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\begin{center}
{\large\textbf{��5�񕡑f��͊wII���K (2011�N5��17�����{)}}
\end{center}
\begin{flushright}
�S������ ���n����/ TA �v�{�q�V�@���{���F
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\bigskip

�ȉ���1�������̉��K�̎��Ԃ́w�͂��߁x�Ƀ��|�[�g�Ƃ��Ē�o���邱��.
\bigskip

\noindent
{\bf [R6]} $f(x)$��$\R$��̗L�E�A���֐��Ƃ���B
$$
F(x+iy)=\frac{1}{\pi}\int_{-\infty}^\infty\frac{y \,f(t)}{(x-t)^2+y^2}dt
$$
�ɂ��㔼���$H$��̊֐����`����(�ϕ����������邱�Ƃ��m���߂�)�B
���̂Ƃ�$U(z)$��$H$��Œ��a�ł���A
$$
\lim_{y\to+0}F(x+iy)=f(x)
$$
�����藧���Ƃ������B

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\noindent
�ȉ��͋������\�p�̖��ł�. 

\prob
$\mathcal{F}$��̈�$D\subset\C$��Œ�`���ꂽ���a�֐��̑���
$\sup\{|u(z)|:u\in\mathcal{F},z\in D\} <\infty$
�𖞂������̂Ƃ���B$\mathcal{F}$�̔C�ӂ̊֐���͍L�`��l�������镔������܂ނ��Ƃ������B


\prob
$f(z)=z^{-1}\log(1+z)$, $u(z)=\log|1+z|$�Ƃ���D

(a) $z=0$��$f$�̏����\���ٓ_�ł��邱�Ƃ������D

(b) $\Gamma_{\delta}=\left\{e^{i\theta}\, ;\, -\pi+\delta<\theta
<\pi-\delta\right\}$�Ƃ����D
$\displaystyle{\lim_{\delta\to 0}\int_{\Gamma_{\delta}}f(z)\, dz}$
���v�Z����D

�@�@�܂�$0\le r\le 1$�ɑ΂��āC
$\displaystyle{u(0)=\frac{1}{2\pi}\int_{0}^{2\pi}u(re^{i\theta})\,d\theta}$
�������D

(c)  $\displaystyle{\int_{0}^{\pi}\log\sin\theta\,d\theta}$���v�Z����. 


\prob $h(z)$��$\Delta^*=\Delta\setminus\{0\}$��̎����l���a�֐�
�Ƃ���B

(a) ����萔$A$�����݂���$h(z)+A\log|z|$��$\Delta^*$��̐����֐��̎�����
�Ȃ邱�Ƃ������B

(b) $h(z)$���L�E�ł����$\Delta$��̒��a�֐��Ɋg������邱�Ƃ������B


\prob
�萔$a,b,c$�ɑ΂���
$\C$��Œ�`���ꂽ������p�f
$$
 L=a\frac{\partial^2}{\partial x^2}+
 b\frac{\partial^2}{\partial y^2}+c\frac{\partial^2}{\partial x\partial y}
$$
���l����B�C�ӂ�$C^2$�֐�$f$�ƔC�ӂ̉�]�ʑ�$\rho_\theta(z)=e^{i\theta}z$�ɑ΂���$(Lf)\circ\rho_\theta=L(f\circ\rho_\theta)$
�����藧�Ƃ���B���̂Ƃ�$L$�̃��v���V�A���̒萔�{�ł��邱�Ƃ������B

\prob �L�E�A���֐�$f(x)$�ɂ������ď㔼��ԏ�̊֐�
$$
F_+(x+iy)=\int_{0}^\infty\frac{y \,f(t)}{(x-t)^2+y^2}dt
$$
���l����B$F_+$��$H\ni z\to0$�ł̋����𒲂ׂ�B

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