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{\large\textbf{��2�񐔗����I���K (2010�N10��15�����{)}}
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�S������ ���n����/ TA �����`�j
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\prob $z=x+iy$�Ƃ����Ƃ����̊֐��͐�����?

(a) $x^2+iy$

(b) $(x^2-y^2+3x)+i(2xy+3y)$

\prob
���f��$z=x+iy\in \C$�Ǝ��Q�������x�N�g��$\displaystyle{\begin{pmatrix}x\\ y\end{pmatrix}}\in \R^2$�𓯈ꎋ����D���̂Ƃ����`�ʑ�
$$
f\colon \begin{pmatrix}x\\ y\end{pmatrix}\mapsto \begin{pmatrix}a & b\\ c & d\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}
$$
��$\C$��̊֐��Ƃ��Ă݂�D���̂Ƃ����͓��l�ł��邱�Ƃ������F

(a) $f$�͐���

(b) $f$�͌�����ۂ����ϊ��ł���

(c) $A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}$, $J=\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$�Ƃ�����$AJ=JA$.


\prob
���f���̋ɕ\��$z=r (\cos \theta+i\sin\theta)$���l����D$C^1$���֐�$f(z)=u(z)+i \,v(z)$�ɂ�������
$$
\frac{\partial f}{\partial \overline{z}}=0  \iff  \left(
\frac{\pa u}{\pa r}=\frac{1}{r}\frac{\pa v}{\pa \theta},\quad
\frac{\pa u}{\pa\theta}=-r\frac{\pa v}{\pa r}
\right)
$$

\prob $\C\setminus (-\infty,0]$�ɂ����ċɕ\��$z=r(\cos \theta+i \sin \theta)$, $(r,\theta)\in (0,\infty)\times (-\pi,\pi)$��p���ĕ��f�ϐ��̑ΐ��֐���
$$
\log z=\log r+ i\theta.
$$
�Œ�`����D$\log z$�͐����ł��邱�Ƃ������D

\prob
$U,V\subset\C$���J�W���C$f\colon U\to V$��$g\colon V\to \C$���S�����\�ł����
$h=g\circ f$��
$$
\frac{\pa h}{\pa z}=\frac{\pa g}{\pa z}\frac{\pa f}{\pa z}+\frac{\pa g}{\pa \overline z}\frac{\pa\overline f}{\pa z},\qquad
\frac{\pa h}{\pa \overline z}=\frac{\pa g}{\pa z}\frac{\pa f}{\pa\overline z}+\frac{\pa g}{\pa \overline z}\frac{\pa\overline f}{\pa\overline z}
$$
�𖞂������Ƃ������D

\prob
$f(z)$�������ł����$g(z)=\overline{f(\overline{z})}$�������ł��邱�Ƃ������D




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���K����
{\tt http://www.ms.u-tokyo.ac.jp/$\sim$hirachi/courses/}
����_�E�����[�h�ł��܂��D�u�`�������ڂ��Ă��܂��D


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