\documentclass[12pt]{article}
\pagestyle{empty}
\usepackage[letterpaper,left=1in,right=1in,top=.9in,bottom=1in, headsep=.25in]{geometry}
\usepackage{ntheorem}
\newtheorem{theorem}{Theorem}
\usepackage{amsmath}

\DeclareMathOperator\Res{Res}
\newcommand*\diff{\mathop{}\!\mathup{d}}


\usepackage{mathtools}
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}

\newcommand{\envert}[1]{\left\lvert#1\right\rvert}

\DeclareMathOperator*{\esssup}{ess\,sup}
\DeclareMathOperator{\meas}{meas}
\newcommand{\wh}{\widehat}
\newcommand{\wt}[1]{\widetilde{#1}}

\usepackage[tracking]{microtype}

%\usepackage{resizegather}

\setlength{\parindent}{0pt} % Removes all indentation globally
\usepackage{unicode-math}

% Set GFS NeoHellenic as your main document font
\setmainfont{GFS Neohellenic}

% Use the designated mathematical extension for GFS NeoHellenic
\setmathfont{GFS Neohellenic Math}
\begin{document}
\begin{theorem}[Residue Theorem]
Let $f$ be analytic in the region $G$ except for the isolated
singularities $a_1,a_2,\dots,a_m$. If $\gamma$ is a closed
rectifiable curve in $G$ which does not pass through any of the
points $a_k$ and if $\gamma\approx 0$ in $G$, then
\[
\frac{1}{2\pi i}\int_\gamma\! f = \sum_{k=1}^m
n(\gamma;a_k)\Res(f;a_k)\,.
\]
\end{theorem}
\[
\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
\hdotsfor[2]{4}\\
-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt
\end{pmatrix}
\]
%
\begin{multline}
\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
\sum_{I\subseteq\mathbf{n}-\{l \}}
(-1)^{\envert{I}}\mathbf{A}^{(\lambda)}(I|I)
\det\mathbf{A}^{(\lambda)}
(\overline I\cup\{l \}|\overline I\cup\{l \}).
\label{sum-ali}
\end{multline}
%
\[v^{k}_{i}=
\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\
0 &\text{otherwise.} \end{cases}
\]
%
\[
\frac{\partial x}{\partial y}
\pmb{\bigg\vert}
\frac{\partial y}{\partial z}
\]
%
\[
\sum_{\substack{i<B\\\text{$i$ odd}}}
\prod_\kappa \kappa F(r_i)\qquad
\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
\mathop{\pmb{\prod}}_\kappa \kappa(r_i)
\]
%
\begin{align*}
\overrightarrow{\psi_\delta(t) E_t h}&
=\underrightarrow{\psi_\delta(t) E_t h}\\
\overleftarrow{\psi_\delta(t) E_t h}&
=\underleftarrow{\psi_\delta(t) E_t h}\\
\overleftrightarrow{\psi_\delta(t) E_t h}&
=\underleftrightarrow{\psi_\delta(t) E_t h}
\end{align*}
%
Then we have the series $A_1,A_2,\dotsc$, the regional sum $A_1+A_2+\dotsb$, the orthogonal product $A_1A_2\dotsm$,
and the infinite integral
\[
\int_{A_1}\int_{A_2}\dotsi
\]
%
\[
\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad
\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad
\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad
\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad
\Bar{\Bar{B}}\quad\Vec{\Vec{V}}
\]
%
\[
\dddot{Q}\qquad\ddddot{R}
\]
%
\[
\sqrt[\leftroot{-2}\uproot{2}\beta]{k}
\]
%
\[
\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}
\]
%
\[
0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
\xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}
\]
%
\[
\sideset{_*^*}{_*^*}\prod_k\qquad
\sideset{}{'}\sum_{0\le i\le m} E_i\beta x
\]
%
\[
\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad
y'_k=\delta_k y_{\tau(k)}
\]
%
\[
\norm{f}_\infty=
\esssup_{x\in R^n}\abs{f(x)}
\]
%
\[
\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\}
=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\}
\quad \forall\alpha>0.
\]
%
\begin{align}
&\varlimsup_{n\rightarrow\infty}
\mathcal{Q}(u_n,u_n-u^{\#})\le0\\
&\varliminf_{n\rightarrow\infty}
\left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\
&\varinjlim (m_i^\lambda\cdot)^*\le0\\
&\varprojlim_{p\in S(A)}A_p\le0
\end{align}
%
\begin{align}
x&\equiv y+1\pmod{m^2}\\
x&\equiv y+1\mod{m^2}\\
x&\equiv y+1\pod{m^2}
\end{align}
%
\begin{equation}
\begin{split}
\sum_{\gamma\in\Gamma_C} I_\gamma&
=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\
&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l}
+\dots+(-1)^k\\
&=(2-1)^k=1
\end{split}
\end{equation}
%
\begin{align*}
\frac {\partial ^2 u}{\partial t^2} &= c^2 \left ( \frac {\partial ^2 u}{\partial r^2} + \frac {1}{r} \frac {\partial u}{\partial r} +\frac {1}{r^2} \frac {\partial ^2 u}{\partial \theta ^2} \right )
