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{\large\textbf{��7�񐔗����I���K (2010�N11��26�����{)}}
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�S������ ���n����/ TA �����`�j
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\prob
${\C}$����$2$�̗̈�$\Omega_1=\{|z|<2\},\Omega_2=\{|z|>1\}$���l����B
���̂Ƃ�$\Omega_1\cap\Omega_2$��̐����֐�$f$��$\Omega_j$��̐����֐�$f_j\hspace{0.5em}(j=1,2)$��p����$f=f_1+f_2$�̂悤�ɕ\����邱�Ƃ������B

�q���g�F $1+\epsilon<|z|<2-\epsilon$��Cauchy�̐ϕ��\�����s���D

\prob
(a)
$$
\frac{1}{1+z^2}=1-z^2+z^4-z^6+\cdots
$$
��ϕ����邱�Ƃɂ��$\arctan z$��$z=0$�ɂ�����e�[���[�W�J�𓱂��D����ɁC���̎������a�����Ƃ߂�D

(b) $\sin z$��$\cos z$�̃e�[���[�W�J��$\tan z=\dfrac{\sin z}{\cos z}$�ɑ�����邱�Ƃɂ��$\tan z$��$z=0$�ɂ�����e�[���[�W�J��$z^5$�܂ŋ��߂�D����ɁC���̃e�[���[�����̎������a�����Ƃ߂�D

\prob $\log(1+z+z^2)$��$z=0$�ɂ�����e�[���[�W�J�ƁC���̎������a�����Ƃ߂�D

�q���g�F$z^3-1$�̈����������g���ƌv�Z���ȒP�ɂȂ�D

\prob
(a) ����
$$
g(z)=z-\frac{z^2}{2}+\frac{z^3}{3}-\cdots+(-1)^{n-1}\frac{z^n}{n}z+\cdots
$$
�̎������a�����Ƃ߂�D

(b) $g'(z)$�����Ƃ߂�D

(c) �����~�̓�����$\exp(g(z))=1+z$�����藧���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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