https://courses.pzhou.org/ 2026-04-17T08:33:53+00:00 https://courses.pzhou.org/ https://courses.pzhou.org/lib/tpl/dokuwiki/images/favicon.ico text/html 2026-02-21T14:41:08+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math185-s23:hw5-extra&rev=1771684868&do=diff hw5.5 For $n \geq 2$, let $z_1, \cdots, z_n$ be $n$ distinct points on $\C$, with $|z_i| < 1$. Let $P(z) = (z-z_1)\cdots (z - z_n)$. Prove that $$ \oint_{|z|=1} \frac{1}{P(z)} dz = 0. $$ Hint: change variable $w = 1/z$, so that the integrand in the interior of disk $|w| < 1$ has no poles. text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math185-s23:hw6-extra&rev=1771684869&do=diff hw6 #4,#5 4. How many roots does the equation $z^4 - 6z + 3 = 0$ have in the disk $|z| < 2 $? and in $|z|<1$? 5. Recall that the definition of a general open map $f: X \to Y$ is that for any open set $U \subset X$, $f(U)$ is open in $Y$. Are the following maps open? You may sketch your reason. text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math185-s23:hw12&rev=1771684869&do=diff HW 12 Fix $\tau = i$. Consider elliptic function with period $1$ and $\tau$. #3 Fix any two distinct points $p_1,p_2$ in the fundamental domain $\{a+bi \mid a, b\in [0,1)\}$. * Can you write a series expression for a function with exactly two (simple) poles at $p_1, p_2$? text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math185-s23:mt1-review&rev=1771684869&do=diff Review for Midterm 1 $\gdef\D{\mathbb{D}}$ In the past month, we have covered first two chapters of Stein's book. Preliminaries First, we talked about complex numbers and reviewed some concepts of topologies. For complex numbers, you should be familiar with the following terms text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math185-s23:start&rev=1771684869&do=diff Math 185: Complex Analysis UC Berkeley, Spring 2023, Section 5. Course Number: 23829 Lecture: MWF 10:00A-10:59A at Etcheverry 3107 Final Exam: Tue, May 9 • 3:00P - 6:00P • Etcheverry 3107 Instructor: Peng Zhou, pzhou.math@berkeley.edu Office hour: 753 Evans Hall, MWF 11-12,