+1,$$
so if $x>0,$ $v>1.$

\noindent{\bf 5.}

$$w = (1-i)z = (1-i)(x+yi) = x - ix + yi + y = (x+y) + (y-x)i = u+vi,$$
so by manipulating $u=x+y$ and $v=y-x,$
$$y = (u+v)/2 > 1 \to u+v > 2 \leftrightarrow v > 2-u.$$

\hrule
%page 305
\noindent{\bf 3.}

$$y = {-v\over u^2+v^2} > c_2 \to c_2(u^2+v^2) + v < 0,$$
which if $c_2>0$ becomes
$$c_2u^2 + c_2(v+1/2c_2)^2 < 1/c_2^2,$$
which is the interior of a circle.

If $c_2 = 0,$ this is $v<0,$ or a half-plane, and if $c_2<0,$ the
relation is inverted, and it is transformed to the exterior of a circle.

\noindent{\bf 7.}

This is first a leftward translation of $1$ and then the $w = 1/z$
transform, or an inversion across $|z-1| = 1,$ then a reflection across
the x-axis.

\noindent{\bf 13.}

With $z = r_0e^{i\theta},$

$$w = z + {1\over z} = r_0e^{i\theta} + {1\over r_0}e^{-i\theta} =
r_0(\cos\theta + i\sin\theta) + {1\over r_0}(\cos\theta - i\sin\theta) =
u + vi$$

$$u = (r_0+1/r_0)\cos\theta,\quad v = (r_0-1/r_0)\sin\theta.$$

\hrule
%page 311

\noindent{\bf 3.}

Since $0\mapsto\infty,$ $c\neq0$ and $d=0,$ giving
$$w = {az+b\over cz}.$$
$\infty\mapsto0,$ so $a=0,$ and $i\mapsto i$ gives
$$i = {b\over ci} \to -c = b,$$
thus
$$w = {-c\over cz} = -{1\over z}$$

\noindent{\bf 9.}

If $f(0) = 0,$ and with $AB \neq 0,$
$${w(w_2-w_3)\over (w-w_3)w_2} = {z(z_2-z_3)\over(z-z_3)z_2} \to
Aw(z-z_3) = Bzw - Bzw_3 $$$$ w(A(z-z_3)-Bz) = - Bzw_3 \to w =
{z\over (B-A)z/(Bw_3) - z_3/w_3},$$
QED.

\hrule
%page 325

\noindent{\bf 4.}

$$w = e^z = e^{x+iy} = e^xe^{iy} = \rho e^{i\phi} \to \rho = e^x,\quad
\phi = y.$$

From the original constraints,
$$0\leq\phi\leq\pi, \quad \rho\geq1.$$

\def\li{\leavevmode\llap{\hbox to \parindent{\hfil$\bullet$\hfil}}}
\noindent This gives boundaries:

\li the inner semicircle of radius untiy, corresponding to the $x=0$
boundary,

\li the positive x-axis, corresponding to the $y=0$ boundary, and

\li the negative x-axis, corresponding to the $y=\pi$ boundary.

\noindent{\bf 9.}

Using the following parametric representations,
$$u = \sin x\cosh y,\quad v = \cos x\sinh y,$$
the bottom of the rectangular region $y = a$ and $-\pi\leq x\leq\pi$
becomes $u = \cosh a\sin x,$ $v = \sinh a\cos x,$ which is the
parametrization of a complete ellipse with minor and major axes $\sinh
a$ and $\cosh a,$ respectively.
Similary, the upper bound parametrizes an ellipse with minor and major
axes $\sinh b$ and $\cosh b.$
This implies that the region between them continuously fills the space
between them.

The cut is between $x=\pi$ and $x=-\pi$ because $\sin\pi = \sin(-\pi)$
and $\cos\pi=\cos(-\pi),$ meaning these boundaries overlap in the image.

\bye
