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\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\def\H{{\mathbb H}}
\def\N{{\mathbb N}}
\def\Z{{\mathbb Z}}
\def\Re{\mathop{\mathrm{Re}}\nolimits}
\def\Im{\mathop{\mathrm{Im}}\nolimits}
\def\Res{\mathop{\mathrm{Res}}\nolimits}

\begin{document}

\begin{center}
{\large\textbf{��12�񕡑f��͊wI���K���� (2007�N1��26�����{)}}
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\begin{flushright}
�S������ ���n����/ TA �����M��Y�E�˖{�׎O
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[1]

\noindent
$z^{\lambda} = \exp (\lambda \log z)$�Œ�`����D��`��́C�Ⴆ��$\C \setminus \{ z \in \C | \Re z \le 0, \Im z =0 \}$���Ƃ�΂悢���C���̗̈��
$\log z$�͐���������C$z^{\lambda}$�������ł���D��ʂɂ͌��_���܂܂Ȃ��P�A���̈���`��Ƃ��邱�Ƃ��ł���B

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[2] 

(a)$f=gz^m,m \in \N,  g \neq 0$�Ə����ƁC$\partial \Delta$��$|g|=1$������C�ő�l�̌������$g = \mathrm{const}$�D

(b)�K���Ȉꎟ�����ϊ�$\phi \in \mathrm{Aut}(\Delta)$���Ƃ�΁C$\phi \circ f$��(a)�̏����𖞂����C$m=1$�ł��邩��C

$\phi \circ f\in \mathrm{Aut}(\Delta)$�C�����$f \in \mathrm{Aut}(\Delta)$�C�܂�$f$�͈ꎟ�����ϊ��D

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[3]

\noindent
$\gamma' = f \circ \gamma$�Ƃ�����$\gamma'$��$\Delta(1,1)$���̕Ȑ��ł���C
\[ \int_{\gamma} \frac{f'(z)}{f(z)}dz = \int_{\gamma'} \frac{1}{z}dz \]

�����CCauchy�̒藝���C�����$0$�ł���D

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[4]

\noindent
\[ I(\xi)=\frac{e^{-\xi^2}}{2}\int_{-\infty}^{\infty}e^{-(x-i\xi)^2}dx + \frac{e^{-\xi^2}}{2}\int_{-\infty}^{\infty}e^{-(x+i\xi)^2}dx\]
$R>0$�ɂ�������$R$����$R+i\xi$�ɂ����������$L_R$�C$-R+i\xi$����$-R$�ɂ����������$L'_R$�Ƃ���ƁC
\[ \int_{-R}^{R}e^{-(x-i\xi)^2}dx - \int_{-R}^{R}e^{-x^2}dx = \int_{L_R}e^{-(x-i\xi)^2}dx + \int_{L'_R}e^{-(x-i\xi)^2}dx. \]
\[ \bigg| \int_{L_R}e^{-(x-i\xi)^2}dx \bigg| \le |\xi|e^{-R^2+\xi^2} \to 0 \quad (R \to \infty) \]
\[ \bigg| \int_{L'_R}e^{-(x-i\xi)^2}dx \bigg| \le |\xi|e^{-R^2+\xi^2} \to 0 \quad (R \to \infty) \]
����āC
\[ \int_{-\infty}^{\infty}e^{-(x-i\xi)^2}dx = \int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt\pi. \]
���l�ɂ���
\[ \int_{-\infty}^{\infty}e^{-(x+i\xi)^2}dx = \int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt\pi. \]
���킩�邩��C$I(\xi)=\sqrt\pi e^{-\xi^2}$�D


[5]

(a)�O�p�s�������$|z|=1$�̂Ƃ�$|f(z)-3z^6| < 3$�ł��邩��C$f$��$|z| < 1$��$6$�̗�_�����D�܂�C$f$��$|z| \ge 1$���_�������Ȃ��D 

(b)$w=1/z$�ƕϐ��ϊ�����ƁC
\[ \int_{|z|=1} \frac{z^5}{3z^6-z^2-z+1}dz=-\int_{w=1} \frac{1}{w(3-w^4-w^5+w^6)}dw \]
�����C(a)�̌��ʂƗ����藝���C�ϕ��̒l��$-2\pi i/3$�D

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\textbf{���f��͊wI�T�C�g�ł͉��K�Ŕz�z����v�����g��u�`�������ڂ��Ă��܂�:
\begin{center}
http://www.ms.u-tokyo.ac.jp/\~{}hirachi/courses/complex1-2006/
\end{center}}
\vfill
\hfill (�𓚍쐬: �˖{�׎O)
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