math.sty

\ProvidesPackage{prettytex/math}
\RequirePackage{prettytex/base}

%%----------------------------------------------------------------------------------------------------------------------
% package loading
%%----------------------------------------------------------------------------------------------------------------------
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{esint}
\usepackage{bm}
\usepackage{upgreek}
\usepackage{nicematrix}

%%----------------------------------------------------------------------------------------------------------------------
% general / auxilliary macros / formatting
%%----------------------------------------------------------------------------------------------------------------------
\newcommand{\qqtext}[1]{\qquad\text{#1}\qquad}
\newcommand{\matrixcolor}{black}
\newcommand{\operatorcolor}{blue!65!black}

\newcommand{\basetrace}{trace}
\newcommand{\baserank}{rank}

\renewcommand{\epsilon}{\varepsilon} % the default epsilon is ugly :shrug:
\newcommand{\feq}{\overset{!}{=}} % should be equal to (force-equal)
\renewcommand{\Re}{\mathfrak{Re}}
\renewcommand{\Im}{\mathfrak{Im}}
\newcommand{\eqv}{\Longleftrightarrow}

\newcommand{\floorl}[1]{\left\lfloor#1\right\rfloor}
\newcommand{\floor}[1]{\lfloor#1\rfloor}
\newcommand{\ceil}[1]{\lceil#1\right}
\newcommand{\ceill}[1]{\left\lceil#1\right\rceil}
\renewcommand{\bar}[1]{\overline{#1}}
\newcommand{\ubar}[1]{\underline{#1}}
\newcommand{\oset}[2]{\overset{#1}{#2}}
\newcommand{\uset}[2]{\underset{#1}{#2}}
\newcommand{\obra}[1]{\overbrace{#1}}
\newcommand{\ubra}[1]{\underbrace{#1}}
\newcommand{\parenth}[1]{\left(#1\right)}

% for small calligraphic letters
\DeclareFontFamily{OT1}{pzc}{}
\DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [1.10] pzcmi7t}{}
\DeclareMathAlphabet{\mathscall}{OT1}{pzc}{m}{it}
\newcommand{\sco}{\mathscall{o}}

%%----------------------------------------------------------------------------------------------------------------------
% package options
%%----------------------------------------------------------------------------------------------------------------------
\DeclareOption{vargreek}{
  \renewcommand{\phi}{\varphi}
  \renewcommand{\rho}{\varrho}
}
\DeclareOption{nocols}{\renewcommand{\operatorcolor}{black}}
\DeclareOption{de}{
  \renewcommand{\basetrace}{spur}
  \renewcommand{\baserank}{rang}
}
\ProcessOptions\relax

%----------------------------------------------------------------------------------------------------------------------
% constants
%----------------------------------------------------------------------------------------------------------------------
\renewcommand{\i}{\mathrm{i}} % imaginary unit
\renewcommand{\j}{\mathrm{j}} % jmaginary unit
\newcommand{\e}{\mathrm{e}} % euler's number
\newcommand{\Coulomb}{\mathrm{k}_C}


%%----------------------------------------------------------------------------------------------------------------------
% renewed commands
%%----------------------------------------------------------------------------------------------------------------------
\renewcommand{\complement}{\mathsf{c}}

%%----------------------------------------------------------------------------------------------------------------------
% sets, topology and measure theory
%%----------------------------------------------------------------------------------------------------------------------
\newcommand{\set}[1]{{\left\{#1\right\}}}
\newcommand{\sm}{\setminus}
\newcommand{\compl}{^\complement}
\newcommand{\eset}{\emptyset}
\newcommand{\sset}{\subseteq}
\newcommand{\inv}{^{-1}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\I}{\mathbb{I}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\T}{\mathbb{T}}
\renewcommand{\S}{\mathbb{S}}
\renewcommand{\H}{\mathbb{H}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\F}{\mathbb{F}}

\DeclareMathOperator{\interior}{\qopname \relax m{\col{\operatorcolor}{int}}}
\DeclareMathOperator{\cls}{\qopname \relax m{\col{\operatorcolor}{cls}}}
\DeclareMathOperator{\diam}{\qopname \relax m{\col{\operatorcolor}{diam}}}
\DeclareMathOperator{\ext}{\qopname \relax m{\col{\operatorcolor}{ext}}}
\newcommand{\inter}{^\circ}
\DeclareMathOperator{\supp}{\qopname \relax m{\col{\operatorcolor}{supp}}}

\newcommand{\fc}{\mathfrak{c}}
\newcommand{\fU}{\mathfrak{U}}

\newcommand{\leb}{\uplambda}
\newcommand{\borel}{\upbeta}
\DeclareMathOperator{\vol}{\qopname \relax m{\col{\operatorcolor}{vol}}}