\end{align*}
%
\begin{align*}
u\left ( r,\theta ,0\right ) & =0\\
u_{t}\left ( r,\theta ,0\right ) & =\left \{ \begin {array} [c]{ccc}\frac {1}{\pi \epsilon ^{2}} & & \text {if }r\leq \epsilon \\
0 & & \text {otherwise}\end {array} \right .
\end{align*}
%
\[
\cfrac{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+\dotsb
}}}}}
\]
%
\begin{equation} P_{r-j}=
\begin{cases}
0& \text{if $r-j$ is odd},\\
r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
\end{cases}
\end{equation}
%
\[
\begin{matrix}
\vartheta& \varrho\\\varphi& \varpi
\end{matrix}\quad
\begin{pmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{pmatrix}\quad
\begin{bmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{bmatrix}\quad
\begin{Bmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{Bmatrix}\quad
\begin{vmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{vmatrix}\quad
\begin{Vmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{Vmatrix}
\]
%
This is a small matrix
\begin{math}
\bigl( \begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix} \bigr)
\end{math}
%
\[
W(\Phi)= \begin{Vmatrix}
\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\
\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&
\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\
\hdotsfor{5}\\
\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&
\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&
\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&
\dfrac{\varphi}{(\varphi_n,\varepsilon_n)}
\end{Vmatrix}
\]
%
\[
\sum_{\begin{subarray}{l}
0\le i\le m\\ 0<j<n
\end{subarray}}
P(i,j)
\]
%
\[\biggl(\mathbf{E}_{y}
\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
\biggr)
\]
%
{\Large
\[\biggl(\mathbf{E}_{y}
\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
\biggr)
\]}
%
\begin{equation}
\begin{split}
f_{h,\varepsilon}(x,y)
&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
-t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
\biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}
\varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
L_{x,y_\varepsilon(\varepsilon s)}
\varphi(x)\,ds\biggr)\biggr]\\
&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
\end{split}
\end{equation}
%
\begin{align}
\begin{split}\abs{I_1}
&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
&\le C_3\left[\int_\Omega\left(\int_{a}^x
g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\
&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}
\left(\int_{a}^x cu_t\,d\xi\right)^2\right\}
c\Omega\right]^{1/2}\\
&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
\left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
(\Omega;\Gamma_r,T)\right\rvert\right\rvert.
\end{split}\label{eq:A}\\
\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)
-\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}
\int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
&\le C_6\left\lvert \left\lvert f\int_\Omega
\left\lvert \wt{S}^{-1,0}_{a,-}
W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
\left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
(\Omega;\Gamma_r,T)\right\rvert\right\rvert.
\end{split}
\end{align}
%
\begin{multline}\label{eq:E}
\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
-2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
=\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
\int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
\end{multline}
%
\begin{gather}
D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
seg(a,r)\equiv\{z\in\mathbf{C}\colon
\Im z= \Im a,\ \abs{z-a}<r\},\notag\\
c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
\end{gather}
%
\begin{align}
\gamma_x(t)&=(\cos tu+\sin tx,v),\\
\gamma_y(t)&=(u,\cos tv+\sin ty),\\
\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
-\frac\beta\alpha\sin tu+\cos tv\right).
\end{align}
%
\begin{align*}
\gamma_x(t)&=(\cos tu+\sin tx,v),\\
\gamma_y(t)&=(u,\cos tv+\sin ty),\\
\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
-\frac\beta\alpha\sin tu+\cos tv\right).
\end{align*}
%
\begin{align}
