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{\large\textbf{��9�񕡑f��͊wII���K (2011�N6��14�����{)}}
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�S������ ���n����/ TA �v�{�q�V�@���{���F
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�ȉ��̖������̉��K�̎��Ԃ́w�͂��߁x�Ƀ��|�[�g�Ƃ��Ē�o���邱��.


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{\bf [R12]}\ �ȉ��������F

(a) �ʑ����$X$��Hausdorff $\iff$ �Ίp���W��$\Delta X=\{(x,x):x\in X\}$��$X\times X$�̕W��

(b) �����֐��̑w�� Hausdorff ��Ԃł���D 

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�ȉ��͋������\�p�̖��ł�. 

\prob $\alpha,\beta,\gamma\in\C$, $\gamma\ne0,-1,-2,\dots$�ɑ΂��Ē��􉽋�����
$$
F(\alpha,\beta,\gamma;z)=1+\sum_{n=1}^\infty \frac{\alpha(\alpha+1)\cdots(\alpha+n-1)\beta(\beta+1)\cdots (\beta+n-1)}{n!\gamma(\gamma+1)\cdots(\gamma+n-1)}z^n
$$
�ɂ���Ē�`����D

(a) $F(\alpha,\beta,\gamma;z)$��$|z|<1$�Ŏ������邱�Ƃ������D���̊֐��𒴊􉽊֐��Ƃ����D

(b) $|z|<1$�ɂ����Ď��������F
$$F(\alpha,\beta,\gamma;z)=\frac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)}
\int_0^1 t^{\beta-1}(1-t)^{\gamma-\beta-1}(1-zt)^{-\alpha}dt
$$
�i�q���g: $(1-zt)^{-\alpha}$���ׂ������W�J����j

(c) ���􉽊֐��͐��̎����ɉ����ĉ�͐ڑ����\�ł��邱�Ƃ������D


\prob
$\Omega$ ��̈�Ƃ��C$u$ �� $\Omega$ ��̒��a�֐��Ƃ���D
���܁C$D \subset \Omega$ ���J�~�Ƃ��C$f$ ��
 $u = \textrm{Re}\, f$ �𖞂��� $D$ ��̐����֐��Ƃ���ƁC
$f$ �� $\Omega$ ���̂��ׂĂ̋Ȑ��ɉ����ĉ�͐ڑ��ł��邱�Ƃ�
�ؖ�����D

\prob
$\Omega = \mathbb{C} \setminus \{ 0 \}$ �Ƃ���D

(a) $\boldsymbol{f} = \mathrm{Log}\, z$ �Ƃ���ƁC
Riemann �� $\cal{R}(\boldsymbol{f})$ �� 
$\{ (z, e^z) : z \in \Omega \} \subset \mathbb{C} \times \mathbb{C}$ 
�ƈʑ����^�ł��邱�Ƃ��ؖ�����D

(b) $\boldsymbol{g} = \sqrt[n]{z}$�i������ $n \in \mathbb{N}$�j�ɂ���
 (a) �Ɠ��l�̋c�_������D

\prob
$\Omega \subset \mathbb{C}$ ��̈�C$a \in \Omega$�C
$\boldsymbol{f}$ ����I��͊֐��Ƃ���D���̂Ƃ��C
\[ \mathfrak{s}_a = \{ P(a) : P \in \boldsymbol{f}, \pi (P) = a \} \]
�ƒ�`����D

(a) $\mathfrak{s}_a$ �͍��X�Z�ȏW���ł��邱�Ƃ������D

(b) $\mathfrak{s}_a$ �̔Z�x�� $a \in \Omega$ �ɂ�炸
���ł��邱�Ƃ������D

\prob
(a) �P�ʉ~�� $\Delta$ ��̐����֐� $f$ �ŁC
$2 \exp \{ f(z)^2 \} = 1+z$ �𖞂������̂����݂��邱�Ƃ������D

(b) $\gamma_1 : [0,1] \ni t \mapsto -1+e^{2\pi i t}$, 
$\gamma_2 : [0,1] \ni t \mapsto 1-e^{2\pi i t}$ �� 2 �̋Ȑ��Ƃ���D
$\tilde{\gamma}_1, \tilde{\gamma}_2$ �����ꂼ��֐��v�f $f_0$ ���n�_�Ƃ���
$\gamma_1,\gamma_2$ �� lift �Ƃ���D���̂Ƃ��C
$\tilde{\gamma}_1(1) \ne \tilde{\gamma}_2(1)$ �������D

(c) $\Omega = \mathbb{C} \setminus \{ -1,1 \}$ �Ƃ���Ƃ��C
�Ȑ� $\gamma_1 \gamma_2$ �� $\gamma_2 \gamma_1$ �� $\Omega$ ����
homotopic �ł͂Ȃ����Ƃ��ؖ�����D
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\prob
$\alpha >0$ �ɑ΂��C�Ȑ� $\gamma_{\alpha} : (0,3) \to \mathbb{C}$ ��
\[ \gamma_{\alpha}(t) = \left\{
     \begin{array}{ll}
      \displaystyle \frac{\alpha}{t}-\pi i, & \quad (0< t \le 1) \\
      \displaystyle \alpha + 2\pi i \left( t-\frac{3}{2} \right),
                                           & \quad (1 \le t \le 2) \\
      \displaystyle \frac{\alpha}{3-t}+\pi i & \quad (2 \le t <3)
     \end{array}
�@�@�@   \right. \]
�ƒ�`����D���̂Ƃ��C$z \not\in \gamma_{\alpha}$ �ɑ΂��āC
\[ f_{\alpha}(z) = 
      \int_{\gamma_{\alpha}} \frac{\exp (e^w)}{z-w} \, dw \]
�Ƃ���D���������D

(a) $z \in \mathbb{C}$ �ɑ΂��C$\alpha, \beta > |z|$ �Ȃ�΁C
$f_{\alpha}(z) = f_{\beta}(z)$.

�i�����C���鐮�֐� $f$ �����݂��āC
$|z|<\alpha$ �Ȃ�� $f(z)=f_{\alpha}(z)$ ����������D�j

(b) $e^{i\theta} \ne 1$ �Ȃ�΁C
$\displaystyle \lim_{r \to \infty} f(r e^{i\theta}) = 0$.

(c) $f$ �͒萔�֐��ł͂Ȃ��D

(d) $\displaystyle \lim_{n \to \infty} g(nz) = 
\left\{ \begin{array}{ll}
        1, & \quad (z =0) \\
        0 & \quad (z \ne 0)
        \end{array} \right.$ 
�𖞂������֐� $g$ �����݂���D