%%----------------------------------------------------------------------------------------------------------------------
% calculus
%%----------------------------------------------------------------------------------------------------------------------
\newcommand{\FD}{\mathrm{d}}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\idd}{\;\mathrm{d}}
\newcommand{\ddx}{\mathrm{d} x} % dx
\newcommand{\ddy}{\mathrm{d} y} % dy
\newcommand{\ddz}{\mathrm{d} z} % dz
\newcommand{\ddu}{\mathrm{d} u} % du
\newcommand{\ddv}{\mathrm{d} v} % dv
\newcommand{\ddr}{\mathrm{d} r} % dr
\newcommand{\dds}{\mathrm{d} s} % ds
\newcommand{\ddt}{\mathrm{d} t} % dt
\DeclareMathOperator{\hess}{\qopname \relax m{\col{\operatorcolor}{H}}}
\DeclareMathOperator{\jac}{\qopname \relax m{\col{\operatorcolor}{J}}}

\newcommand{\derx}[1]{\frac{\dd #1}{\dd x}}
\newcommand{\nderx}[1]{\frac{\dd^n #1}{\dd x^n}}
\newcommand{\kderx}[1]{\frac{\dd^k #1}{\dd x^k}}
\newcommand{\nder}[1]{#1^{(n)}}
\newcommand{\kder}[1]{#1^{(k)}}

\newcommand{\abs}[1]{|#1|}
\newcommand{\labs}[1]{\left|#1\right|}

\let\lim\relax
\let\sup\relax
\let\inf\relax
\let\min\relax
\let\max\relax
\let\arg\relax
\let\liminf\relax
\let\limsup\relax
\let\div\relax

\DeclareMathOperator*{\div}{\qopname \relax m{\col{\operatorcolor}{div}}}
\DeclareMathOperator*{\minimise}{\qopname \relax m{\col{\operatorcolor}{minimise}}}
\DeclareMathOperator*{\curl}{\qopname \relax m{\col{\operatorcolor}{curl}}}
\DeclareMathOperator*{\grad}{\qopname \relax m{\col{\operatorcolor}{grad}}}
\DeclareMathOperator*{\lim}{\qopname \relax m{\col{\operatorcolor}{lim}}}
\DeclareMathOperator*{\sup}{\qopname \relax m{\col{\operatorcolor}{sup}}}
\DeclareMathOperator*{\inf}{\qopname \relax m{\col{\operatorcolor}{inf}}}
\DeclareMathOperator*{\min}{\qopname \relax m{\col{\operatorcolor}{min}}}
\DeclareMathOperator*{\max}{\qopname \relax m{\col{\operatorcolor}{max}}}
\DeclareMathOperator*{\arg}{\qopname \relax m{\col{\operatorcolor}{arg}}}
\DeclareMathOperator*{\liminf}{\qopname \relax m{\col{\operatorcolor}{lim\:inf}}}
\DeclareMathOperator*{\limsup}{\qopname \relax m{\col{\operatorcolor}{lim\:sup}}}
\DeclareMathOperator*{\argmin}{\qopname \relax m{\col{\operatorcolor}{arg\:min}}}
\DeclareMathOperator*{\argmax}{\qopname \relax m{\col{\operatorcolor}{arg\:max}}}
\DeclareMathOperator*{\uniflim}{\qopname \relax m{\col{\operatorcolor}{unif\:lim}}}
\DeclareMathOperator*{\apexlim}{\qopname \relax m{\col{\operatorcolor}{apex\:lim}}}
\newcommand{\laplace}{{\triangle}}

\newcommand{\limh}{\lim_{h\to 0}}
\newcommand{\limt}{\lim_{t\to 0}}
\newcommand{\limx}{\lim_{x\to x_0}}
\newcommand{\limz}{\lim_{z\to z_0}}
\newcommand{\limn}{\lim_{n\to \infty}}
\newcommand{\limk}{\lim_{k\to \infty}}

\newcommand{\seqn}[1]{(#1_n)_{n\in\N}}
\newcommand{\seqnz}[1]{(#1_n)_{n\in\N_0}}
\newcommand{\seqk}[1]{(#1_k)_{k\in\N}}
\newcommand{\seqkz}[1]{(#1_k)_{k\in\N_0}}

