harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_{j_{1},m_{1},j_{2},m_{2}}^{j_{1}+j_{2}-1,m_{1}+m_{2}}=\frac{N(j_{3},m_{3})(j_{3}-1)!\sqrt{j_{3}}}{N(j_{1},m_{1})(j_{1}-1)!\sqrt{j_{1}}N(j_{2},m_{2})(j_{2}-1)!\sqrt{j_{2}}}(j_{2}m_{1}-j_{1}m_{2}).
\begin{array}{ll}{\alpha_{1}={\frac{1}{2}}(e_{1}-e_{2}-e_{3}-e_{4}+e_{5}-\sqrt3e_{6}),\quad}&{\alpha_{2}=e_{4}-e_{5},}\\{\alpha_{3}=e_{3}-e_{4},}&{\alpha_{4}=e_{4}+e_{5},}\\{\alpha_{5}={\frac{1}{2}}(e_{1}-e_{2}-e_{3}-e_{4}-e_{5}+\sqrt3e_{6}),}&{\alpha_{6}=e_{2}-e_{3},}\\{\alpha_{0}=-(e_{1}+e_{2}).}&{}\\\end{array}
{\calM}\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{dx}{2\pi}\displaystyle\frac{1}{2}e^{-x}\,{\calQ}(x)\doteq\displaystyle\frac{1}{L}\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{dx}{2\pi}e^{-x}\,{\calQ}_{-}(x)\,.
H^{(p)}=\frac{1}{p!}\intdx_{1}\ldotsdx_{p}\left[\frac{\phi^{1}}{2|z|^{2}}\left\{z_{\alpha}\frac{\vec{\delta}_{L}}{\deltaz_{\alpha}}+z_{\alpha}^{*}\frac{\vec{\delta}_{L}}{\deltaz_{\alpha}^{*}}\right\}\right]^{p}H_{0}
\begin{array}{l}{\hat{\bf\omega}^{1}(\lambda_{1}\hat{\bfA}_{1}+\lambda_{2}\hat{\bfA}_{2})=\lambda_{1}\hat{\bf\omega}^{1}(\hat{\bfA}_{1})+\lambda_{2}\hat{\bf\omega}^{1}(\hat{\bfA}_{2}),}\\{\forall\lambda_{1},\lambda_{2}\inR,\quad\hat{\bfA}_{1},\hat{\bfA}_{2}\in\hat{\bfT}_{\Phi_{p}}.}\\\end{array}
W=\frac{m_{0}\Lambda_{1}^{4}\Omega_{(2,\,r-1)}}{\Omega_{(1,\,r-1)}}+\frac{m_{0}\Lambda_{r}^{4}\Omega_{(1,\,r-2)}}{\Omega_{(1,\,r-1)}}\pm2\frac{m_{0}(\prod_{i=1}^{r}\Lambda_{i}^{4})^{1/2}}{\Omega_{(1,\,r-1)}}-\frac{b^{2}}{4m_{0}}\Omega_{(1,\,r-1)}-m_{0}M_{0}-b\,U_{(0,\,r-1)}.
V(\varphi_{0},\beta)=\frac{2}{\beta}\sum_{s=1}^{\infty}\sum_{n=-\infty}^{+\infty}\int\frac{d^{d}p}{(2\pi)^{d}}\frac{(-1)^{s}}{s}\frac{(M+g\varphi_{0})^{2s}}{(\omega_{n}^{2}+q^{2})^{s}}.
\beta(g)~=~(\mu-2)g~+~\frac{Ng^{2}}{2}~+~g^{2}\beta_{1}\!\left(\frac{gN}{2}\right)~+~\frac{g^{2}}{N}\beta_{2}\!\left(\frac{gN}{2}\right)~+~O\left(\frac{1}{N^{3}}\right)
A_{i}^{\prime}=e^{-iga_{0}x_{0}\tau_{3}}\Omega_{\mathrm{D}}A_{i}^{(1)}\Omega_{\mathrm{D}}^{\dagger}e^{iga_{0}x_{0}\tau_{3}}+e^{-iga_{0}x_{0}\tau_{3}}s_{i}e^{iga_{0}x_{0}\tau_{3}}+\partial_{i}a_{0}x_{0}\tau_{3},
+\left.{\frac{1}{60}}F_{\mu\rho}~^{||\rho}F^{\mu\lambda}~_{||\lambda}-{\frac{1}{72}}(F^{\mu\nu}F_{\mu\nu})\bar{R}-{\frac{1}{180}}\bar{R}^{\lambda\nu\mu\rho}F_{\lambda\nu}F_{\mu\rho}-{\frac{1}{90}}\bar{R}_{\mu\nu}F^{\rho\mu}F_{\rho}~^{\nu}\right)~~,
\displaystyle\frac{1}{(q)_{\infty}}\sum_{r=-\infty}^{\infty}(-1)^{r}q^{r((2k+1)r+1-2jk)/2}=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k-1}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+j(n_{1}+\cdots+n_{k-1})}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}.
G(x,x^{\prime})=\frac1{(2\pi)^{3}}\sum_{n=-\infty}^{\infty}e^{in(\theta-\theta^{\prime})}\int_{-\infty}^{\infty}dke^{ik(z-z^{\prime})}\int_{-\infty}^{\infty}d\omegae^{i\omega(\tau-\tau^{\prime})}g_{n}(r,r^{\prime})\.
