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--- math121a-f23:september_11_monday [2023/09/10 16:15]
pzhou created
+++ math121a-f23:september_11_monday [2026/02/21 14:41] (current)
@@ Line 1: / Line 1: @@
 ====== Holomorphic Function, Meromorphic Function ======
+tl;dr
+A function $f: \C \to \C \cup \{\infty\}$ is holomorphic at $z_0$ if there exists $\epsilon>0$, and complex numbers $a_0,a_1,\cdots$ that for all $|z - z_0|<\epsilon$, we can write
+$$ f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n. $$
+A function $f: \C \to \C \cup \{\infty\}$ is meromorphic at $z_0$ with order $m \geq 1$ pole, if there exists $\epsilon>0$, and complex numbers $a_{-m}, \cdots , a_0,a_1,\cdots$ that for all $0 < |z - z_0|<\epsilon$, we can write
+$$ f(z) = \sum_{n=-m}^\infty a_n (z-z_0)^n. $$
 ==== Real function Differentiability ====
@@ Line 16: / Line 27: @@
 x^2 & x \leq 0 \end{cases}
 $$
 Is it differentiable at $x=0$? Can you plot $f'(x)$?
-(this is an example, where the function $f(x)$ is differentiable, but the derivative $f'(x)$ is not continuous, hence we cannot define $f''(x)$ at $x=0$, hence $f(x)$ is not a smooth function on $\R$).
+(this is an example, where the function $f(x)$ is differentiable, but the derivative $f'(x)$ is not continuous, hence we cannot define $f^{"}(x)$ at $x=0$, hence $f(x)$ is not a smooth function on $\R$).
@@ Line 48: / Line 62: @@
 The Taylor expansion of $f$ centered at point $p \in \Omega$ is the following identity. If $B_r(p) \In \Omega$, then for any $z \in B_r(p)$, we have
-$$ f(z) = f(p) + f'(p) (z-p) + f''(p) \frac{ (z-p)^2}{2!} + \cdots + f^{(n)}(p) \frac{ (z-p)^2}{n!} + \cdots. $$
+$$ f(z) = f(p) + f'(p) (z-p) + f"(p) \frac{ (z-p)^2}{2!} + \cdots + f^{(n)}(p) \frac{ (z-p)^2}{n!} + \cdots. $$
+===== Point Singularity =====
+Suppose $f: \C \RM \{0\} \to \C$ is holomorphic function, you cannot help but wonder, what goes wrong at $0$? You might find
+  * a removable singularity (i.e. a false alarm, $f$ is all good at $z=0$)
+  * a pole, $z^n f(z)$ is holomorphic
+  * an essential singularity: anything else, like $e^{1/z}$, $e^{1/z^2}$.
+If you had a pole, we can apply that Taylor expansion formula (at $z=0$) to $z^n f(z)$, then divide out by $z$.
+===== Exercises =====
+Find the first two terms in these expansions.
+. Taylor expand $(z+1)(z+2)$ around $z=3$.
+. Laurent expand $1/[(z-1)(z-2)]$ around $z=1$. And do it again, this time around $z=2$.
+.5 (Optional) Laurent expand $e^{1/z + z}$ around $z=0$.
+. You may have heard about Cauchy Riemann equation: let $f(z)$ be a $\C$-valued function on $\C$, and let  $z = x+iy$, $f = u+iv$, then we can view $u,v$ as real valued functions depending on $x,y$.
+If $f$ is holomorphic, then we have
+$$ \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)
-===== Singularity =====

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