%%----------------------------------------------------------------------------------------------------------------------
% algebra
%%----------------------------------------------------------------------------------------------------------------------
\newcommand{\Hom}{\col{\operatorcolor}{\mathrm{Hom}}}
\DeclareMathOperator{\im}{\qopname \relax m{\col{\operatorcolor}{im}}}
\newcommand{\id}{\col{\operatorcolor}{\mathrm{id}}}
\newcommand{\sign}{\col{\operatorcolor}{\mathrm{sign}}}
\newcommand{\GL}{\col{\operatorcolor}{\mathrm{GL}}} % general linear group
\newcommand{\SL}{\col{\operatorcolor}{\mathrm{SL}}} % special linear group
\newcommand{\SO}{\col{\operatorcolor}{\mathrm{SO}}} % special orthogonal group
\newcommand{\epimorph}{\twoheadrightarrow} % mapping is epimorphism (injective)
\newcommand{\monmorph}{\rightarrowtail} % mapping is monomorphism (surjective)
\newcommand{\isomorph}{\twoheadrightarrowtail} % mapping is isomorphism (bijective)
\newcommand{\Aut}{\col{\operatorcolor}{\mathrm{Aut}}} % set of automorphisms
\newcommand{\End}{\col{\operatorcolor}{\mathrm{End}}} % set of endomorphisms
\DeclareMathOperator{\ran}{\qopname \relax m{ran}}
\DeclareMathOperator{\dom}{\qopname \relax m{dom}}
\DeclareMathOperator{\codom}{\qopname \relax m{co\:dom}}
\DeclareMathOperator{\proj}{\qopname \relax m{proj}}
\DeclareMathOperator{\coker}{\qopname \relax m{coker}}

\newcommand{\dspan}{\col{\operatorcolor}{\mathrm{span}}}
\let\dim\relax
\let\ker\relax
\let\deg\relax
\DeclareMathOperator{\dim}{\qopname \relax m{\col{\operatorcolor}{dim}}}
\DeclareMathOperator{\ker}{\qopname\relax m{\col{\operatorcolor}{ker}}}
\DeclareMathOperator{\deg}{\qopname \relax m{\col{\operatorcolor}{deg}}}

%%----------------------------------------------------------------------------------------------------------------------
% linear algebra
%%----------------------------------------------------------------------------------------------------------------------
\newcommand{\transpose}{\mathsf{t}}
\newcommand{\tran}{^\transpose}
\newcommand{\itran}{^{-\transpose}}

\newcommand{\dotp}[2]{\left\langle#1,#2\right\rangle}

\renewcommand{\vec}[1]{{\bm{#1}}}
\newcommand{\hvec}[1]{{\hat{\vec{#1}}}} % vector with hat
\newcommand{\tvec}[1]{{\tilde{\vec{#1}}}} % vector with tilde
\newcommand{\bvec}[1]{{\bar{\vec{#1}}}} % vector with bar
\newcommand{\uvec}[1]{{\ubar{\vec{#1}}}}
\newcommand{\zvec}{\vec{0}}

\newcommand{\norm}[1]{{\|#1\|}}
\newcommand{\normc}{\norm{\cdot}}
\newcommand{\lnorm}[1]{{\left\|#1\right\|}}
\newcommand{\inorm}[1]{{\|#1\|_\infty}}
\newcommand{\inormc}{\inorm{\cdot}}
\newcommand{\linorm}[1]{{\left\|#1\right\|_\infty}}

\newcommand{\bmat}[2][1]{{\renewcommand{\arraystretch}{#1}\begin{bmatrix}#2\end{bmatrix}}}
\newcommand{\fbmat}[1]{\begin{bNiceMatrix}#1\end{bNiceMatrix}}

\newcommand{\diag}{\col{\operatorcolor}{\mathrm{diag}}} % diagonal of matrix / diagonal matrix (definition dependent)
\newcommand{\Eig}{\col{\operatorcolor}{\mathrm{Eig}}} % eigendata of matrix
\newcommand{\Gram}{\col{\operatorcolor}{\mathrm{Gram}}} % Gram matrix
\newcommand{\detm}[2]{\left\vert\begin{array}[#1]#2\end{array}\right\vert} % shorthand determiant
\newcommand{\hateq}{\overset{\scriptscriptstyle\wedge}{=}} % matrices are congruent
\let\det\relax
\DeclareMathOperator{\det}{\col{\operatorcolor}{\mathrm{det}}}
\DeclareMathOperator{\spec}{\qopname \relax m{\col{\operatorcolor}{spec}}} % set of eigenvalues of linear map
\DeclareMathOperator{\Ann}{\col{\operatorcolor}{\mathrm{Ann}}} % annihilator of matrix
\DeclareMathOperator{\Adj}{\col{\operatorcolor}{\mathrm{Adj}}} % annihilator of matrix
\DeclareMathOperator{\Cof}{\col{\operatorcolor}{\mathrm{Cof}}} % annihilator of matrix

\newcommand{\mat}[1]{\mathrm{\col{\matrixcolor!95!black}{#1}}} % matrix typing
\newcommand{\tmat}[1]{\tilde{\mat{#1}}} % matrix with tilde
\newcommand{\hmat}[1]{\hat{\mat{#1}}} % matrix with hat
\newcommand{\barmat}[1]{\bar{\mat{#1}}}
\newcommand{\umat}[1]{\ubar{\mat{#1}}}