S_{0}=\frac{1}{2}\int_{0}^{1}d\tau[~\frac{\dot{x}^{2}}{e}+em^{2}+\chi\psi_{a}\dot{x}^{a}/e-\psi_{a}\dot{\psi}^{a}+\psi_{S}\dot{\psi_{S}}-m\psi_{S}\chi~],
D^{\mu_{1}\cdot\cdot\cdot\mu_{k}}(\partialn)=\frac1{m!}\epsilon^{\mu_{1}\cdot\cdot\cdot\mu_{k}\mu_{k+1}\cdot\cdot\cdot\mu_{n}}\epsilon_{a_{1}\cdot\cdot\cdota_{m}}\partial_{\mu_{k+1}}n^{a_{1}}\cdot\cdot\cdot\partial_{\mu_{n}}n^{a_{m}}
\left\{\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{12}}\\{j_{3}}&{j}&{j_{23}}\\\end{array}\right\}^{S}=\sum_{j_{13}}(-1)^{[j_{12}+j_{23}+j_{13}]+2j_{13}I_{13}}\left\{\begin{array}{ccc}{j_{1}}&{j_{3}}&{j_{13}}\\{j_{2}}&{j}&{j_{23}}\\\end{array}\right\}^{S}~\left\{\begin{array}{ccc}{j_{2}}&{j_{1}}&{j_{12}}\\{j_{3}}&{j}&{j_{13}}\\\end{array}\right\}^{S}\,.
\left(\frac{d\xi_{0F}}{d\xi_{i}}\right)^{-1}\sum_{m=-1}^{\infty}e_{n}^{pE}E_{nm}\left(V_{0E}^{-1}V_{0F}\right)(\xi_{0F})^{m+1}=\left(\frac{d\xi_{0j}}{d\xi_{i}}\right)^{-1}\sum_{m=-1}^{\infty}e_{n}^{pE}E_{nm}\left(V_{0E}^{-1}V_{0j}\right)(\xi_{0j})^{m+1}\.
Q^{l_{1}l_{2}l_{3}}={\frac{(-{\frac{1}{2}}+l_{1}+l_{3})!(-{\frac{1}{2}}+l_{2}+l_{3})!}{(l_{1}+l_{3}+{\frac{3}{2}})!(l_{2}+l_{3}+{\frac{3}{2}})!({\frac{1}{2}}+l_{1}-l_{3})!({\frac{1}{2}}+l_{2}-l_{3})!({\frac{1}{2}}+l_{3}-l_{1})!({\frac{1}{2}}+l_{3}-l_{2})!}}
\sum_{\lambda}\epsilon^{\mu}(\lambda)\epsilon^{*\nu}(\lambda)=-g_{\perp}^{\mu\nu}+{\frac{1}{q^{+}}}\bigl(\eta^{\mu}q_{\perp}^{\nu}+\eta^{\nu}q_{\perp}^{\mu}\bigr)+{\frac{{\bfq}_{\perp}^{2}}{(q^{+})^{2}}}\eta^{\mu}\eta^{\nu}\;,
T_{12}\,(\,p_{120}^{\prime},\,p_{120},\,P_{0}-p_{30})\,\to\,\,T_{12}\,(\,p_{120}^{\prime},\,s_{12}\,\mp\,{\frac{P_{0}-S}{2}}\,\pm\,{\frac{p_{30}-h_{3}}{2}},\,P_{0}-p_{30})\,
\begin{array}{c}{R(v/u)(K_{l}^{t}(u)\otimes{\bf1})R(u^{-1}v^{-1}q^{-1})({\bf1}\otimesK_{r}^{t}(v))=}\\{({\bf1}\otimesK_{r}^{t}(v))R(u^{-1}v^{-1}q^{-1})(K_{r}^{t}(u)\otimes{\bf1})R(v/u)}\\\end{array}
{{M_{T}}_{\rho\sigma}}^{\alpha\beta}=\delta_{\rho\sigma}^{\alpha\beta}(-\Box)+2\left(1-\frac{1}{\alpha}\right)\delta_{(\rho}^{(\alpha}\nabla_{\sigma)}\nabla^{\beta)}-\left(1-\frac{1}{\alpha}\right)g^{\alpha\beta}\nabla_{(\rho}\nabla_{\sigma)}-{P_{\rho\sigma}}^{\alpha\beta}\,
W_{i_{1}j}^{(1)}(x,y)=\left(\begin{array}{cc}{\widetilde{W}_{ij}^{(0)}}\\\hline{}\\{\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{\kappa}&{-{\sqrt\kappa}}&{0}\\{0}&{-\kappa}&{0}&{0}&{-{\sqrt\kappa}}\\\end{array}}\\\end{array}\right)\delta^{2}(x-y),
\displaystyleV_{\mathrm{eff}}(\Phi)=\displaystyle\frac{m^{2}}{2}\Phi^{2}+\frac{\lambda}{4!}\Phi^{4}-\frac{1}{12\pi}\left(M_{H}(\Phi)^{3}+M_{W}(\Phi)^{3}+M_{A\phi}(\Phi)^{3}-2m^{3}\right).
