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\begin{document}

{\gt
�ŵ����ر齬���� {\bf 2000}ǯ{\bf 11}��{\bf 16}�� ʬ
��������ʹֵ����������ʤ��ä�ʬ��
}

\vspace{1zh}
\newcommand{\xys}{x^2+y^2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\fbox{\begin{minipage}{0.95\textwidth}
[4]\quad
������ʬ��
\begin{equation}
 v(x,y)= 2y+\frac{x}{\xys}
\label{eq:v}
\end{equation}
��Ϳ��������§�ؿ��μ¿���ʬ$u(x,y)$ ����衣
�ޤ�������§�ؿ�$f(z)=u+iv$ �� $z=x+iy$ �δؿ��Ȥ���ɽ����
\end{minipage}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1zh}

�����������꡼�ޥ������
\begin{equation}
u_x = v_y
\label{eq:ux=vy}
\end{equation}
\begin{equation}
u_y = -v_x
\label{eq:uy=-vx}
\end{equation}
�����Ѥ��롣

�ޤ� $u_x, \ u_y$ ��׻����Ƥ�����

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[
\left(
\begin{minipage}{0.95\textwidth}
�ޤ���ʬ������ʬ��
\begin{equation}
\left(\frac{f}{g}\right)' = \frac{f'g-fg'}{g^2}
\end{equation}
�Ǥ��뤳�Ȥ�����Ȥ���Ƴ���Ƥ�����
������Ѥ����
\begin{equation}
 \frac{\partial}{\partial x}\left(\frac{x}{\xys}\right) = 
 \frac{1\cdot(\xys)-x(2x)}{(\xys)^2} = \frac{-x^2+y^2}{(\xys)^2}
 \label{eq:x/dx}
\end{equation}
����� $x, y$ �����촹����С��虜�虜�׻����ʤ��Ƥ�
\begin{equation}
 \frac{\partial}{\partial y}\left(\frac{y}{\xys}\right) = 
 \frac{-y^2+x^2}{(\xys)^2}
 \label{eq:y/dy}
\end{equation}
�������롣
Ʊ�ͤˤ���
\begin{equation}
 \frac{\partial}{\partial x}\left(\frac{y}{\xys}\right) =
  \frac{2xy}{(\xys)^2}
 \label{eq:y/dx}
\end{equation}
��׻����Ƥ�����
\begin{equation}
 \frac{\partial}{\partial y}\left(\frac{x}{\xys}\right) =
  \frac{2xy}{(\xys)^2}
 \label{eq:x/dy}
\end{equation}
�������롣
�֤ʤ�Ǥ���ʷ׻��������ǤƤ���Ρ��פȻפä����⤷��ʤ�����
��ǻȤ��Τǡ������ǤޤȤ�Ʒ׻������Ƥ����������ʾС�
\end{minipage}
\right)
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

��(\ref{eq:v})��$x$������ʬ����ȡ���(\ref{eq:x/dx})���顤
\begin{equation}
 v_x = \frac{-x^2+y^2}{(\xys)^2}
\label{eq:vx}
\end{equation}
�Ȥʤ롣$y$������ʬ����ȡ���(\ref{eq:x/dy})���顤
\begin{equation}
 v_y = 2 + \frac{2xy}{(\xys)^2}
\label{eq:vy}
\end{equation}
�Ȥʤ롣

��(\ref{eq:ux=vy}), (\ref{eq:uy})���顤
\begin{equation}
 u_x = 2 + \frac{2xy}{(\xys)^2}
\end{equation}
����ξ�դ� $x$ ��������ʬ����� $u$ ����ޤ�Τ�����
�虜�虜�ִ���ʬ����Ԥ�ʤ��Ƥ⡤��(\ref{eq:y/dx})�򸫤��
\begin{equation}
 u = 2x + \frac{y}{\xys} + F(y)
\end{equation}
�ȵ����롣
����� $y$ ������ʬ����ȡ��ʼ�(\ref{eq:y/dy})�����
\begin{equation}
 u_y = \frac{-y^2+x^2}{(\xys)^2} + F'(y)
\label{eq:uy}
\end{equation}
�Ȥʤ롣
������ꡤ��(\ref{eq:uy=-vx})������Ω�ĤΤǡ���(\ref{eq:uy})�ȼ�(\ref{eq:vx})
����Ӥ���ȡ�
\begin{equation}
 F'(y) = 0 
\end{equation}
���ʤ��
\begin{equation}
 F(y) = c
\end{equation}
�Ȥʤꡤ
\begin{equation}
 u = 2x + \frac{y}{\xys} + c
\label{eq:u}
\end{equation}
�Ǥ��뤳�Ȥ�ʬ���롣
��������$c$ ��Ǥ�դμ¿���

����ǡ���(\ref{eq:u})�ȼ�(\ref{eq:v})�Ȥ�ꡤ
\begin{equation}
 f(z) = u + iv = 2x + \frac{y}{\xys} + c + i\left( 2y+\frac{x}{\xys}\right)
\end{equation}
�Ȥʤ롣
����� $z= x+iy$ ��ɽ���褦�˹ͤ���Τ�����
���礨�� $x+iy$ �ΤޤȤޤ��Ĥ��äƤ����Ф褤��
��̤�Ϥ�����Ÿ������ȡ� ��$i(-i)=1$ ���Ѥ����
\begin{eqnarray*}
 f(z) &=& 2x + \frac{y}{\xys} + c + 2iy + i\frac{x}{\xys}
\\
      &=& 2(x + iy) + \frac{y+ix}{\xys} + c
\\
      &=& 2(x + iy) + \frac{i(-iy-i^2x)}{\xys} + c
\\
      &=& 2(x + iy) + i\frac{(x-iy)}{\xys} + c
\end{eqnarray*}
�Ȥʤ롣2���ܤ�ʬ��� $\xys=(x+iy)(x-iy)$ �Ȱ���ʬ��Ǥ���Τǡ�
\begin{eqnarray*}
 f(z) &=& 2(x + iy) + i\frac{(x-iy)}{(x+iy)(x-iy)} + c
\\
      &=& 2(x + iy) + i\frac{1}{x+iy} + c
\\
      &=& 2z + i\frac{1}{z} + c
\end{eqnarray*}
�Ȥʤ롣�����ޤ���


\vspace{4zh}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\fbox{\begin{minipage}{0.95\textwidth}
[6]\quad
$\sin z$���¿��ͤ���Τ�$z$ʿ�̤Τ����ʤ��礫��$\sin z = 0$��
��­���� $z$ ���ͤ���衣
\end{minipage}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1zh}

�ޤ���$x, \ y$ ���¿��ξ�硤
\begin{equation}
 \cos(ix) = {\rm cosh} \: x, \quad \sin(ix) = i \: {\rm sinh} \: x
  \label{eq:cossin1}
\end{equation}
\begin{equation}
 {\rm cosh}(ix) = \cos x, \quad {\rm sinh}(ix) = i\sin x
  \label{eq:cossin2}
\end{equation}
�θ�����פ��Ф����ʤ��������˺�줺�ˡ���
����ʣ�ǿ� $z, \  w$ ���Ф���
\begin{equation}
 \sin(z+w) = \sin z\cos w + \cos z\sin w
  \label{eq:sin_z+w}
\end{equation}
\begin{equation}
 \cos(z+w) = \cos z\cos w - \sin z\sin w
  \label{eq:cos_z+w}
\end{equation}
�θ�����פ��Ф���

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[
\left(
\begin{minipage}{0.95\textwidth}
��(\ref{eq:sin_z+w}), ��(\ref{eq:cos_z+w})�θ�����˺��Ƥ��ޤä�����
�����ͤ��롣

��(\ref{eq:sin_z+w}), ��(\ref{eq:cos_z+w})�� $z, \ w$ ���¿��ξ��Ǥ�
����Ω�ĤΤǡ��¿��ξ��θ�����Ƴ���Ƥ�פ��Ф��롣

�����顼�θ���
\begin{equation}
 e^{i\theta} = \cos\theta + i\sin\theta
\end{equation}
�ʤ����˺��ʤ��ǡ��Ȥ������˺�ɸ�ǹͤ���лפ��Ф��롣��
���Ѥ����
\begin{equation}
 e^{i(x+y)} = \cos(x+y) + i\sin(x+y)
\label{eq:e_x+y1}
\end{equation}
�ޤ����ؿ�����Ÿ������
\begin{equation}
 e^{ix+iy} = e^{ix}e^{iy} = \left(\cos x + i\sin x\right) \left(\cos y + i\sin y\right)
\label{eq:e_x+y2}
\end{equation}
�Ȥ�ͤ��뤳�Ȥ�����롣
���Ȥϼ�(\ref{eq:e_x+y2})�α��դ�Ÿ�����ơ������ȵ�����ʬ�����塤
��(\ref{eq:e_x+y1})�α��դ���Ӥ����
��(\ref{eq:sin_z+w}), ��(\ref{eq:cos_z+w})�θ�����Ƹ��Ǥ��롣
\end{minipage}
\right)
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$z=x+iy$ ��ɽ���ȼ�(\ref{eq:sin_z+w})��ꡤ
\begin{equation}
 \sin z = \sin(x+iy) = \sin x \: \cos(iy) + \cos x \: \sin(iy)
\end{equation}
�ޤ�����(\ref{eq:cossin1}), ��(\ref{eq:cossin2})����
\begin{equation}
 \sin z = \sin x \ {\rm cosh}\, y + i \cos x \ {\rm sinh}\, y
\end{equation}
������0�Ǥ��뤫�顤
\begin{equation}
 \cos x \ {\rm sinh}\, y  = 0
\end{equation}
��ä�
\begin{equation}
 \cos x = 0 \quad �ޤ��� \quad {\rm sinh}\, y  = 0
\end{equation}
�Ĥޤ�
\begin{equation}
 x = \frac{1}{2}\pi + k\pi  \quad (k=0, \pm1, \pm2, \ldots) \quad
  �ޤ��� \quad y  = 0
  \label{eq:imag=0}
\end{equation}
�λ��� $\sin z$ �ϼ¿��Ȥʤ롣

\vspace{1zh}

$\sin z=0$ �ξ��Ϲ��˼����� 0 �ʤΤǡ�
\begin{equation}
 \sin x \ {\rm cosh}\, y = 0
\end{equation}
${\rm cosh}\, y \ne 0$ �ʤΤǡ�
\begin{equation}
 \sin x = 0
\end{equation}
����ȼ�(\ref{eq:imag=0})�ξ����碌���
\begin{equation}
 x = y = 0
\end{equation}
���ʤ�� $z=0$ �ξ��ΤߤǤ��롣

\end{document}