\newcommand{\spur}{\col{\operatorcolor}{\mathrm{\basetrace}}}
\newcommand{\spr}{\col{\operatorcolor}{\mathrm{\basetrace}}}
\newcommand{\Spur}{\col{\operatorcolor}{\mathrm{\basetrace}}}
\newcommand{\Spr}{\col{\operatorcolor}{\mathrm{\basetrace}}}
\newcommand{\trace}{\col{\operatorcolor}{\mathrm{\basetrace}}}
\newcommand{\Trace}{\col{\operatorcolor}{\mathrm{\basetrace}}}
\newcommand{\Tr}{\col{\operatorcolor}{\mathrm{\basetrace}}}

\newcommand{\rg}{\col{\operatorcolor}{\mathrm{\baserank}}}
\newcommand{\Rg}{\col{\operatorcolor}{\mathrm{\baserank}}}
\newcommand{\rang}{\col{\operatorcolor}{\mathrm{\baserank}}}
\newcommand{\Rang}{\col{\operatorcolor}{\mathrm{\baserank}}}
\newcommand{\rk}{\col{\operatorcolor}{\mathrm{\baserank}}}
\newcommand{\Rk}{\col{\operatorcolor}{\mathrm{\baserank}}}
\newcommand{\rank}{\col{\operatorcolor}{\mathrm{\baserank}}}
\newcommand{\Rank}{\col{\operatorcolor}{\mathrm{\baserank}}}


\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{
  \hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{#1}}
\makeatother

\newcommand*{\longleadsto}[1]{
  \tikzexternaldisable
  \begin{tikzpicture}
    \draw [->, line join=round, decorate, decoration={
          zigzag,
          segment length=4,
          amplitude=.9,
          post=lineto,
          post length=2pt}] (0,0) -- (#1,0);
  \end{tikzpicture}
  \tikzexternalenable
}

\newcommand{\rtrf}[1]{%
  \tikzexternaldisable%
  \begin{tikzpicture}%
    \draw [->, line join=round, decorate, decoration={%
          zigzag,%
          segment length=4,%
          amplitude=.9,%
          post=lineto,%
          post length=2pt}] (0,0) -- (1.5cm,0)%
    node[midway,above]{$\substack{#1}$};%
  \end{tikzpicture}%
  \tikzexternalenable%
} % matrix row transformation

%----------------------------------------------------------------------------------------------------------------------
% number theory
%----------------------------------------------------------------------------------------------------------------------
\newcommand{\ggt}{\col{\operatorcolor}{\mathrm{ggT}}} % greatets common divisor (de)
\newcommand{\kgv}{\col{\operatorcolor}{\mathrm{kgV}}} % least common multiple (de)
\renewcommand{\gcd}{\col{\operatorcolor}{\mathrm{gcd}}} % gcd (en)
\newcommand{\lcm}{\col{\operatorcolor}{\mathrm{lcm}}} % lcm (en)
\newcommand{\stirlingone}[2]{\begin{bmatrix}#1\\#2\end{bmatrix}} % stirling number first kind
\newcommand{\stirlingtwo}[2]{\begin{Bmatrix}#1\\#2\end{Bmatrix}} % striling number second kind
\newcommand{\cycle}[1]{\begin{pmatrix}#1\end{pmatrix}} % cycle notation

%----------------------------------------------------------------------------------------------------------------------
% exotic trig functions
%----------------------------------------------------------------------------------------------------------------------
\newcommand{\Arsinh}{\mathrm{Arsinh}} % Area sinus hyperbolicus
\newcommand{\Arcosh}{\mathrm{Arcosh}} % Area cosinus hyperbolicus
\newcommand{\Artanh}{\mathrm{Artanh}} % Area tangens hyperbolicus
\newcommand{\Arcoth}{\mathrm{Arcoth}} % Area cotangens hyperbolicus
\newcommand{\Arsech}{\mathrm{Arsech}} % Area secans hyperbolicus
\newcommand{\Arcsch}{\mathrm{Arcsch}} % Area cosecancs hyperbolicus
\newcommand{\sinc}{\mathrm{sinc}} % sinus cardinalis (de, not normalised)
\newcommand{\sinnc}{\mathrm{si}} % sinus cardinalis (de, normalised)
\newcommand{\Si}{\mathrm{Si}} % integral sinus
\newcommand{\Li}{\mathrm{Li}} % integral logarithmus naturlis
\newcommand{\arccot}{\mathrm{arccot}}
\newcommand{\arcsec}{\mathrm{arcsec}}
\newcommand{\arccsc}{\mathrm{arccsc}}
\newcommand{\sech}{\mathrm{sech}}
\newcommand{\csch}{\mathrm{csch}}
\DeclareMathOperator{\atan}{\qopname \relax m{atan2}}
\DeclareMathOperator{\ld}{\mathrm{ld}}