feat: maths folder -> newtykins/maths

.husky/pre-commit

@@ -2,5 +2,4 @@
 . "$(dirname "$0")/_/husky.sh"
 
 npx pretty-quick --staged
-.husky/scripts/pdfToPng.sh
 git add .

.husky/scripts/pdfToPng.sh

@@ -1,22 +0,0 @@
-#!/bin/bash
-find . -print0 | while IFS= read -r -d '' file
-do
-  if [ -f "$file" ]; then
-	if [[ $file == *pdf ]]; then
-		mkdir temp
-		gm convert "$file" +adjoin temp/temp%02d.png
-
-		for temp in ./temp/*.png
-		do
-			gm convert "$temp" -fuzz 80% -trim +repage -bordercolor white -border 50x25 "$temp"
-		done
-
-		readarray -d . -t arr <<< $file
-
-		gm convert -append ./temp/*.png "${arr[1]:1}.png"
-		
-		rm -rf temp
-		rm "$file" "${arr[1]:1}.synctex.gz"
-	fi
-fi
-done

.vscode/extensions.json

@@ -1,3 +0,0 @@
-{
-    "recommendations": ["James-Yu.latex-workshop"]
-}

.vscode/settings.json

@@ -12,7 +12,6 @@
         "commitlint.config.js": true,
         "**/pnpm-lock.yaml": true
     },
-    "C_Cpp.errorSquiggles": "Disabled",
     "files.associations": {
         "*.tcc": "cpp",
         "complex": "cpp",
@@ -33,7 +32,5 @@
         "ostream": "cpp",
         "ratio": "cpp",
         "streambuf": "cpp"
-    },
-    "latex-workshop.latex.autoBuild.run": "onSave",
-    "latex-workshop.latex.autoClean.run": "onBuilt"
+    }
 }

maths/halley's method.tex

@@ -1,53 +0,0 @@
-\documentclass{style}
-\usepackage{amsmath}
-\usepackage{breqn}
-\usepackage{mathtools}
-
-\begin{document}
-\begin{center}
-	Suppose we have an $n$th degree polynomial, such that $f(x) = c_{n}x^{n} + c_{n-1}x^{n-1} + c_{n-2}x^{n-2} + ... + c_{0} = \sum_{i=0}^{n}c_{n-i}x^{n-i}$
-
-	\hfill
-
-	By the power rule of differentiation, we can conclude that the first derivative of $f(x)$ is as follows
-
-	\begin{dmath*}
-		f'(x) = nc_{n}x^{n-1} + (n-1)c_{n-1}x^{n-2} + (n-2)c_{n-2}x^{n-3} + ... + c_{1} = \sum_{i=0}^{n-1}(n-i)c_{n-i}x^{n-i-1}
-	\end{dmath*}
-
-	And the second derivative is
-
-	\begin{dmath*}
-		f''(x) = n(n-1)c_{n}x^{n-2} + (n-1)(n-2)c_{n-1}x^{n-3} + (n-2)(n-3)c_{n-2}x^{n-4} + ...+ c_{2} = \sum_{i=0}^{n-2}(i-n)(i-n+1)a_{n-i}x^{n-i-2}
-	\end{dmath*}
-
-	After starting with an initial guess, the next iteration of Halley's method is given by $x_k - \frac{2f(x_k)f'(x_k)}{2[f'(x_k)]^2 - f(x_k)f''(x_k)}$.
-	
-	This means that we must first find $f(x)f'(x)$, $[f'(x)]^2$, and $f(x)f''(x)$. These all consist of the multiplication of two series - there is a nice general form to this problem stated below
-
-	\begin{dmath*}
-		(\sum_{i=0}^{n}x_{i})(\sum_{j=0}^{m}y_{j}) = \sum_{i=0}^{n}\sum_{j=0}^{m}x_iy_j
-	\end{dmath*}
-
-	\newpage
-	From this, we can conclude that:
-
-	\begin{dmath*}
-		f(x)f'(x) = (\sum_{i=0}^{n}c_{n-i}x^{n-i})(\sum_{i=0}^{n-1}(n-i)c_{n-i}x^{n-i-1}) = \sum_{i=0}^{n}\sum_{j=0}^{n-1}(n-j)c_{n-i}c_{n-j}x^{2n-i-j-1}
-	\end{dmath*}
-
-	\begin{dmath*}
-		[f'(x)]^2 = (\sum_{i=0}^{n-1}(n-i)c_{n-i}x^{n-i-1})(\sum_{i=0}^{n-1}(n-i)c_{n-i}x^{n-i-1}) = \sum_{i=0}^{n-1}\sum_{j=0}^{n-1}(n-i)(n-j)c_{n-i}c_{n-j}x^{2(n-1)-i-j}
-	\end{dmath*}
-
-	\begin{dmath*}
-		f(x)f''(x) = (\sum_{i=0}^{n}c_{n-i}x^{n-i})(\sum_{i=0}^{n-2}(i-n)(i-n+1)a_{n-i}x^{n-i-2}) = \sum_{i=0}^{n}\sum_{j=0}^{n-2}(j-n)(j-n+1)c_{n-i}c_{n-j}x^{2(n-1)-i-j}
-	\end{dmath*}
-
-	And all that is left to do is to plug it into the formula for Halley's method, leaving us with the following:
-
-	\begin{align*}
-		x_{k+1} = x_{k} - \dfrac{2\sum_{i=0}^{n}\sum_{j=0}^{n-1}(n-j)c_{n-i}c_{n-j}x^{2n-i-j-1}}{\splitdfrac{2\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}(n-i)(n-j)c_{n-i}c_{n-j}x^{2(n-1)-i-j}}{- \sum_{i=0}^{n}\sum_{j=0}^{n-2}(j-n)(j-n+1)c_{n-i}c_{n-j}x^{2(n-1)-i-j}}}
-	\end{align*}
-\end{center}
-\end{document}

