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\def\Re{\mathop{\mathrm{Re}}\nolimits}
\def\Im{\mathop{\mathrm{Im}}\nolimits}

\begin{document}

\begin{center}
{\large\textbf{��2�񕡑f��͊wI���K���� (2006�N10��13�����{)}}
\end{center}

\begin{flushright}
�S������ ���n����/ TA �����M��Y�E�˖{�׎O
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[1]

\noindent
(1) �񒲘a��$(\varphi(z),i,-1,1)
=(z,1,i,-1)$�ł��邱�Ƃ���C$\varphi(z)=\frac{(1+2i)z+1}{z+(1-2i)}$�D

\noindent
(2) (1)�Ɠ��l�D$\psi(z)=\frac{6iz+2}{z+3i}$�D

\noindent
(3) 1���ϊ���3�_�̑����w�肷��Έ�ӓI�ɒ�܂邱�Ƃ���C
���قȂ�3�_��s���_�Ƃ���1���ϊ��͍P���ʑ�$z$�Ɍ�����D

\noindent
(4) �C�ӂ�3�_���C�ӂ�3�_�ɂ���1���ϊ����K�����݂��C�܂��C
3�_�̑��ɂ����1���ϊ��͈�ӓI�Ɍ��܂�D
����$a,b$�͕s���_�ł��邽�߁C�Ⴆ�΁C$0$�̑���$\lambda\in\hat{\C}
\setminus\{a,b\}$��
����悤��1���ϊ�$\tau$�����߂�΂悢�D
$\lambda=\infty$, $\lambda\neq \infty$�ɉ����āC���ꂼ��
\begin{align*}
\frac{\tau(z)-a}{\tau(z)-b}
=\frac{z-a}{z-b},\quad
\frac{\tau(z)-a}{\tau(z)-b}
\left/\frac{\lambda-a }{\lambda-b}\right.
=\frac{z-a}{z-b}
\end{align*}
�ł��邪�C��1����$\frac{\lambda-a }{\lambda-b}=1$�Ɖ��߂������̂�
���Ȃ�Ȃ��̂ŁC���ǁC���߂�1���ϊ���
\[\frac{\tau(z)-a}{\tau(z)-b}=\mu\frac{z-a}{z-b}, \quad
\mu\in \C\setminus\{0\}\]
�ł��ׂĐs�������D
$\tau(z)=\frac{(a-\mu b)z+(\mu-1)ab}{(1-\mu )z+a\mu-b}$, 
$\mu\in \C\setminus\{0\}$.



\medskip

[2] 
{\bf ����}
(1)�ɂ�����$\alpha$�����ł��邩�𖾋L���Ă��܂���ł����D
��蕶�́u$\Delta (1)=\left\{|z|<1\right\}$�Ƃ���D�v�Ƃ����ӏ����C
�u$\Delta (1)=\left\{|z|<1\right\}$, $\alpha\in \Delta(1)$�Ƃ���D�v
�ɒ������܂��D�\���󂠂�܂���ł����D

\noindent
(1) $\alpha$, $\bar{\alpha}^{-1}$�͒P�ʉ~�Ɋւ��āC�݂��ɋ����ł��邩��C
$\varphi_\alpha\left(\bar{\alpha}^{-1}\right)=\infty$�ł���D
����āC�Ⴆ�΁C$\varphi_\alpha\left(z\right)=
\frac{z-\alpha}{-\bar{\alpha}z+1}$�Ƃ����D
�����
\[|z|< 1\iff 1-\left|\varphi_\alpha(z)\right|^2
=\frac{\left(1-\left|\alpha\right|\right)^2\left(1-\left|z\right|\right)^2
}{\left|\bar{\alpha}z-1\right|^2}>0\]
�ł��邩��C���߂�1���ϊ�������ꂽ�D

\noindent
(2) �q���g�ɏ]����$\tau\circ\varphi_{\tau^{-1}(0)}^{-1}$��
$0,\infty$�����ꂼ��$0,\infty$�ɂ����̂ŁC$\lambda z$�̌`�ł���D
����ɒP�ʉ~��P�ʉ~�ɂ������Ƃ���C
$e^{i\theta}z$, $\theta\in\R$�̌`�ɂȂ�C
$\tau(z)=e^{i\theta}\varphi_{\tau^{-1}(0)}
=e^{i\theta}\frac{z-\alpha}{-\bar{\alpha}z+1}$, 
$\alpha=\tau^{-1}(0)\in\Delta(1)$�𓾂�.


