From 1d700ab734aa9b6711eb31796beb25cb7659d8e0 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 20 Dec 2022 20:11:25 +0100
Subject: More improvements to knowledge base

---
 source/know/concept/residue-theorem/index.md | 11 +++++++----
 1 file changed, 7 insertions(+), 4 deletions(-)

(limited to 'source/know/concept/residue-theorem')

diff --git a/source/know/concept/residue-theorem/index.md b/source/know/concept/residue-theorem/index.md
index a0f515e..49a6f7a 100644
--- a/source/know/concept/residue-theorem/index.md
+++ b/source/know/concept/residue-theorem/index.md
@@ -12,7 +12,7 @@ A function $$f(z)$$ is **meromorphic** if it is
 [holomorphic](/know/concept/holomorphic-function/)
 except in a finite number of **simple poles**,
 which are points $$z_p$$ where $$f(z_p)$$ diverges,
-but where the product $$(z - z_p) f(z)$$ is non-zero
+but where the product $$(z - z_p) f(z)$$ is nonzero
 and still holomorphic close to $$z_p$$.
 In other words, $$f(z)$$ can be approximated close to $$z_p$$:
 
@@ -26,7 +26,8 @@ represents the rate at which $$f(z)$$ diverges close to $$z_p$$:
 
 $$\begin{aligned}
     \boxed{
-        R_p = \lim_{z \to z_p} (z - z_p) f(z)
+        R_p
+        \equiv \lim_{z \to z_p} (z - z_p) f(z)
     }
 \end{aligned}$$
 
@@ -37,7 +38,8 @@ purely depends on the simple poles $$z_p$$ enclosed by $$C$$:
 
 $$\begin{aligned}
     \boxed{
-        \oint_C f(z) \dd{z} = i 2 \pi \sum_{z_p} R_p
+        \oint_C f(z) \dd{z}
+        = i 2 \pi \sum_{z_p} R_p
     }
 \end{aligned}$$
 
@@ -48,7 +50,8 @@ we know that we can decompose $$f(z)$$ like so,
 where $$h(z)$$ is holomorphic and $$z_p$$ are all its poles:
 
 $$\begin{aligned}
-    f(z) = h(z) + \sum_{z_p} \frac{R_p}{z - z_p}
+    f(z)
+    = h(z) + \sum_{z_p} \frac{R_p}{z - z_p}
 \end{aligned}$$
 
 We integrate this over a contour $$C$$ which contains all poles, and apply
-- 
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