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--- math121a-f23:september_11_monday [2023/09/13 05:59]
pzhou
+++ math121a-f23:september_11_monday [2026/02/21 14:41] (current)
@@ Line 79: / Line 79: @@
 . Taylor expand $(z+1)(z+2)$ around $z=3$.
-. Laurent expand $1/[(z+1)(z+2)]$ around $z=1$. And do it again, this time around $z=2$.
+. Laurent expand $1/[(z-1)(z-2)]$ around $z=1$. And do it again, this time around $z=2$.
 .5 (Optional) Laurent expand $e^{1/z + z}$ around $z=0$.
-. You may have heard about Cauchy Riemann equation: it says the following: if $f(z)$ is a $\C$-valued function on $\C$, and let  $z = x+iy$, $f = u+iv$, then we can view $u,v$ as real valued functions depending on $x,y$. If $f$ is holomorphic, then we have
+. You may have heard about Cauchy Riemann equation: let $f(z)$ be a $\C$-valued function on $\C$, and let  $z = x+iy$, $f = u+iv$, then we can view $u,v$ as real valued functions depending on $x,y$.
+If $f$ is holomorphic, then we have
 $$ \frac{\d u(x,y)}{\d x} = \frac{\d v(x,y)}{\d y}, \quad \frac{\d v(x,y)}{\d x} = - \frac{\d u(x,y)}{\d y}$$
-Either prove this, or verify that this is true for your favorite holomorphic function (don't choose $f$ to be a constant, too boring)
+Your task: either prove this in general if you feel strong, or verify that this is true for your favorite holomorphic function (don't choose $f$ to be a constant, too boring)

Lecture Notes