maths/nth root.tex

@@ -1,12 +0,0 @@
-\documentclass{style}
-\usepackage{amsmath}
-\begin{document}
-\begin{gather*}
-	f(x) = x^k - s \\
-	f'(x) = kx^{k - 1}
-\end{gather*}
-
-\begin{gather*}
-	x_{n + 1} = x_n - \frac{x_n^k - s}{kx_n^{k - 1}} = \frac{x_n(k - 1) + sx_n^{1 - k}}{k}
-\end{gather*}
-\end{document}

maths/style.cls

@@ -1,11 +0,0 @@
-\LoadClass[17pt]{extarticle}
-\pagenumbering{gobble}
-
-\usepackage{geometry}
-\geometry{a4paper, portrait, margin=1in}
-
-\newcommand{\euler}{\begin{gather*}
-	\text{By Euler's formula:} \\
-	e^{i\theta} = cos(\theta) + i\sin(\theta) \\
-	e^{-i\theta} = cos(\theta) - i\sin(\theta)
-\end{gather*}}

maths/trigometric functions/cos.tex

@@ -1,27 +0,0 @@
-\documentclass{../style}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\begin{document}
-\euler
-
-\begin{gather*}
-	\therefore \cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}
-\end{gather*}
-
-\begin{gather*}
-	\text{let} \quad \cos(\theta) = x \\
-	2x = e^{i\theta} + e^{-i\theta} \\
-	2xe^{i\theta} = (e^{i\theta})^2 + 1 \\ 
-	(e^{i\theta})^2 + (-2x)e^{i\theta} + 1 = 0
-\end{gather*}
-
-\begin{gather*}
-	e^{i\theta} = \frac{-(-2x) \pm \sqrt{(-2x)^2 - 4}}{2} = x \pm \sqrt{x^2 - 1} \\
-	i\theta = \ln(x \pm \sqrt{x^2 - 1}) \\
-	\theta = -i\ln(x \pm \sqrt{x^2 - 1})
-\end{gather*}
-
-\begin{gather*}
-	\therefore \arccos(\theta) = -i\ln(\theta \pm \sqrt{\theta^2 - 1})
-\end{gather*}
-\end{document}

maths/trigometric functions/cot.tex

@@ -1,29 +0,0 @@
-\documentclass{../style}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\begin{document}
-\euler
-
-\begin{gather*}
-	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
-	\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \\
-	\tan(\theta) = \frac{\sin(\theta)}{\cos{\theta}} = -\frac{i(-1 + e^{2i\theta})}{1 + e^{2i\theta}} \\
-	\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{1 + e^{2i\theta}}{i(-1 + e^{2i\theta})}
-\end{gather*}
-
-\begin{gather*}
-	\text{let} \quad \cot(\theta) = x \\
-	x(i + ie^{2i\theta}) = -1(1 + e^{2i\theta}) \\
-	ix + ixe^{2i\theta} = -e^{2i\theta} - 1 \\
-	ixe^{2i\theta} + e^{2i\theta} = -1 - ix \\
-	(ix + 1)e^{2i\theta} = -1 - ix \\
-	e^{2i\theta} = -\frac{1 - ix}{1 + ix} \\
-	2i\theta = \ln(\frac{x + i}{x - i}) \\
-	i\theta = \frac{1}{2}\ln(\frac{x + i}{x - i}) \\
-	\theta = -\frac{i}{2}\ln(\frac{x + i}{x - i})
-\end{gather*}
-
-\begin{gather*}
-	\therefore \text{arccot}(\theta) = -\frac{i}{2}\ln(\frac{\theta + i}{\theta - i})
-\end{gather*}
-\end{document}

