KU Math 800: Lecture 1/21/2026

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[definition=%counter%] Let $\Omega \subseteq \CC$ be an open set. We say $f:\Omega \to \CC$ is holomorphic at $z_0 \in \Omega$ if

\[f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}.\]

[/definition]

[example=%counter%]

  1. $f \in \CC[z]$.
  2. $f \in \CC(z)$. [/example]

[example=%counter%] A non-example of a holomorphic function is $f(z) = \overline{z}$. This can be explicitly proven by showing that $f$ does not satisfy the CR equations. [/example]

[proposition=%counter%] Holomorphic at $z_0$ implies continuous at $z_0$. [/proposition]

[proof] Since

\[f(z) = \frac{f(z) - f(z_0)}{z - z_0}(z - z_0) + f(z_0),\]

we can just take the limit as $z \to z_0$ to get $f(z) \to f(z_0)$. [/proof]

[proposition=%counter%] If $f, g$ are holomorphic in $\Omega \subseteq \CC$, then

  1. $f + g, fg$ are holomorphic $(f + g)’ = f’ + g’$ and $(fg)’ = f’g + fg’$.
  2. If $g(z_0) \neq 0$, then $f/g$ is holomorphic at $z_0$ with the usual quotient rule applying. [/proposition]

Via the prototypical bijection $\RR^2 \leftrightarrow \CC$, we can write $f$ as

\[f = u + iv\]

where $u,v: \RR^2 \to \RR$. This gives rise to the following:

[theorem=%counter%] Let $f = u + iv$ be holomorphic at $z_0$. Then the Cauchy Riemann equations

\[\pdv{u}{x} = \pdv{v}{y} \quad\text{ and }\quad \pdv{u}{y} = -\pdv{v}{x}\]

[/theorem]

[corollary] $u$ and $v$ are harmonic. [/corollary]

[proposition=%counter%] If $f$ is holomorphic at $z_0 = x_0 + iy_0$, then

\[\det J(x_0, y_0) = |f'(z_0)|^2.\]

[/proposition]

[remark=%counter%] Notice that this says that $f$ being holomorphic means that $f$ has to preserve orientation because the Jacobian is nonnegative. [/remark]