E_{0}(\rho_{1},n_{1})={\frac{n_{1}^{2}}{\2\rho_{1}}}{\frac{\Gamma(2-{\frac{1}{2n_{1}}})}{\Gamma({\frac{1}{2n_{1}}})}}+\rho_{1}^{2}{\frac{\Gamma({\frac{5}{2n_{1}}})}{\Gamma({\frac{1}{2n_{1}}})}}.
g_{AB}=\left(\begin{array}{ccc}{-n^{2}(1+2A)}&{a^{2}B_{,i}}&{nA_{y}\nonumber}\\{a^{2}B_{,j}}&{a^{2}\left[(1+2{\cal{R}})\delta_{ij}+2E_{,ij}\right]}&{a^{2}B_{y,i}\nonumber}\\{nA_{y}}&{a^{2}B_{y,i}}&{b^{2}(1+2A_{yy})\nonumber}\\\end{array}\right)\,.
\Psi_{E}(r,\theta;\nu)=\sqrt{{\frac{M}{2\pi\hbar^{2}}}}\left\{\sum_{j=-\infty}^{[\nu]}e^{-i\pi(\nu+j)/2}\,J_{\nu-j}(kr)\,e^{ij\theta}+\sum_{j=[\nu]+1}^{\infty}e^{i\pi(\nu+j)/2}\,J_{j-\nu}(kr)\,e^{ij\theta}\right\}.
\Gamma^{(PV)}(A)=\frac{1}{2}\left.\frac{d}{d~s}(\zeta(s)-\zeta^{P}(s))\right|_{s=0}=i\frac{\pi}{2}\eta(0)-i\sum_{k=0}^{\infty}(-1)^{k}\frac{m^{(2k+1)}}{2k+1}\eta(2k+1),
\left(\theta_{m,2k}\pm\theta_{2k-m,2k}\right)\;\left(\theta_{m^{\prime},2k}\pm\theta_{2k-m^{\prime},2k}\right)\:(\tau)=\theta_{\frac{m-m^{\prime}}{2},k}\;\theta_{\frac{m+m^{\prime}}{2},k}\:(\tau)\:\pm\:\theta_{k-\frac{m+m^{\prime}}{2},k}\;\theta_{k-\frac{m-m^{\prime}}{2},k}\:(\tau).
F^{{\mu}{\nu}{\sigma}}=-G^{{\mu}{\nu}{\sigma}}+{\lambda}^{{\rho}[{\mu}}{{\calG}_{\rho}}^{{\nu}{\sigma}]}+{\frac{1}{3!}}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\lambda}_{{\theta}[{\rho}}{{\calG}^{\theta}}_{{\eta}{\delta}]}.
\frac{d^{2}E}{d\omegad\Omega}=2{\calG}\frac{\omega^{D-2}}{(2\pi)^{D-4}}\,\Lambda_{ij,\,lm}(\hat{k})\,T^{*\,ij}(\omega,{\bfk})\,T^{\,ij}(\omega,{\bfk})\,,
\kappa(u)=u^{-(n-2)/n}\frac{(u^{2};t^{2})_{\infty}(t^{2}u^{-2};t^{2})_{\infty}}{(q^{2};t^{2})_{\infty}(t^{2}q^{-2};t^{2})_{\infty}}\bar{\kappa}(u),
\begin{array}{lcl}{\mathrm{SU}_{V}(4)\otimes\mathrm{SU}_{S}(4/1)}&{\longrightarrow}&{~\mathrm{Sp}_{V}(4)\otimes\mathrm{SU}_{S}(2)\otimes\mathrm{SU}_{S}(2/1)}\\{}&{\longrightarrow}&{~[\mathrm{SU}_{V}(2)]^{2}\otimes[\mathrm{SU}_{S}(2)]^{2}}\\{}&{\longrightarrow}&{~[\mathrm{SU}_{V+S}(2)]^{2}~\approx~\mathrm{SO}_{V+S}(4).}\\\end{array}
(M-\overline{{M}})\:T_{[0]}^{(-1)}\:T_{[0]}^{(0)}\:\overline{{T_{[0]}^{(-1)}}}\;=\;0\;=\;(N-\overline{{N}})\:T_{[0]}^{(-1)}\:T_{[0]}^{(0)}\:\overline{{T_{[0]}^{(-1)}}}
\pi^{1(\underline{\alpha})}=\left(\pi_{q}^{1+}\atop\pi_{\dot{q}}^{1-}\right)=\left(\nabla\theta_{q}^{1+}-\frac12\gamma_{q\dot{r}}^{i}\theta_{\dot{r}}^{1-}\Omega^{+2i}\atop\nabla\theta_{\dot{q}}^{1-}-\frac12\tilde{\gamma}_{\dot{q}r}^{i}\theta_{r}^{1+}\Omega^{-2i}\right),
U(g_{1})|\bbox{w}\pm\rangle\simeq(\alpha_{\bbox{v}}^{\pm}(g_{\bbox{w}^{\prime}}^{-1}g_{1}g_{\bbox{w}}))^{-1/2}|_{\bbox{v}=\bbox{0}}|{\bbox{w}}_{g_{1}}\pm\rangle\qquadg\in{\calG}.
\Omega_{\alpha\beta}^{(0)}=\left(\begin{array}{cccc}{0}&{-m\delta_{ij}}&{e\delta_{ij}\delta(\vec{y}-\vec{q})}&{0}\\{m\delta_{ij}}&{0}&{0}&{0}\\{-e\delta_{ij}\delta(\vec{y}-\vec{q})}&{0}&{2\theta\epsilon_{ij}\delta(\vec{x}-\vec{y})}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right)
\overline{u}(p^{\prime})\Gamma_{\mathrm{a}}^{a,\mu}u(p)=\frac{g^{3}}{\Theta}[T^{b}T^{a}T_{b}]\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\varepsilon_{\rho\sigma\lambda}{\bar{k}}^{\lambda}\bar{u}(p^{\prime})[\gamma^{\sigma}(\not\!p^{\prime}-\not\!k+m)\gamma^{\mu}(\not\!p-\not\!k+m)\gamma^{\rho}]u(p)}{[(p-k)^{2}-m^{2}+i\epsilon][(p^{\prime}-k)^{2}-m^{2}+i\epsilon][-{\bfk}^{2}]}.
