Improve knowledge base

1 files changed, 17 insertions, 10 deletions

diff --git a/source/know/concept/holomorphic-function/index.md b/source/know/concept/holomorphic-function/index.md
index 976758b..db3bdfb 100644
--- a/source/know/concept/holomorphic-function/index.md
+++ b/source/know/concept/holomorphic-function/index.md
@@ -28,7 +28,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-We decompose $$f$$ into the real functions $$u$$ and $$v$$ of real variables $$x$$ and $$y$$:
+We decompose $$f$$ into the real functions $$u$$ and $$v$$
+of real variables $$x$$ and $$y$$:
 
 $$\begin{aligned}
     f(z)
@@ -36,7 +37,8 @@ $$\begin{aligned}
     = u(x, y) + i v(x, y)
 \end{aligned}$$
 
-Since we are free to choose the direction of $$\Delta z$$, we choose $$\Delta x$$ and $$\Delta y$$:
+Since we are free to choose the direction of $$\Delta z$$,
+we choose $$\Delta x$$ and $$\Delta y$$:
 
 $$\begin{aligned}
     f'(z)
@@ -53,9 +55,13 @@ we thus arrive at the **Cauchy-Riemann equations**:
 
 $$\begin{aligned}
     \boxed{
-        \pdv{u}{x} = \pdv{v}{y}
-        \qquad
-        \pdv{v}{x} = - \pdv{u}{y}
+        \begin{aligned}
+            \pdv{u}{x}
+            &= \pdv{v}{y}
+            \\
+            \pdv{v}{x}
+            &= - \pdv{u}{y}
+        \end{aligned}
     }
 \end{aligned}$$
 
@@ -85,7 +91,8 @@ Just like before, we decompose $$f(z)$$ into its real and imaginary parts:
 $$\begin{aligned}
     \oint_C f(z) \dd{z}
     &= \oint_C (u + i v) \dd{(x + i y)}
-    = \oint_C (u + i v) \:(\dd{x} + i \dd{y})
+    \\
+    &= \oint_C (u + i v) \:(\dd{x} + i \dd{y})
     \\
     &= \oint_C u \dd{x} - v \dd{y} + i \oint_C v \dd{x} + u \dd{y}
 \end{aligned}$$
@@ -137,10 +144,10 @@ $$\begin{aligned}
 {% include proof/end.html id="proof-int-formula" %}
 
 
-Similarly, **Cauchy's differentiation formula**,
-or **Cauchy's integral formula for derivatives**
-gives all derivatives of a holomorphic function as follows,
-and also guarantees their existence:
+Similarly, **Cauchy's differentiation formula**
+or **integral formula for derivatives**
+gives the $$n$$th-order derivative of a holomorphic function as follows,
+and guarantees its existence:
 
 $$\begin{aligned}
     \boxed{