\noindent
(3) $i$�̎����Ɋւ��鋾��$-i$��$\infty$�ɂ��邱�Ƃɒ��ӂ���D
�Ⴆ�΁C$\psi(z)=\frac{z-i}{z+i}$�D

\noindent
(4) $\sigma$�͏㔼���ʂ��㔼���ʂɂ���1���ϊ��ł���Ƃ���D
$\psi\circ \sigma\circ \psi^{-1}$��$\Delta(1)$��$\Delta(1)$�̏��
�����̂ŁC
$\psi\circ \sigma\circ \psi^{-1}=\varphi_\alpha$�Ə�����D
�䂦��$\sigma=\psi^{-1}\circ \psi\circ \sigma\circ \psi^{-1}\circ \psi
=\psi^{-1}\circ \varphi_\alpha\circ \psi$�Ə�����D

\medskip

[3]

\noindent
\begin{gather*}
\sum_{n=0}^{N}\left|c_n\right|
\le \sum_{0\le n+k\le N}|a_n||b_k|
\le \sum_{0\le n\le N,\  0\le k\le N}|a_n||b_k|
\le
\sum_{n=0}^{\infty}\left|a_n\right|\sum_{n=0}^{\infty}\left|b_n\right|,\\
\begin{split}
\left|\sum_{n=0}^{2N}a_n\sum_{n=0}^{2N}b_n
-\sum_{n=0}^{2N}\sum_{k=0}^n a_{n-k}b_k\right|
&\le \sum_{2N<n+k,\atop n\le 2N,k\le 2N}|a_n||b_k|
\le \sum_{0\le n\le 2N,\atop 0\le k\le 2N}|a_n||b_k| 
-\sum_{0\le n\le N,\atop  0\le k\le N}|a_n||b_k|\\
&
= \sum_{n=0}^{2N}|a_n|\sum_{n=0}^{2N}|b_n| 
-\sum_{n=0}^N\left|a_n\right|\sum_{n=0}^N\left|b_n\right|
\end{split}
\end{gather*}
�̂��ꂼ��̕s��������C��Ύ������邱�Ƃ��킩��C�a�����܂�D


\smallskip
\noindent
{\bf (�R�����g)}
[1] 
(1),(2) �񒲘a���p������@�̑��ɁC$1,i,-1$��$0,1,\infty$�ɂ���1���ϊ��ƁC
$i,-1,1$��$0,1,\infty$�ɂ���1���ϊ��̋t�ϊ�������������@������܂��D
�܂�$\varphi(z)=\frac{az+b}{cz+d}$�Ƃ����āC��������$a,b,c,d$��
���肷�邱�Ƃ��o���܂��D

\noindent
(3) 2���������ɋA������𓚂������Ƃ��Ă��܂����C���̏ꍇ�C
�����ɂ�$\infty$���s���_�ɂȂ��Ă���ꍇ�������Ă��܂��܂��D
(�܂�$\frac{a\infty+b}{c\infty+d}=\infty$�̈Ӗ�
���l����K�v���łĂ��Ă��܂��܂��D)
�񒲘a���p���Ă��ؖ��ł��܂��D

\noindent
[2]
(1) $\varphi_\alpha$�̉��ɕt���Ă���$\alpha$�̈Ӗ���������Ȃ��Ƃ���
���₪����܂����D�P�ʉ~��P�ʉ~�ɂ���1���ϊ�$\varphi$��
�����ɑ��݂��܂����C
���ɂ���悤�ɁC�P�ʉ~���̂ǂ̓_��$0$�ɂ��邩���w�肷��΁C
�i���̂悤�Ȃ��͕̂K�����݂��āC�X�Ɂj��ӓI�Ɍ��܂��Ă��܂��܂��D
�܂�P�ʉ~��P�ʉ~�ɂ���1���ϊ�$\varphi$��$\Delta(1)$��
�_$\alpha$��p���ăp�����[�^�t���ł��邽�߁C
�����$\varphi_\alpha$�Ə����Ă���̂ł��D

\noindent
(5) �㔼���ʂ��㔼���ʂɉf��1���ϊ��͈�ʂ�
\[\sigma(z)=\frac{az+b}{cz+d},\quad
\left(
\begin{array}{cc}
a&b\\
c&d
\end{array}
\right)
\in SL\left(2,\R\right)
\]
�Ə�����܂��D������$SL\left(2,\R\right)$��2�����s��ōs�񎮂�$1$�ł���
���̑S�̂̏W���ł��D

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\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 (�����쐬: �ɓ����ꂳ��̃t�@�C����˖{���ҏW)
\end{document}