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\begin{document}
\begin{center}
{\large\textbf{��6�񕡑f��͊wII���K (2011�N5��24�����{)}}
\end{center}
\begin{flushright}
�S������ ���n����/  TA �v�{�q�V�@���{���F
\end{flushright}

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�ȉ��̖������̉��K�̎��Ԃ́w�͂��߁x�Ƀ��|�[�g�Ƃ��Ē�o���邱��.

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\noindent
{\bf [R7]}
$u$��P�A���̈�$\Omega$��̎����l���a�֐��Ƃ���D$\Omega$��̐����֐�$f$��$u=\text{Re}f$�𖞂������̂����݂��邱�Ƃ������D(�q���g�F$\partial u/\partial z$�͐����֐�)

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\noindent
{\bf [R8]}
 $|z|^a$, ($a\ge0$)�����$\log(1+|z|^2)$�͗򒲘a�ł��邱�Ƃ������D
 
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�ȉ��͋������\�p�̖��ł�. 

\prob
�֐�$u\colon[0,1)\to\R$��$u(|z|)$��
$\Delta\setminus\{0\}$��Œ��a�֐��ɂȂ���̂����肹��D

\prob
 �����֐�$f(z)$�ɑ΂���$\log(1+|f(z)|^2)$�͗򒲘a�ł��邱�Ƃ������D

\prob (�򒲘a�֐��̋Ǐ���)
$u$�𕡑f�̈�$\Omega$��̘A���֐��Ƃ���D�C�ӂ�$z\in\Omega$�ɑ΂�����$r$�����݂���$u|_{\Delta(z,r)}$���򒲘a�ł����$u$��$\Omega$��ŗ򒲘a�ł��邱�Ƃ������D

\prob $C^2$ �֐�$f$��
$$
\Delta f(z)=\lim_{r\to0}\frac{2}{r^2}\left(\frac{1}{2\pi}\int_{0}^{2\pi}f(z+re^{i\theta})d\theta}-f(z)\right)
$$
�𖞂������Ƃ������D�i�q���g�F$f$��$z$�ł̂Q���܂ł̃e�[���[�W�J���l����D�j
�����p����
$f$ ���򒲘a�ł���� $\Delta f \ge 0$ �ł��邱�Ƃ������D

\prob $C^2$ �֐�$f$��
$$
r\frac{d}{dr}\int_{0}^{2\pi}f(z+re^{i\theta})d\theta}
=\iint_{\Delta(z,r)}\Delta f(x+iy)dxfy
$$
�𖞂������Ƃ������D
�����p����
$\Delta f \ge 0$�ł����$f$�͗򒲘a�ł��邱�Ƃ������D


\prob  $\mathcal F$ ��̈� $\Omega$ �ɂ�����
�򒲘a�֐��̈�̑��Ƃ���D

(a) $\mathcal F$ ���܂� Perron ���S�̂̋��ʕ��� 
$\overline{\mathcal F}$ �� Perron ���ł��邱�Ƃ������D

(b) $\mathcal F$ �� $\Omega$ ��ň�l�L�E�ł���� 
$\Omega$ ��̒��a�֐� $u$ �ŔC�ӂ� $v\in\mathcal F$ 
�ɑ΂��� $v\le u$ �𖞂������̂���������݂��邱�Ƃ������D

\prob
$\Omega=\Delta\setminus\{0\}$ �̋��E��̊֐� $f$ �� 
$f|_{\partial\Delta}=0$, $f(0)=1$ �Œ�`����D
$\Omega$ ��̗򒲘a�֐� $u$ �ŋ��E��� $u\le f$ �ł���
���̂� $\Omega$ ��� $u\le0$ �𖞂������Ƃ������D

\prob
(a) $\Omega = \mathbb{C} \setminus (-\infty, 0]$ �Ƃ���Ƃ��C
$\partial \Omega$ �̊e�_�ł� barrier �����D

(b) $\mathbb{C}$ ���̔C�ӂ̗L�E�̈�ɑ΂��āCbarrier ������
���E��̓_�����Ȃ��Ƃ� 1 ���݂��邱�Ƃ������D
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