x& =y && \text {by eq:C}\\
x'& = y' && \text {by eq:D}\\
x+x' & = y+y' && \text {by Axiom 1.}
\end{align}
%
\begin{gather}
\begin{split} \varphi(x,z)
&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
\end{split}\\[6pt]
\begin{align*}
\zeta^0 &=(\xi^0)^2,\\
\zeta^1 &=\xi^0\xi^1,\\
\zeta^2 &=(\xi^1)^2,
\end{align*}
\end{gather}
%
\begin{gather*}
\begin{split} \varphi(x,z)
&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
\end{split}\\[6pt]
\begin{align} \zeta^0&=(\xi^0)^2,\\
\zeta^1 &=\xi^0\xi^1,\\
\zeta^2 &=(\xi^1)^2,
\end{align}
\end{gather*}
%
\begin{alignat}{3}
V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
& \qquad U_i & = u_i,
\qquad \text{for $i\ne j$;}\label{eq:B}\\
V_j & = v_j, & \qquad X_j & = x_j,
& \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
\end{alignat}
%
\begin{align*}
u_{\theta \theta }+\frac {v^2}{1-\frac {v^2}{c^2}} u_{vv} + v u_v&=0
\end{align*}
%
\begin{align*}
u \left (\theta , v\right ) &= \frac {{\mathrm e}^{\frac {-4 \sqrt {\textit {\_c}_{1}}\, \theta \,c^{2}+v^{2}}{4 c^{2}}} \left (\operatorname {WhittakerW}\left (-\frac {\textit {\_c}_{1}}{2}+\frac {1}{2}, \frac {i \sqrt {\textit {\_c}_{1}}}{2}, \frac {v^{2}}{2 c^{2}}\right ) c_{4} +\operatorname {WhittakerM}\left (-\frac {\textit {\_c}_{1}}{2}+\frac {1}{2}, \frac {i \sqrt {\textit {\_c}_{1}}}{2}, \frac {v^{2}}{2 c^{2}}\right ) c_{3} \right ) \left (c_{1} {\mathrm e}^{2 \sqrt {\textit {\_c}_{1}}\, \theta }+c_{2} \right )}{v}
\end{align*}
%
\begin{equation}
u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left ( D_{n}\cos \left ( c\frac {n\pi }{L}t\right ) +E_{n}\sin \left ( c\frac {n\pi }{L}t\right ) \right ) \Phi _{n}\left ( x\right ) \tag {1}
\end{equation}
%
\begin{align*}
\int _{0}^{L}f\left ( x\right ) \Phi _{n}\left ( x\right ) dx & =D_{n}\int _{0}^{L}\Phi _{n}^{2}\left ( x\right ) dx\\
    & =\frac {L}{2}D_{n}
\end{align*}
%
\begin{align*}
\int _{0}^{L}g\left ( x\right ) \Phi _{n}\left ( x\right ) dx & =E_{n}c\frac {n\pi }{L}\int _{0}^{L}\Phi _{n}^{2}\left ( x\right ) dx\\
     & =\frac {L}{2}E_{n}c\frac {n\pi }{L}\\
     & =\frac {1}{2}E_{n}cn\pi
\end{align*}
%
\begin{align}
\Delta F_0 &= \sqrt{\sum_{i=1}^n\left(\frac{\delta F_0}{\delta x_i}\Delta x_i\right)^2}\\[0.2cm]
\Delta F_0 &= \sqrt{6.044 \cdot  10^{-6}\text{m}^2}
\end{align}
%
\begin{gather}\tag{1}
\begin{aligned}
    a &= b + c \\
    d &= e + f + g +r+ c +e + f + g +r+ c+e + f + g +r\\
    h + i &= j
\end{aligned}
\end{gather}
%
\begin{equation}
\frac {du}{dt}=\frac {\partial u}{\partial x}\frac {dx}{dt}+\frac {\partial u}{\partial t} \tag {2}
\end{equation}
%
Integrating the above w.r.t $t$ gives
\begin{align*}
\int \left(x^{\prime} x^{\prime \prime}+6 x^{\prime} x^{5}\right)d t &= 0 \\
\frac{{x^{\prime}}^{2}}{2}+x^{6} &= c_1
\end{align*}
%
\begin{gather*}\begin{aligned}
\int (1-x)^{20} x^4 \, dx&=-\frac {1}{21} (1-x)^{21}+\frac {2}{11} (1-x)^{22}-\frac {6}{23} (1-x)^{23}+\frac {1}{6} (1-x)^{24}-\frac {1}{25} (1-x)^{25}
\end{aligned}\end{gather*}
%
And
\begin{gather*}\begin{aligned}
\int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx&=\frac {x}{(-5+x) \left (3-x+3 (3+x) \log ^2\left (\log \left (x^2\right )\right )\right )}
\end{aligned}\end{gather*}
%
%
\begin{gather*}
\frac {d}{dx}\phi \left ( x,y\right ) =0
\end{gather*}
Hence
\begin{equation}
\frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B}
\end{equation}
%
Integrating (1) w.r.t. $x$ gives
\begin{align*}
\int \frac{\partial \phi}{\partial x} \mathop{\mathrm{d}x} &= \int M\mathop{\mathrm{d}x}\\
\int \frac{\partial \phi}{\partial x} \mathop{\mathrm{d}x} &= \int -2 x -1\mathop{\mathrm{d}x}\\
\phi &= -x^{2}-x+ f(y)\tag{3}
\end{align*}
Where $f(y)$ is used for the constant of integration since $\phi$ is a function of both $x$ and $y$.
Taking derivative of equation (3) w.r.t $y$ gives
\begin{align*}
\frac{\partial \phi}{\partial y} = 0+f'(y)\tag{4}
\end{align*}
But since $\phi$ itself is a constant function, then let $\phi=c_2$ where $c_2$ is new constant and combining $c_1$ and $c_2$ constants into the constant $c_1$ gives the solution as
\begin{align*}
c_1 &= -x^{2}-x +y
\end{align*}
%

\end{document}