S[\psi]=-\intd^{2}x\left\{\frac{1}{2}\sum_{j=1}^{N}\bar{\psi}^{j}\gamma\cdot\partial\psi^{j}+\frac{\lambda}{8N}\sum_{ij=1}^{N}\bar{\psi}^{i}*\psi^{i}*\bar{\psi}^{j}*\psi^{j}\right\}
T_{M_{2}}l\intd^{2}\xi(2\pi\alpha^{\prime})\epsilon^{ij}\partial_{i}\varrho\\left({\tilde{\calG}}_{j}^{(1)}-{\textstyle\frac{1}{2\pi\alpha^{\prime}}}{\tilde{A}}^{(1)}{}_{j}+{\textstyle\frac{m}{2}}b_{j}\right)\,.
f_{(\eta)}^{\vec{\alpha}\beta\gamma}=f^{\alpha[1]\betab[1]}\,\,f^{b[1]\alpha[2]b[2]}\,f^{b[2]\alpha[3]b[3]}\,\cdots\,\,f^{b[\eta-2]\alpha[\eta-1]b[\eta-1]}f^{b[\eta-1]\alpha[\eta]\gamma}\;,
\gamma_{ij}(\sigma)=G_{\mu\nu}\frac{\partialX^{\mu}}{\partial\sigma^{i}}\frac{\partialX^{\nu}}{\partial\sigma^{i}}=G_{\mu\nu}^{\prime}\frac{\partialX^{\mu}}{\partial\sigma^{i}}\frac{\partialX^{\nu}}{\partial\sigma^{i}}.
dS^{2}={H_{7}}^{-1/2}[dudv-\mu^{2}\sum_{i=1}^{4}{z^{i}}^{2}du^{2}+\sum_{i=1}^{4}dz^{i}dz^{i}+d\tilde{x}^{2}+d\tilde{y}^{2}]+{H_{7}}^{1/2}(dr^{2}+r^{2}d{\theta}^{2}).
ds^{2}=(d\rho_{1}^{2}+\rho_{1}^{2}d\psi_{1}^{2})+(d\rho_{2}^{2}+\rho_{2}^{2}(d\bar{\psi}_{2}+d\psi_{1})^{2})+d\bar{s}_{d-4}^{2}.
{\mathbbA}^{4\vert2N-2,2N}={\frac{\mathrm{SU}(2,2/N)}{\{K,S,\bar{S},M,D,T,R,Q^{1}\}}}=(x^{\mu},\theta_{2}^{\alpha},\ldots,\theta_{N}^{\alpha},\bar{\theta}_{\dot{\alpha}}^{1},\ldots,\bar{\theta}_{\dot{\alpha}}^{N})\;.
L_{\mathrm{g}}(q^{r},\dot{q}^{r},v,\mu^{r})=L_{\mathrm{q}}(q^{r},v)+\mu_{a}^{s_{a}}(\dot{q}_{a}^{s_{a}}-q_{a}^{s_{a}+1})+\mu_{a}^{n_{a}}(\dot{q}_{a}^{n_{a}}-v_{a}).
{\calP}_{x^{-}}e^{i\int_{-L}^{L}dy^{-}\big[A_{-}(x^{+},y^{-},{\vec{x}}_{\perp})-i{\vec{\alpha}}_{\perp}\big({\vec{\partial}}_{\perp}+i{\vec{A}}_{\perp}(x^{+},y^{-},{\vec{x}}_{\perp})\big)\big]}=\pm1\,\,,
S_{WZ}^{1}=\int\!\mathrm{d}^{2}\sigma\,\Big[-i\epsilon^{ij}e_{i}{}^{r}\big(\bar{\theta}_{1}\Gamma_{r}\hat{D}_{j}\theta_{1}-\bar{\theta}^{2}\Gamma_{r}\hat{D}_{j}\theta^{2}\big)+\epsilon^{ij}\big(\bar{\theta}_{1}\Gamma^{r}\hat{D}_{i}\theta_{1}\big)\big(\bar{\theta}_{2}\Gamma_{r}\hat{D}_{j}\theta_{2}\big)\Big]\,,
\begin{array}{llll}{a(\gamma(u))\;\:=}&{\oint_{\gamma_{1^{\prime}}}\lambda^{\gamma}(u)\;=}&{-\left(\frac{2}{1-u}\right)^{1/2}\left(a_{D}(u)+\frac{u+1}{\sqrt2\pi}b_{2}(u)\right)\equiv}&{a_{D}^{\gamma}(u)}\\{a_{D}(\gamma(u))=}&{\oint_{\gamma_{2^{\prime}}}\lambda^{\gamma}(u)=}&{-\left(\frac{2}{1-u}\right)^{1/2}\left(a(u)+\frac{u+1}{\sqrt2\pi}b_{1}(u)\right)\;\;\equiv}&{a^{\gamma}(u)}\\\end{array}
\langleT(x)T(0)\rangle=\frac{N^{2}}{\lambda^{(2-\alpha)}}\frac{1}{|x|^{2\alpha+p+1}}=\frac{N^{\alpha}}{g_{YM}^{2(2-\alpha)}}\frac{1}{|x|^{2\alpha+p+1}}\,~~~~~~~\alpha=\frac{7-p}{5-p}\.
