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\begin{document}

\begin{center}
{\large\textbf{��4�񕡑f��͊wI���K���� (2006�N10��27�����{)}}
\end{center}

\begin{flushright}
�S������ ���n����/ TA �����M��Y�E�˖{�׎O
\end{flushright}

\bigskip

[1]

\noindent
(1)\ 
\begin{align*}
\int_{|z|=1} |z-1|\,|dz|
=\int_0^{2\pi}\left|e^{i\theta}-1\right|\,d\theta
=\int_0^{2\pi}\sqrt{\left(e^{i\theta}-1\right)
\left(e^{-i\theta}-1\right)}\,d\theta
=\int_0^{2\pi}2\sin\frac{\theta}{2}\,d\theta=8�D
\end{align*}

\noindent
(2)\ $n\le0$�̂Ƃ���Cauchy�̐ϕ��藝���$0$. $n>0$�̂Ƃ�
$e^z=\sum_{j=0}^\infty z^j/j!$�͐ϕ��H��ň�l��������̂�
\begin{align*}
\int_{|z|=1} e^z z^{-n}\,dz
=\sum_{j=0}^\infty \int_{|z|=1}\frac{z^{j-n}}{j!}\,dz
=\frac{2\pi i}{(n-1)!},\quad
\text{($j=n-1$�ȊO�̍���$0$)}
\end{align*}

\noindent
(3)\ ���������������Čv�Z����DCauchy �̐ϕ�������p�����
\begin{align*}
\int_{|z|=2} \frac{dz}{z^2+1}
=\frac{1}{2i}\int_{|z|=2} \frac{dz}{z-i}
-\frac{1}{2i}\int_{|z|=2} \frac{dz}{z+i}
=\pi-\pi=0�D
\end{align*}

\noindent
(4)\ $\alpha=0$�̂Ƃ�
\[\int_{|z|=1}|z-\alpha|^{-4}|dz|
=\int_0^{2\pi}1\, d\theta
=2\pi.\]
$\alpha\neq 0$�̂Ƃ�
\begin{align*}
\int_{|z|=1}|z-\alpha|^{-4}|dz|
&=\int_0^{2\pi}\frac{d\theta}{\left(e^{i\theta}-\alpha\right)^2
\left(e^{-i\theta}-\bar{\alpha}\right)^2}
=\frac{1}{i\bar{\alpha}^2}
\int_0^{2\pi}\frac{e^{i\theta}\cdot ie^{i\theta}d\theta}{
\left(e^{i\theta}-\alpha\right)^2
\left(e^{i\theta}-\frac{1}{\bar{\alpha}}\right)^2}\\
&=\frac{1}{i\bar{\alpha}^2}
\int_{|z|=1}\frac{z\,dz}{
\left(z-\alpha\right)^2
\left(z-\frac{1}{\bar{\alpha}}\right)^2}
\end{align*}
�ƂȂ�D$|\alpha|<1$�̂Ƃ��C$z\left(z-\frac{1}{\bar{\alpha}}\right)^{-2}$��
$|z|\le 1$���܂ޗ̈�Ő����ł��邱�Ƃ���CCauchy�̐ϕ������ɂ��C
\begin{align*}
\int_{|z|=1}|z-\alpha|^{-4}|dz|
=\frac{2\pi i}{i\bar{\alpha}^2}\left. \frac{d}{dz}\right|_{z=\alpha}
z\left(z-\frac{1}{\bar{\alpha}}\right)^{-2}
=2\pi\left(1+\left|\alpha\right|^2\right)
\left(1-\left|\alpha\right|^2\right)^{-3}
\end{align*}
�ł���D�܂�$|\alpha|>1$�Ƃ������l�ɂ��āC
\begin{align*}
\int_{|z|=1}|z-\alpha|^{-4}|dz|
=\frac{2\pi i}{i\bar{\alpha}^2}\left. \frac{d}{dz}
\right|_{z=\frac{1}{\bar{\alpha}}}
z\left(z-\alpha\right)^{-2}
=2\pi\left(1+\left|\alpha\right|^2\right)
\left(\left|\alpha\right|^2-1\right)^{-3}.
\end{align*}
�܂Ƃ߂�ƁC$|\alpha|\lessgtr 1$�ɉ����Ă��ꂼ��
\[\int_{|z|=1}|z-\alpha|^{-4}|dz|=
\pm 2\pi\left(1+\left|\alpha\right|^2\right)
\left(1-\left|\alpha\right|^2\right)^{-3}\]

