Theorem 1 (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 rectiffable curve in \(G\) which does not pass through any of the points \(a_k\) and if
\(\gamma \approx 0\) in \(G\), then
\[ \frac {1}{2\pi i}\int _\gamma \! f = \sum _{k=1}^m n(\gamma ;a_k)\Res (f;a_k)\,. \]
\[ \begin {pmatrix} D_1t&-a_{12}t_2&\dots& -a_{1n}t_n\\ -a_{21}t_1&D_2t&\dots& -a_{2n}t_n\\ \hdotsfor [2]{4}\\ -a_{n1}t_1&-a_{n2}t_2&\dots&D _nt \end {pmatrix} \]
\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^{\#})\le 0\\ &\varliminf _{n\rightarrow \infty } \left \lvert a_{n+1}\right \rvert /\left \lvert a_n\right \rvert =0\\ &\varinjlim (m_i^\lambda \cdot )^*\le 0\\ &\varprojlim _{p\in S(A)}A_p\le 0 \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}
\[ W(\Phi )= \begin {Vmatrix} \dfrac \varphi {(\varphi _1,\varepsilon _1)}&0&\dots& 0\\ \dfrac {\varphi k_{n2}}{(\varphi _2,\varepsilon _1)}& \dfrac \varphi {(\varphi _2,\varepsilon _2)}&\dots& 0\\ \hdotsfor {5}\\ \dfrac {\varphi k_{n1}}{(\varphi _n,\varepsilon _1)}& \dfrac {\varphi k_{n2}}{(\varphi _n,\varepsilon _2)}&\dots& \dfrac {\varphi k_{n\,n-1}}{(\varphi _n,\varepsilon _{n-1})}& \dfrac {\varphi }{(\varphi _n,\varepsilon _n)} \end {Vmatrix} \]
\[ \sum _{\begin {subarray}{l} 0\le i\le m\\ 0<j<n \end {subarray}} P(i,j) \]
\[\biggl (\mathbf {E}_{y} \int _0^{t_\varepsilon }L_{x,y^x(s)}\varphi (x)\,ds \biggr ) \]
\[\biggl (\mathbf {E}_{y} \int _0^{t_\varepsilon }L_{x,y^x(s)}\varphi (x)\,ds \biggr ) \]
\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*}