\eta^{2}(\eta^{2}+1){\frac{B^{\prime\prime}}{B}}+2\eta\left(\eta^{2}+{\frac{2}{2-\gamma}}\right){\frac{B^{\prime}}{B}}+{\frac{\gamma}{2-\gamma}}\left[\eta^{2}(\eta^{2}+1)\left({\frac{B^{\prime}}{B}}\right)^{2}+1\right]=0,
\bar{D}^{\dot{\alpha}}J_{\alpha\dot{\alpha}}-\frac{N_{f}}{48\pi^{2}}\left(1-\frac{3N_{c}}{N_{f}}-\gamma(g)\right)D_{\alpha}W^{2}=0,{}~~~~\bar{D}^{2}K-\frac{N_{f}}{2\pi^{2}}W^{2}=0,
{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[{B}_{i}^{v^{\prime}(2)}\right]=-\frac{1}{2}{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[\frac{\partial}{\partialv^{i}}[{\Gamma^{\prime}}^{(2)}]\right]+{K}_{\hat{0}\alpha}^{v^{\prime}},
A=\frac{\gamma^{2}(-\varphi^{2}+\sigma^{2})+2\gamma\sigma+1}{\gamma^{2}(\varphi^{2}+\sigma^{2})+2\gamma\sigma+1},\;\;\;\;\;B=\frac{2\gamma^{2}(\varphi\sigma)+2\gamma\varphi}{\gamma^{2}(\varphi^{2}+\sigma^{2})+2\gamma\sigma+1}.
I\left[A_{\alpha}^{\;\;(\lambda)}\right]=\varepsilon^{\alpha\beta\gamma}\intd^{3}xA_{\alpha(\lambda)}\left(\partial_{\beta}A_{\gamma}^{\;\;(\lambda)}+\frac{g}{3}\varepsilon^{\lambda\mu\nu}A_{\beta(\mu)}A_{\gamma(\nu)}\right),
\left<T_{\mu\nu}(x)\right>_{\mu^{2}}^{{\mathbbH}^{9}}=\frac{m^{2}}{30240\pi^{4}a^{7}}\left(36-49a^{2}b+14a^{4}b^{2}-a^{6}b^{3}\right)a\sqrtb\,\,g_{\mu\nu}^{{\mathbbH}^{9}}(x).
\Big\langleJ_{\mu}^{(l)}(x)J_{\nu}^{(l^{\prime})}(y)\Big\rangle=\delta_{ll^{\prime}}\;\;\frac{1}{\pi}\varepsilon_{\mu\rho}\partialx_{\rho}\,\varepsilon_{\nu\sigma}\partialy_{\sigma}\;\Big\langle\varphi^{(l)}(x)\varphi^{(l)}(y)\Big\rangle_{B}\;,
\left(C_{\mathbf{N}\,\,K}^{\mathbf{P}}\,\,\,\right)^{2}\,\propto\gamma_{i}^{\left(D\right)}\left(a,b,c\right)\mathcal{N}_{i}\left(a,b,c\right)\mathcal{N}_{n+1-i}\left(\overline{{a}},b,c\right).
\langle:\!\!\phi^{2}(0)\!\!:\rangle\simeq\sum_{n,n_{1}}\prod_{i=1}^{n}\sum_{\underline{{m}}_{i}}\,^{\prime}\prod_{j=1}^{n_{1}}\sum_{\underline{{l}}_{j}}\,^{\prime}\sum_{k=1}^{n}\frac{2K}{m_{k}}\left|\psi^{(n,n_{1})}\right|^{2}\,.
\tilde{D}_{T}^{[r,0]}\left(x_{M}^{2}\right)=\frac{i\Lambda^{2}}{(2\pi)^{2}}\sum_{s=0}^{(r-1)/2}\,2Re\left\{\left[\varrho_{r,4s+2}\sigma_{r,4s+3}\right]\frac{K_{1}\left(i\sqrt{\sigma_{r,4s+3}}\Lambda\sqrt{x_{M}^{2}}\right)}{i\sqrt{\sigma_{r,4s+3}}\Lambda\sqrt{x_{M}^{2}}}\right\}\,,
\left(1-z^{\prime}\right)^{-\frac{n^{\prime}}{(e^{\prime}{\calP}_{L})}(e^{\prime}Q)}\left(1-zz^{\prime}\right)^{-\frac{4nn^{\prime}}{(e{\calP}_{L})(e^{\prime}{\calP}_{L})}(ee^{\prime})}\left[U^{i}(z)V^{i^{\prime}}(z^{\prime})+\delta^{ii^{\prime}}\frac{zz^{\prime}}{(1-zz^{\prime})^{2}}\right]
q_{-}^{0}(\tau)=-\frac{\Omega\vartheta_{1}^{\prime}(0)}{2\pi}~\frac{e^{\Omega\tau\frac{\vartheta_{1}^{\prime}}{\vartheta_{1}}(\Omegay)}\vartheta_{1}(\Omega(y-\tau))+e^{-\Omega\tau\frac{\vartheta_{1}^{\prime}}{\vartheta_{1}}(\Omegay)}\vartheta_{1}(\Omega(y+\tau))}{\vartheta_{1}(\Omega\tau)\vartheta_{1}(\Omegay)},
M(s,t,u)=\sum_{i=0}^{\infty}{\alpha}_{i}^{(t)}(t){\nu}_{t}^{i}+\sum_{R_{u}}^{}\frac{\rho^{(tu)}(t)}{(\sigma-2M_{R}^{2})-t+{\nu}_{t}}+\sum_{R_{s}}^{}\frac{\rho^{(ts)}(t)}{(\sigma-2M_{R}^{2})-t-{\nu}_{t}}\;
\partial_{\zeta}^{2}(\zeta^{\frac{1}{2}}A_{0})+Q(\zeta)(\zeta^{\frac{1}{2}}A_{0})=\frac{1}{\zeta^{2}}(\zeta^{\frac{1}{2}}B)-\frac{2}{\zeta}\partial_{\zeta}(\zeta^{\frac{1}{2}}B)
P\circ\Bigl(\bigotimes_{s\in\Lambda^{(4)}}Z_{s}^{(V),(\epsilon_{s})}\Bigr)=\Phi\,\Bigl[P\circ\Bigl(\bigotimes_{s\in\Lambda^{(4)}}Z_{\tau(s)}^{(V),(\epsilon_{\tau(s)})}\Bigr)\Bigr]\,\Phi^{-1}.