\medskip

[2] 

$|f(\gamma(t)) \gamma'(t)|$ �� $I =[0, 1]$ ��L�E�Ȃ̂� $< R_0$ �Ȃ�$R_0$������D
$z$,$w$��$\C$�̌��Ƃ��C$z$��C�ӂɌŒ肷��D$C=\gamma(I)$�Ƃ�����
\[
\left|\frac{1}{(\gamma(t) - z)(\gamma(t) - w)}\right| < \frac{1}{\text{dist} (z, C) \text{dist} (w, C)}
\]
�ł���
$\text{dist} (w, C)$��$w$�Ɋւ���A���֐��Ȃ̂ŁC����$\epsilon_0$,$R_1$�������āC$|z - w| < \epsilon_0$�Ȃ�
\[
 \frac{1}{\text{dist} (z, C) \text{dist} (w, C)} < R_1.
\]
$\delta$���C�ӂɗ^����ꂽ�Ƃ��C$\epsilon_1 R_0 R_1 < \delta$�Ȃ�$\epsilon_1$���Ƃ�ƁC$|z - w| < \min (\epsilon_0, \epsilon_1)$ �Ȃ��$|F(z) - F(w)| < \delta$�ł���D

$F(\infty) = 0$�Ƃ����Ƃ�$\infty$�ł�$F$�͘A���ɂȂ�D���� $\delta$���C�ӂɗ^����ꂽ�Ƃ���D$C$�͗L�E�Ȃ̂ŁC����$R_2$�������āC $|z|$��$R_2$���傫���Ƃ�$\text{dist}(z, C) > R_0 / \delta$�ƂȂ��Ă���C���̂Ƃ�$|F(z)| < \delta$.

\medskip

[3]

\noindent
$\sin$�̃e�[���[�W�J��p�����$z \ne 0$�̂Ƃ�
$f(z)=1-\frac{1}{3!} z^2+\frac{1}{5!} z^4
-\cdots+(-1)^{n}\frac{1}{(2n+1)!}z^{2n}+\cdots
$�ƂȂ�D�������C$z=0$�ł��C���̓����͐�������.
����āC���̓����̍��ӂ͎������a��$\infty$������C
$f$��$\C$��̐����֐����߂Ă���Ƃ�����D

[4]

\noindent
\begin{itemize}
\item[{(1)}]
Cauchy�̐ϕ�������莟�𓾂�.
\[
\frac{1}{2\pi i}\int_{|\zeta|=r} \frac{f(\zeta)}{\zeta -0} d\zeta
=
f(0).
\]
���̍��ӂ�$\zeta$��$\zeta = r e^{i \theta}$�Ƃ��ĕ\���΋��߂铙���𓾂�.
\item[{(2)}]
�����͂��̒ʂ�.
\begin{align*}
\int \int_{|z|\leq R} f(z) dxdy 
&= \int_0^R \int_0^{2\pi} f(re^{i \theta})r d\theta dr && \text{(�ɍ��W�\��)}\\
&= \int_0^R r \int_0^{2\pi} f(re^{i \theta}) d\theta dr  \\
&= 2\pi f(0) \int_0^R r dr && \text{($\because$ (1)���)} \\
&= \pi R^2 f(0).
\end{align*}
\end{itemize}

\bigskip

\textbf{���f��͊wI�T�C�g�ł͉��K�Ŕz�z����v�����g��u�`�������ڂ��Ă��܂�:
\begin{center}
http://www.ms.u-tokyo.ac.jp/\~{}hirachi/courses/complex1-2006/
\end{center}}

\hfill (�𓚍쐬: �˖{�׎O)
\end{document}
