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zMDXW-hA*w6JrD1h>&a&BP!Puj#(#(lW#T9oLsucH0!!lYcP6$_KaHE}kFA~`s03?> zuuR7*on@~5;4Td5J5nPrpT$b3W(-(IYzgdbEvb;1Zx^-w0}x0N zsJS`ZPEbi4ZWz*})HyovJ%u@)#KP41)ciah&p1@yG-0!IWFXAz2huv=&n#y6)x52m z`iJT#?IsDoHwWpd);EjT?Sh={d>DG9T=RA3dqB$wfAlYKEjGjMp0{{6*l&kK^LfJF8mQPl83qF`_C#GAg;KpXsW8f zywIK=8tQNlG~5H>jqp}eQ%C$|uBhuG|35|k%+xwc7Zp zJ(H81pQ)O!2V1%{zpQfq1CeL8%dj+&tJEPDh!`B~{#Km!#97vLr0i(R2;ChC!7Ku{-@God_hUHZ*bryfTo{zmgtgOrFq}xSo^=Mda zb=-HD*6N!j3O; 0,\] + 的所有点$z = x + \imath y$.请分$\lambda$的三种情况(1) $\lambda<1$, (2) $\lambda >1$,(3) $\lambda = 1$分别绘制图形. +\end{problem} + +\begin{problem} + 若$|z|=1$, $a,b$为任意复数,试证明 + $$ + \left| \frac{a z + b}{\bar{b} z + \bar{a}} \right| = 1 . + $$ +\end{problem} + +\begin{problem} + 试用$\cos\varphi, \sin\varphi$ 表示$\cos 4\varphi$. +\end{problem} + +\begin{problem} + 求下列方程的根,并在复平面上画出它们的位置. + \begin{enumerate} + \item $z^4 + 1 = 0$, + \item $z^2 + 2 z \cos \lambda + 1 = 0, \quad 0 < \lambda < \pi$. + \end{enumerate} +\end{problem} + +\begin{problem} +验证 +$$ +\tan^{-1}(z)=\frac{\imath }{2}[\ln (1-\imath z)-\ln (1+\imath z)] . +$$ +\end{problem} + + +\begin{problem} + 试证明极坐标下的柯西-黎曼方程 + $$ + \left\{\begin{array}{l} + \frac{\partial u}{\partial \rho}=\frac{1}{\rho} \frac{\partial v}{\partial \varphi} ,\\ + \frac{1}{\rho} \frac{\partial u}{\partial \varphi}=-\frac{\partial v}{\partial \rho} . + \end{array}\right. + $$ +\end{problem} + +% \begin{problem} +% 已知解析函数 $f(z)$ 的实部 $u(x, y)$或虚部$ v(x, y)$, 求该解析函数. +% \begin{enumerate} +% \item $u = e^x \sin{y}$, +% \item $u = x^2 - y^2 + xy$, $f(0) = 0$. +% \end{enumerate} +% \end{problem} + +% \begin{proof}[Solution] +% Write a solution here +% \begin{align*} +% x &= y +% \end{align*} +% \begin{tabular}{| >{\centering\arraybackslash}m{1in} | >{\centering\arraybackslash}m{1in} | >{\centering\arraybackslash}m{1in} | >{\centering\arraybackslash}m{1in} |>{\centering\arraybackslash}m{1in} |} +% \hline +% \textbf{A} & \textbf{B} & \textbf{C} & \textbf{D} &\textbf{E} \\[8pt] +% \hline +% a & b & c & d & e \\[8pt] +% \hline +% f & g &h & i & j \\[8pt] +% \hline +% \end{tabular} +% \end{proof} + +\end{document} \ No newline at end of file