\def\Re{\mathop{\rm Re}\nolimits}
\def\Im{\mathop{\rm Im}\nolimits}
\def\Log{\mathop{\rm Log}\nolimits}
\let\rule\hrule
\def\hrule{\medskip\rule\medskip}

%page 95
\noindent{\bf 8.}

$$\log z = i\pi/2 \to z = e^{i\pi/2} = i,$$ by Euler's formula.

\hrule
%page 99
\noindent{\bf 2.}

\noindent{\it (a)}

Where $z_n = r_ne^{i\theta_n},$ for all values of $\theta_n,$
$$\log\left({z_1\over z_2}\right) = \ln|z_1/z_2| + i\arg(z_1/z_2)$$$$ =
(\ln(r_1) - \ln(r_2)) + i(\theta_1-\theta_2) = (\ln(r_1) + i\theta_1) -
(\ln(r_2) + i\theta_2) = \log z_1 - \log z_2,$$
because $\ln(a-b) = \ln a - \ln b$ and $\arg(z_1/z_2) = \arg(z_1) -
\arg(z_2).$

\noindent{\it (b)}

$1/z = \overline z/|z|^2.$ $$\log(1/z) = \ln|1/z| + i\arg(1/z) =
- \ln|z| - i\arg z = - (\ln|z| + i\arg z) = -\log(z),$$ so
$$\log(z*(1/z)) = \log(z) + \log(1/z) = \log(z) - \log(z) = 0.$$

\hrule
%page 103
\noindent{\bf 2.}

\noindent{\it (a)}

The principal value of $(-i)^i$ is $e^{i\Log(i)} = e^{i\pi/2}.$

\noindent{\it (b)}

The base of this power function is $e^(1-2\pi/3) = b \to \Log(b) = 1-2\pi i/3.$
This gives a principal value of $e^{3\pi i(1-2\pi/3)} = e^{3\pi i - 2\pi^2} = 
-e^{2\pi^2}.$

\noindent{\it (c)}

The principal value of $(1-i)^{4i}$ is $e^{\Log(1-i)4i} =
e^{(4i)(\ln(2)/2 - \pi/4)} = e^{(2\ln(2)i)-\pi} = e^{-\pi}(\cos(2\ln(2))
+ i\sin(2\ln(2))).$

\noindent{\bf 8.}

\noindent{\it (a)}

$$z^{c_1}z^{c_2} = e^{c_1\Log(z)}e^{c_2\Log(z)} = e^{\Log(z)(c_1+c_2)} =
z^{c_1+c_2}.$$

\noindent{\it (b)}

$${z^{c_1}\over z^{c_2}} = {e^{\Log(z)c_1}\over e^{\Log(z)c_2}} =
e^{c_1\Log z - c_2\Log z} = e^{\Log(z)(c_1-c_2)}.$$

\noindent{\it (c)}

$$(z^c)^n = (e^{c\Log z})^n = e^{nc\Log z} = z^{cn}.$$

\hrule
%page 107
\noindent{\bf 3.}

$$\sin(z+z_2) = \sin z \cos z_2 + \cos z \sin z_2 \to
\cos(z+z_2)   = \cos z \cos z_2 - \sin z \sin z_2 \to
\cos(z_1+z_2) = \cos z_1\cos z_2-\sin z_1\sin z_2.$$

\noindent{\bf 7.}
% need to do more here

$\sin z = \sin x \cosh y + i\cos x \sinh y.$
$\cos z = \cos x \cosh y - i\sin x \sinh y.$

$$|\sin z|^2 = (\sin x\cosh y)^2 + (\cos x\sinh y)^2
= \sin^2 x(1+\sinh^2 y) + \cos^2 x\sinh^2 y =
\sin^2 x + \sinh^2 y.$$

$$|\cos z|^2 = \cos^2 x \cosh^2 y + \sin^2 x \sinh^2 y
= \cos^2 x \cosh^2 y + (1-\cos^2 x)\sinh^2 y
= \sinh^2 y + \cos^2 x.$$