{a^{n}\Omega_{n+1}}\frac{a^{2}K^{2}}{[(\frac{n}{2}-1)\frac{n}{2}-(N+\frac{n}{2}-1)(N+\frac{n}{2})]^{2}}[C_{N}^{\frac{n-1}{2}}(cos{\theta_{n}^{0}})]^{2}
({\cal{D}}_{\mu}\phi)^{\prime}=(\partial_{\mu}U)U^{-1}\phi^{\prime}+U(\partial_{\mu}\phi)V+\phi^{\prime}V^{-1}(\partial_{\mu}V)+{\tilde{W}}_{\mu}^{\prime}\phi^{\prime}+{\tilde{B}}_{\mu}^{\prime}\phi^{\prime}
\mathrm{I=\int\frac{dq_{0}}{2\pi}\int\frac{dk_{0}}{2\pi}\,\frac{k_{0}}{(k_{0}^{2}-\vec{k}^{2}+i\epsilon)}\,\frac{q_{0}}{(q_{0}^{2}-\vec{q}^{2}+i\epsilon)}\,,\qquad\epsilon>0,}
\nu_{\alpha\beta}^{g_{1}g_{2}}\:=\:\nu_{\alpha\beta}^{g_{2}}\,g_{2}^{*}\nu_{\alpha\beta}^{g_{1}}\,\left(h_{\alpha}^{g_{1},g_{2}}\right)\,\left(h_{\beta}^{g_{1},g_{2}}\right)^{-1}
P_{j,n,s}^{SPF~j^{\prime},n^{\prime},s^{\prime}}=P_{j}^{~j^{\prime}}P_{n}^{~n^{\prime}}P_{s}^{~s^{\prime}}+(-1)^{(2j-n+s)/2}P_{j}^{~k/2-j^{\prime}}P_{n}^{~n^{\prime}+k+2}P_{s}^{~s^{\prime}+2}~,
c_{12}=\frac{4}{\pi}(\sum_{\ell=2}^{4}|b_{\ell}|)\int_{0}^{\infty}d\lambda[\frac{1}{[(\frac{3}{4})^{2}+\lambda^{2}]}+(\frac{5}{3})\frac{1}{[(\frac{5}{4})^{2}+\lambda^{2}]}]\cong112.
S^{\upsilon}=\int\left[p_{\mu}\dot{A}^{\mu}-\mathcal{H}^{\upsilon}\right]dx\,,\;\mathcal{H}^{\upsilon}=p_{\mu}\upsilon^{\mu}-\frac{1}{2}\left[\left(\upsilon^{i}+\partial_{i}A_{0}\right)^{2}-\frac{1}{2}F_{ik}F^{ik}\right]\,.
S_{inv}^{(0)}[\theta]\;=\;\intd^{3}x\,\frac{1}{2}{\tilde{f}}_{\mu}^{a}(\theta){\tilde{f}}_{\mu}^{a}(\theta)\;=\;\intd^{3}x\;\frac{1}{4}f_{\mu\nu}^{a}(\theta)f_{\mu\nu}^{a}(\theta)
{\calA}=\pii\gamma\sum_{r=1}^{N}\left[\left(2R^{2}\frac{\bar{z}_{r}}{1+|z_{r}|^{2}}+\tilde{b}_{r}\right)dz_{r}-\left(2R^{2}\frac{{z}_{r}}{1+|z_{r}|^{2}}+\bar{\tilde{b}}_{r}\right)d\bar{z}_{r}\right].
