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--- math121a-f23:september_11_monday [2023/09/10 16:25]
pzhou
+++ math121a-f23:september_11_monday [2026/02/21 14:41] (current)
@@ Line 71: / Line 71: @@
 If you had a pole, we can apply that Taylor expansion formula (at $z=0$) to $z^n f(z)$, then divide out by $z$.
+===== Exercises =====
+Find the first two terms in these expansions.
+. 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 $z=2$.
+.5 (Optional) Laurent expand $e^{1/z + z}$ around $z=0$.
+. 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
+$$ \frac{\d u(x,y)}{\d x} = \frac{\d v(x,y)}{\d y}, \quad \frac{\d v(x,y)}{\d x} = - \frac{\d u(x,y)}{\d y}$$
+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