\noindent{\bf 12.}

$\sin z$ and $\cos z$ are real with $z$ on the real axis.
Therefore, $\overline{\sin z} = \sin\overline z$ and $\overline{\cos z}
= \cos\overline z.$

\hrule
%page 111
\noindent{\bf 6.}

\noindent{\it (a)}
$$|\cosh z|^2 = \sinh^2 x + \cos^2 y \to |\sinh x|^2 \leq |\cosh z|^2
\to |\sinh x|\leq |\cosh z|.$$
$$|\cosh z|^2 = \cosh^2 x + cos^2 y \to |\cosh z|^2 \leq \cosh^2 x \to
|\cosh z| \leq \cosh x,$$
because $\cosh x > 0.$

\noindent{\it (b)}
%9b of sec 38 would probably be good

$$|\sinh y| \leq |\cos z| \leq \cosh y.$$
We change $z$ to $z' = zi,$ so $y$ becomes $x.$
$$|\sinh x| \leq |\cosh z| \leq \cosh x.$$

\noindent{\bf 16.}
$$\sinh z = \sinh x \cos y + i\cosh x \sin y.$$
$$\cosh z = \cosh x \cos y + i\sinh x \sin y.$$

\noindent{\it (a)}
$$\sinh z = i = \sinh x\cos y + i\cosh x\sin y \to 1 = \cosh x\sin y =
\sin y,$$
because all zeroes of $\cosh x$ are on imaginary axis, so
$x = 0 \to \cosh x = 1.$

$\sin y = 1$ on $y = 2n\pi + \pi/2 \to z = (2n\pi + \pi/2)i.$

\noindent{\it (b)}
$$\cosh z = \cosh(yi) = 1/2 = \cos y \to y = (2n\pi \pm \pi/3)i.$$

\hrule
%page 113
\noindent{\bf 1.}
%learn the derivation of inverse sin

\noindent{\it (c)}

$$\cosh^{-1}(-1) = \log\left[-1+0^{1/2}\right] = \log(1)
= 2n\pi i+\pi.$$

\hrule
%page 119
\noindent{\bf 2.}

\noindent{\it (a)}

$\int_0^1 (1+it)^2 dt = \int_0^1 1 + 2it - t^2 dt =
[t + it^2 - t^3/3]_0^1 = 1 + i - 1/3 = 2/3 + i.$

\noindent{\it (b)}

$\int_1^2 \left({1\over t} -i\right)^2 dt = \int_1^2 {1\over t^2} -
{2i\over t} - 1 dt = [-{1\over t} - 2i\ln t - t]_1^2 =
1-{1\over 2} - 2i\ln2 + 2 - 1 = 3/2 - 2i\ln2.$

\noindent{\it (c)}

$\int_0^{\pi/6} e^{i2t} dt = \int_0^{\pi/6} \cos(2t) + i\sin(2t) dt =
(1/2)[\sin(2t) - i\cos(2t)]_0^{\pi/6} (1/2)[\sqrt3/2 - i/2].$

\noindent{\it (d)}

$\int_0^\infty e^{-zt} dt = \lim_n\to\infty [-e^{-zt}/z]_0^n = 1/z -
\lim_t\to\infty e^{-zt}/z = 1/z.$

\hrule
%page 123
\noindent{\bf 1.}
%do no 2 to confirm knowledge of jordan curve

\noindent{\it (a)}

This is true for a real-valued function, so 
$$\int_{-b}^{-a} u(-t) dt + i\int_{-b}^{-a} v(-t) = \int_a^b u(t) dt +
i \int_a^b v(t) dt,$$
because $u$ and $v$ are real-valued, so the corresponding imaginary and
real components are equal.

\noindent{\it (b)}

This is true of a real function on $(a,b),$ so $$\int_a^b u(t) dt =
\int_\alpha^\beta u(\phi(\tau))\phi'(\tau) d\tau,$$
and similarly for $v(t),$ so
$$\int_a^b u(t)+iv(t) dt =
\int_\alpha^\beta (u(\phi(\tau)) + iv(\phi(\tau))\phi'(\tau) d\tau.$$

\bye