\mathcal{D}_{A_{+},A_{-}}\Phi=\left(\begin{matrix}{-D_{4}+iD_{3}}&{D_{2}+iD_{1}}\\{-D_{2}+iD_{1}}&{-D_{4}-iD_{3}}\\\end{matrix}\right)\left(\begin{matrix}{\phi_{1}}\\{\phi_{2}}\\\end{matrix}\right),
M^{2}=\frac{2}{\kappa_{1}}\frac{d^{2}V(\varphi)}{d\varphi^{2}}\large|_{V=0}=\frac{2}{\kappa_{1}}\frac{1}{\sqrt{V}}{\frac{d^{2}\sqrt{V}}{dT^{2}}}\large|_{V=0}\.
\begin{array}{c}{\Pi_{j}^{i}=(Z_{j},q^{i})+i\delta_{j}^{i}l_{i}\,\quad\bar{\Pi}_{j}^{i}=(\Pi_{j}^{i})^{*}\,\quadi\geqj\,}\\{\Pi=(\psi_{a}\pi^{a})+il_{r}\,\quad\bar{\Pi}=(\Pi)^{*}\,}\\{l_{i}=\sum_{k=i}^{r-1}s_{k}\,\quadl_{r}=2s_{r}}\\\end{array}
P_{a}=\partial_{a},\qquadJ_{ab}=x_{a}\partial_{b}-x_{b}\partial_{a},\qquadS_{\alpha}^{(k)a\cdotsd}=\underbrace{x^{a}\cdotsx^{d}}_{{k-\mathrm{times}}}v_{\alpha},\quadk=0,1,2,\ldots\quad,
{\frac{\partial^{2}\Omega}{\partialx\partial{\tilde{x}}}}{\frac{\partial^{2}\Omega}{\partialy\partial{\tilde{y}}}}-{\frac{\partial^{2}\Omega}{\partialx\partial{\tilde{y}}}}{\frac{\partial^{2}\Omega}{\partialy\partial{\tilde{x}}}}=1\,.
S=\intd^{4}x\sqrt{-g}\left\{\mathrm{e}^{-\Phi}\left[R+\left(\nabla\Phi\right)^{2}-6\left(\nabla\beta\right)^{2}-\frac12\mathrm{e}^{2\Phi}\left(\nabla\sigma\right)^{2}-2\Lambda\right]-\frac{1}{2}Q^{2}\mathrm{e}^{c\beta}\right\},
S=\left({\frac{H_{T}H_{U}}{H_{S}^{2}}}\right)^{1/3}\,,\qquadT=\left({\frac{H_{S}H_{U}}{H_{T}^{2}}}\right)^{1/3}\,,\qquadU=\left({\frac{H_{T}H_{S}}{H_{U}^{2}}}\right)^{1/3}\,.
\zeta(\nu|\beta)={\frac{\varrho^{-2\nu}}{2\pii}}\int_{\mu}^{\infty}d\omega\int_{C}dz\sum_{\sigma_{l}}{\frac{\breve{\Phi}(\omega;z)}{z-\sigma_{l}}}(z^{2}+\omega^{2})^{-\nu}~~~,
S[q^{n}]=\intdt\,L(q^{n},\dot{q}^{n})\\\,\\\L\left(q^{n},\dot{q}^{n}\right)=\dot{q}^{n}p_{n}(q^{n},\dot{q}^{n})-H\left(q^{n},p_{n}(q^{n},\dot{q}^{n})\right)\,
\deltaA_{L}=K_{1}\int\phi_{0}^{2}\left(\int\frac{4A_{0}^{\prime}}{\phi_{0}^{2}}e^{-4A_{0}}\,dy\right)dy+A_{0}^{\prime}k_{2}(t)-K_{1}\inte^{-4A_{0}}\,dy+k_{3}(t)\;.
\{{\calM}{}_{,}^{\otimes}v(y)\}=\Bigl({\calM}T_{\rho}\;{\calG}_{\rho\sigma}(\ell,y)\;-\;T_{\rho}{\calM}\;{\calG}_{\rho\sigma}(0,y)\Bigr)\otimesT_{\sigma}v(y)\quad,
h=\intd^{2}x_{\bot}\;\left[-\frac{g^{2}L}{8}\frac{1}{J(\varphi({x}_{\bot}))}\frac{\delta}{\delta\varphi({x}_{\bot})}J(\varphi({x}_{\bot}))\frac{\delta}{\delta\varphi({x}_{\bot})}+\frac{2}{\eta^{2}g^{2}L}(\nabla_{\bot}\varphi({x}_{\bot}))^{2}\right]\;,
\left(\begin{matrix}{\displaystyle\frac{\partial^{2}L_{\mathrm{q}}}{\partialv_{a}\partialv_{b}}}&{\displaystyle\frac{\partial\omega_{A}}{\partialv_{a}}}\\\end{matrix}\right)=\frac{\partial}{\partialv_{a}}\left(\begin{matrix}{\gamma_{b}}&{\omega_{A}}\\\end{matrix}\right),
\chi=\S_{\begin{array}{l}{{\scriptstylen_{1},...,n_{N}}=0}\\{{\scriptstylem_{1},...,m_{M}=0}}\\{{\scriptstyleq_{1},...,q_{Q}=0}}\\{{\scriptstylet_{1},...,t_{T}=0}}\\\end{array}}^{1}c^{\prime}({\{n_{r}\}}_{1}^{N};{\{m_{r}\}}_{1}^{M};{\{q_{r}\}}_{1}^{Q};{\{t_{r}\}}_{1}^{T})\chi({\{n_{r}\}}_{1}^{N};{\{m_{r}\}}_{1}^{M};{\{q_{r}\}}_{1}^{Q};{\{t_{r}\}}_{1}^{T};{\{\nu_{r}\}}_{1}^{4}),
\begin{array}{c}{\{\Theta_{+},\Theta_{-}\}={\tilde{\calH}},\quad\{\Theta_{\pm},\Theta_{\pm}\}=\pmi{\calZ}}\\{\{{\calZ},\Theta_{\pm}\}=0,\;\;-\{{\tilde{\calH}},\Theta_{\mp}\}=0,\;\;\{{\calZ},{\tilde{\calH}}\}=0.}\\\end{array}
C=-\frac{4i}{g}\frac{1}{I_{2}}[B,F]+\Omega\Bigl(\frac{1}{2}a^{1}\lambda_{1}+\frac{1}{2}a^{2}\lambda_{2}+\frac{1}{2}a^{3}\lambda_{3}+\frac{1}{2}a^{8}\lambda_{8}\Bigr)\Omega^{\dagger}.
