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{\large\textbf{��12�񕡑f��͊wII���K (2011�N7��5�����{)}}
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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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\textbf{[R15]} 
$\C$���̃R���p�N�g�W��$K$�ɑ΂���
$$
\widehat{K}=\{z\in\Omega: |f(z)|\le\|f\|_K,\forall f\in\mathcal{O}(\C)\}
$$
�ƒ�`����D������$\|f\|_K=\max_{z\in K}|f(z)|$�ł���D

(a) $\widehat{K}$�̓R���p�N�g�ł��邱�Ƃ������D

(b) $K=[\Delta(0,5)]\setminus(\Delta(2,1)\cup\Delta(3,1))$�ɑ΂���
$\widehat{K}$�����߂�D������$\Delta(a,r)$�͒��S$a$���a$r$�̊J�~�D

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�ȉ��͋������\�p�̖��ł��D


\prob  $K\subset\C$���R���p�N�g�W���C$\C\setminus K$�͘A���Ƃ���D$f\in\mathcal{O}(K)$�ɑ΂��đ������̗�$\{f_n\}_n$
��
$$
\lim_{n\to\infty}\max_{z\in K}| f(z)-f_n(z)|=0
$$
�𖞂������̂����݂��邱�Ƃ������D

\prob $V$��$\R$��̗L�������x�N�g����ԁC$V$���$\R$�l���`�ʑ��S�̂̂Ȃ��W����$V^*$������D
�R���p�N�g�W��$K\subset V$�ɑ΂���
$$
\widehat{K}=\{v\in V: f(v)\le\max_{w\in K}f(w),\forall f\in V^*\}
$$
�������D$\widehat{K}$��$K$���܂ލŏ��̓ʏW���ł��邱�Ƃ������D



\prob
$\C$��̗L���^�֐�$f(z)$�̋ɂ�$a_k$, $k=1,2,3,\dots$�Ƃ���D�������肷��F

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(a) �ɂ̈ʐ��͂��ׂĂP�ł���
$0<|a_1|\le |a_2|\le |a_3|\le\cdots$�Ƃ���D

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(b) $\lim_{n\to\infty}R_n=\infty$�𖞂������������$\{R_n\}$�ɑ΂���
$
\sup_{n}\big(\sup_{|z|=R_n}|f(z)|\big)<\infty
$

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\noindent
���̓����������F
$$
f(z)=f(0)+\lim_{n\to\infty}\sum_{|a_k|< R_n}\text{Res}(f,a_k)\left(\frac{1}{z-a_k}+\frac{1}{a_k}\right).
$$

\prob $K$�𕡑f�̈�$\Omega$�̃R���p�N�g�W���Ƃ���D����(a), (b)�͓��l�ł��邱�Ƃ������D

(a) $\Omega\setminus K$�̊e�A��������$\Omega$�ő��΃R���p�N�g

(b) $\forall z\in\Omega\setminus K$�ɑ΂���
$
|f(z)|>\|f\|_K
$�𖞂���
$ f\in\mathcal{O}(\Omega)$
�����݂���D



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(b) $f(z)$ �͔��������� $f''(z)=6 \{ f(z) \}^2 -4 f(z)$ ��
���������Ƃ������D\\
�i�q���g�F$\displaystyle \sum_{k=1}^{\infty} \frac{1}{k^2} = 
\frac{\pi^2}{6}$, 
$\displaystyle \sum_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}$ 
�Ȃǂ�p���āC$f(z)$ �� $z=0$ �̂܂��Ń��[�����W�J����D�j

(c) ���̓������ؖ�����D�������C�ϕ��H�͓K���ɑI�Ԃ��̂Ƃ���D
\[ z = \frac{1}{2} \int_{f(z)}^{\infty} \frac{dw}{\sqrt{w^2(w-1)}}. \]