From 05e62532dadf1d3a31c19a3874229a8687167c4d Mon Sep 17 00:00:00 2001
From: Trance-0 <60459821+Trance-0@users.noreply.github.com>
Date: Tue, 11 Feb 2025 11:16:11 -0600
Subject: [PATCH] procedural updating

---
 pages/Math416/Math416_L3.md | 60 ++++++++++++++++++++++++++++++++-----
 pages/Math416/Math416_L4.md | 11 +++----
 2 files changed, 56 insertions(+), 15 deletions(-)

diff --git a/pages/Math416/Math416_L3.md b/pages/Math416/Math416_L3.md
index e730b9d..4c9f11c 100644
--- a/pages/Math416/Math416_L3.md
+++ b/pages/Math416/Math416_L3.md
@@ -6,12 +6,12 @@
 
 #### Definition of differentiability in complex variables
 
-Suppose $G$ is an open subset of $\mathbb{C}$.
+**Suppose $G$ is an open subset of $\mathbb{C}$**.
 
 A function $f:G\to \mathbb{C}$ is differentiable at $\zeta_0\in G$ if
 
 $$
-\lim_{\zeta\to \zeta_0}\frac{f(\zeta)-f(\zeta_0)}{\zeta-\zeta_0}
+f'(\zeta_0)=\lim_{\zeta\to \zeta_0}\frac{f(\zeta)-f(\zeta_0)}{\zeta-\zeta_0}
 $$
 
 exists.
@@ -32,6 +32,8 @@ $$
 \lim_{(x,y)\to (x_0,y_0)}\frac{|R(x,y)|}{|(x,y)-(x_0,y_0)|}=\lim_{(x,y)\to (x_0,y_0)}\frac{|R(x,y)|}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=0.
 $$
 
+_$R(x,y)$ is the immediate result of mean value theorem applied to $u$ at $(x_0,y_0)$_.
+
 > Theorem from 4111?
 > 
 > If $u$ is differentiable at $(x_0,y_0)$, then $\frac{\partial u}{\partial x}(x_0,y_0)$ and $\frac{\partial u}{\partial y}(x_0,y_0)$ exist.
@@ -55,11 +57,25 @@ $$
 
 So $\lim_{(x,y)\to (x_0,y_0)}\frac{|R(x,y)|}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=0$ if and only if $\lim_{(x,y)\to (x_0,y_0)}\frac{a(x-x_0)}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=0$ and $\lim_{(x,y)\to (x_0,y_0)}\frac{b(y-y_0)}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=0$.
 
-On the imaginary part, we have
+On the imaginary part, we proceed similarly. Define
+$$
+S(x,y)=v(x,y)-v(x_0,y_0)-\frac{\partial v}{\partial x}(x_0,y_0)(x-x_0)-\frac{\partial v}{\partial y}(x_0,y_0)(y-y_0).
+$$
+Then the differentiability of $v$ at $(x_0,y_0)$ guarantees that
+$$
+\lim_{(x,y)\to (x_0,y_0)}\frac{|S(x,y)|}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=0.
+$$
+Moreover, considering the definition of the complex derivative of $f=u+iv$, if we approach $\zeta_0=x_0+iy_0$ along different directions we obtain
+$$
+f'(\zeta_0)=\frac{\partial u}{\partial x}(x_0,y_0)+i\frac{\partial v}{\partial x}(x_0,y_0)
+=\frac{\partial v}{\partial y}(x_0,y_0)-i\frac{\partial u}{\partial y}(x_0,y_0).
+$$
+Equating the real and imaginary parts of these two expressions forces
+$$
+\frac{\partial u}{\partial x}(x_0,y_0)=\frac{\partial v}{\partial y}(x_0,y_0),\quad \frac{\partial u}{\partial y}(x_0,y_0)=-\frac{\partial v}{\partial x}(x_0,y_0).
+$$
 
-...
-
-Conclusion (The Cauchy-Riemann equations):
+#### Theorem 2.6 (The Cauchy-Riemann equations):
 
 If $f=u+iv$ is complex differentiable at $\zeta_0\in G$, then $u$ and $v$ are real differentiable at $(x_0,y_0)$ and
 
@@ -67,6 +83,12 @@ $$
 \frac{\partial u}{\partial x}(x_0,y_0)=\frac{\partial v}{\partial y}(x_0,y_0),\quad \frac{\partial u}{\partial y}(x_0,y_0)=-\frac{\partial v}{\partial x}(x_0,y_0).
 $$
 
+> Some missing details:
+>
+> The Cauchy-Riemann equations are necessary and sufficient for the differentiability of $f$ at $\zeta_0$.
+>
+> This states that a function $f$ is **complex differentiable** at $\zeta_0$ if and only if $u$ and $v$ are real differentiable at $(x_0,y_0)$ and the Cauchy-Riemann equations hold at $(x_0,y_0)$. That is $f'(\zeta_0)=\frac{\partial u}{\partial x}(x_0,y_0)+i\frac{\partial v}{\partial x}(x_0,y_0)=\frac{\partial v}{\partial y}(x_0,y_0)-i\frac{\partial u}{\partial y}(x_0,y_0)$.
+
 And $u$ and $v$ have continuous partial derivatives at $(x_0,y_0)$.
 