maths/trigometric functions/csc.tex

@@ -1,28 +0,0 @@
-\documentclass{../style}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\begin{document}
-\euler
-
-\begin{gather*}
-	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
-	\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{2i}{e^{i\theta} - e^{-i\theta}}
-\end{gather*}
-
-\begin{gather*}
-	\text{let} \quad \csc(\theta) = x \\
-	\frac{2i}{x} = e^{i\theta} - e^{-i\theta} \\
-	\frac{2i}{x}e^{i\theta} = (e^{i\theta})^2 - 1 \\ 
-	(e^{i\theta})^2 + (-\frac{2i}{x})e^{i\theta} - 1 = 0
-\end{gather*}
-
-\begin{gather*}
-	e^{i\theta} = \frac{-(\frac{2i}{x}) \pm \sqrt{(\frac{2i}{x})^2 - 4(-1)}}{2} = x^{-1}i \pm \sqrt{1 - x^2} \\
-	i\theta = \ln(x^{-1}i \pm \sqrt{1 - x^2}) \\
-	\theta = -i\ln(x^{-1}i \pm \sqrt{1 - x^2})
-\end{gather*}
-
-\begin{gather*}
-	\therefore \text{arccsc}(\theta) = -i\ln(i\theta^{-1} \pm \sqrt{1 -\theta^2})
-\end{gather*}
-\end{document}

maths/trigometric functions/sec.tex

@@ -1,29 +0,0 @@
-\documentclass{../style}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\begin{document}
-\euler
-
-\begin{gather*}
-	\therefore \cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \\
-	\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{2}{e^{i\theta} + e^{-i\theta}}
-\end{gather*}
-
-\begin{gather*}
-	\text{let} \quad \sec(\theta) = x \\
-	x(e^{i\theta} + e^{-i\theta}) = 2 \\
-	e^{i\theta} + e^{-i\theta} = \frac{2}{x} \\
-	(e^{i\theta})^2 + 1 = \frac{2}{x}e^{i\theta} \\
-	(e^{i\theta})^2 + (-\frac{2}{x})e^{i\theta} + 1 = 0
-\end{gather*}
-
-\begin{gather*}
-	e^{i\theta} = \frac{-(-\frac{2}{x}) \pm \sqrt{(-\frac{2}{x})^2 - 4}}{2} = x^{-1} \pm \sqrt{x^{-2} - 1} \\
-	i\theta = \ln(x^{-1} \pm \sqrt{x^{-2} - 1}) \\
-	\theta = -i\ln(x^{-1} \pm \sqrt{x^{-2} - 1})
-\end{gather*}
-
-\begin{gather*}
-	\therefore \text{arcsec}(\theta) = -i\ln(\theta^{-1} \pm \sqrt{\theta^{-2} - 1})
-\end{gather*}
-\end{document}

maths/trigometric functions/sin.tex

@@ -1,27 +0,0 @@
-\documentclass{../style}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\begin{document}
-\euler
-
-\begin{gather*}
-	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}
-\end{gather*}
-
-\begin{gather*}
-	\text{let} \quad \sin(\theta) = x \\
-	2ix = e^{i\theta} - e^{-i\theta} \\
-	2ixe^{i\theta} = (e^{i\theta})^2 - 1 \\ 
-	(e^{i\theta})^2 + (-2ix)e^{i\theta} - 1 = 0
-\end{gather*}
-
-\begin{gather*}
-	e^{i\theta} = \frac{-(-2ix) \pm \sqrt{(-2ix)^2 - 4(-1)}}{2} = ix \pm \sqrt{1 - x^2} \\
-	i\theta = \ln(ix \pm \sqrt{1 - x^2}) \\
-	\theta = -i\ln(ix \pm \sqrt{1 - x^2})
-\end{gather*}
-
-\begin{gather*}
-	\therefore \arcsin(\theta) = -i\ln(i\theta \pm \sqrt{1 -\theta^2})
-\end{gather*}
-\end{document}

maths/trigometric functions/tan.tex

@@ -1,28 +0,0 @@
-\documentclass{../style}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\begin{document}
-\euler
-
-\begin{gather*}
-	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
-	\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \\
-	\tan(\theta) = \frac{\sin(\theta)}{\cos{\theta}} = -\frac{i(-1 + e^{2i\theta})}{1 + e^{2i\theta}}
-\end{gather*}
-
-\begin{gather*}
-	\text{let} \quad \tan(\theta) = x \\
-	x(1 + e^{2i\theta}) = -i(-1 + e^{2i\theta}) \\
-	x + xe^{2i\theta} = i - ie^{2i\theta} \\
-	xe^{2i\theta} + ie^{2i\theta} = i - x \\
-	e^{2i\theta}(i + x) = i - x \\
-	e^{2i\theta} = \frac{i - x}{i + x} \\
-	2i\theta = \ln(\frac{i - x}{i + x}) \\
-	i\theta = \frac{1}{2}\ln(\frac{i - x}{i + x}) \\
-	\theta = -\frac{i}{2}\ln(\frac{i - x}{i + x})
-\end{gather*}
-
-\begin{gather*}
-	\therefore \arctan(\theta) = -\frac{i}{2}\ln(\frac{i - \theta}{i + \theta})
-\end{gather*}
-\end{document}