 And let $c=\frac{\partial u}{\partial x}(x_0,y_0)$ and $d=\frac{\partial v}{\partial x}(x_0,y_0)$.
@@ -75,7 +97,7 @@ And let $c=\frac{\partial u}{\partial x}(x_0,y_0)$ and $d=\frac{\partial v}{\par
 
 ### Holomorphic Functions
 
-#### Definition of holomorphic functions
+#### Definition 2.8 (Holomorphic functions)
 
 A function $f:G\to \mathbb{C}$ is holomorphic (or analytic) at $\zeta_0\in G$ if it is complex differentiable at $\zeta_0$.
 
@@ -97,9 +119,31 @@ So polynomials are holomorphic on $\mathbb{C}$.
 
 So rational functions $p/q$ are holomorphic on $\mathbb{C}\setminus\{z\in \mathbb{C}:q(z)=0\}$.
 
+#### Definition 2.9 (Complex partial differential operators)
+
+Let $f:G\to \mathbb{C}$, $f=u+iv$, be a function defined on an open set $G\subset \mathbb{C}$.
+
+Define:
+
+$$
+\frac{\partial}{\partial x}f=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x},\quad \frac{\partial}{\partial y}f=\frac{\partial u}{\partial y}+i\frac{\partial v}{\partial y}.
+$$
+
+And
+
+$$
+\frac{\partial}{\partial \zeta}f=\frac{1}{2}\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)f,\quad \frac{\partial}{\partial \bar{\zeta}}f=\frac{1}{2}\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)f.
+$$
+
+This definition of partial differential operators on complex functions is consistent with the definition of partial differential operators on real functions.
+
+$$
+\frac{\partial}{\partial x}f=\frac{\partial}{\partial \zeta}f+\frac{\partial}{\partial \bar{\zeta}}f,\quad \frac{\partial}{\partial y}f=i\left(\frac{\partial}{\partial \zeta}f-\frac{\partial}{\partial \bar{\zeta}}f\right).
+$$
+
 ### Curves in $\mathbb{C}$
 
-#### Definition of curves in $\mathbb{C}$
+#### Definition 2.11 (Curves in $\mathbb{C}$)
 
 A curve $\gamma$ in $G\subset \mathbb{C}$ is a continuous map of an interval $I$ into $G$. We say $\gamma$ is differentiable if $\forall t_0\in I$, $\gamma'(t_0)=\lim_{t\to t_0}\frac{\gamma(t)-\gamma(t_0)}{t-t_0}$ exists.
 
diff --git a/pages/Math416/Math416_L4.md b/pages/Math416/Math416_L4.md
index f6c3f67..03be436 100644
--- a/pages/Math416/Math416_L4.md
+++ b/pages/Math416/Math416_L4.md
@@ -58,8 +58,7 @@ $$
 >
 > A function $g(h)$ is $o(h)$ if $\lim_{h\to 0}\frac{g(h)}{h}=0$.
 > 
-> <!---TODO: check after lecture-->
-> $f$ is differentiable if and only if $f(z+h)=f(z)+f'(z)h+\frac{1}{2}h^2f''(z)+o(h^3)$ as $h\to 0$.
+> $f$ is differentiable if and only if $f(z+h)=f(z)+f'(z)h+\frac{1}{2}h^2f''(z)+o(h^3)$ as $h\to 0$. (By Taylor expansion)
 
 Since $f$ is holomorphic at $\gamma(t_0)=\zeta_0$, we have
 
@@ -88,13 +87,13 @@ $$
 
 EOP
 
-#### Definition of conformal function
+#### Definition 2.12 (Conformal function)
 
 A function $f:G\to \mathbb{C}$ is called conformal if it preserves the angle between two curves.
 
-#### Theorem of conformal function
+#### Theorem 2.13 (Conformal function)
 
-If $f:G\to \mathbb{C}$ is holomorphic function on open set $G\subset \mathbb{C}$ and $\gamma_1,\gamma_2$ are regular curves in $G$ with $\gamma_1(t_0)=\gamma_2(t_0)=\zeta_0$ for some $t_0\in I_1\cap I_2$, and $f'(\zeta_0)\neq 0$, then $f$ is conformal at $\zeta_0$.
+If $f:G\to \mathbb{C}$ is conformal at $\zeta_0\in G$, then $f$ is holomorphic at $\zeta_0$ and $f'(\zeta_0)\neq 0$.
 
 Example:
 
@@ -104,8 +103,6 @@ $$
 
 is not conformal at $z=0$ because $f'(0)=0$.
 
-
-
 #### Lemma of conformal function
 
 Suppose $f$ is real differentiable, let $a=\frac{\partial f}{\partial \zeta}(\zeta_0)$, $b=\frac{\partial f}{\partial \overline{\zeta}}(\zeta_0)$.