=(-1)^{l_{1}+1}c_{0}\sqrt{\frac{1+(-1)^{l_{1}+l_{0}}}{2(l_{1}+l_{0}+1)(l_{1}-l_{0}-1)}+\frac{1-(-1)^{l_{1}+l_{0}}}{2(l_{1}+l_{0})(l_{1}-l_{0})}},
{\calP}=\prod_{i=1}^{s_{-}}\prod_{n=1}^{k-2}(w_{i}^{(n)})^{m_{1}-j_{1}}(1-w_{i}^{(n)})^{m_{2}-j_{2}}\prod_{i<j}(w_{i}^{(n)}-w_{j}^{(m)}).
A_{\nu\rho\sigma}={\frac{g^{2}}{96\pi^{2}}}[-A_{\nu}^{a}\overleftarrow{\partial}_{\rho}A_{\sigma}^{a}+A_{\nu}^{a}\partial_{\rho}A_{\sigma}^{a}+A_{\rho}^{a}\overleftarrow{\partial}_{\nu}A_{\sigma}^{a}-A_{\rho}^{a}\partial_{\nu}A_{\sigma}^{a}+A_{\nu}^{a}\overleftarrow{\partial}_{\sigma}A_{\rho}^{a}-A_{\nu}^{a}\partial_{\sigma}A_{\rho}^{a}+2f_{abc}A_{\nu}^{a}A_{\rho}^{b}A_{\sigma}^{c}]
Z_{SG}=Z_{BE}\sum_{n=0}^{\infty}\frac{1}{n!^{2}}(\frac{\alpha_{0}}{\lambda^{2}})^{2n}\left\langle\prod_{j=1}^{n}\inte^{i\lambda\varphi(x_{j})}e^{-i\lambda\varphi(y_{j})}d^{2}x_{j}d^{2}y_{j}\right\rangle,
\chi_{F}^{(2k+1)}\left(\psi_{i}^{(-k)},\psi_{j}^{(-k)}\right)={\frac{1}{24(2k+1)\pii}}\oint_{\calC}\psi_{i}^{(-k)}{\calS}_{J_{H}^{-1}}^{(2k+1)}\psi_{j}^{(-k)},\qquadk=0,1,2,\ldots,
\frac{\Theta_{12}}{D_{1}}+\frac{\Theta_{21}}{D_{2}}=\frac{1}{2}\left(\frac{1}{D_{1}}+\frac{1}{D_{2}}\right)+\frac{1}{2}\left(\frac{1}{D_{1}}-\frac{1}{D_{2}}\right)\left(\Theta_{12}-\Theta_{21}\right)\,.
G_{\mu\nu\rho\sigma}(p)=\int\!dp_{1}\int\!dp_{2}\sum_{s_{1}}\sum_{s_{2}}\\delta(p-p_{1}-p_{2})\langle0\vertT_{\mu\nu}^{V}(0)\vertp_{1},s_{1};p_{2},s_{2}\rangle\langlep_{1},s_{1};p_{2},s_{2}\vertT_{\rho\sigma}^{V}(0)\vert0\rangle.
S=\frac{1}{16\piG_{5}}\intd^{5}x\sqrt{g}\lbraceR-\frac{1}{3}(\partial\phi)^{2}-\frac{1}{12}e^{-\frac{4\phi}{3}}H^{2}-\frac{1}{4}e^{-\frac{2\phi}{3}}F^{2}\rbrace
(e_{i}^{g})^{2}\;=\;g_{ab}t^{a}t^{b}(dt)_{|\partialM_{i}}^{2}\;\;\;,\;\;\;\varphi_{i}\;=\;\varphi_{|\partialM_{i}}\;\;\;,\;\;\;x_{i}^{\mu}\;=\;x_{|\partialM_{i}}^{\mu}\;\;\;.
Re\,{\calI}_{\xi}^{(\beta)}(X)\simeq\frac{1}{12}\,c_{\xi}(X)\,{\calT}_{\xi}^{2}(X)-\frac{1}{4\pi}\,\theta\left({\calM}_{\xi}^{2}(X)\right)\left({\calM}_{\xi}^{2}(X)\right)^{1/2}{\calT}_{\xi}(X).
S[A_{\mu},h]=-\frac{1}{2g^{2}}\intdx^{3}\,\mathrm{tr}[F_{\mu\nu}\,F^{\mu\nu}]+\intdx^{3}\left(\mathrm{tr}(D_{\mu}h)^{2}+\frac{\lambda}{4}(2\mathrm{tr}h^{\dagger}h-v^{2})^{2}\right)\,\,\,,
\begin{array}{lcl}{\{p,f\}_{\theta}}&{=}&{f^{\prime}}\\{\{p^{2},f\}_{\theta}}&{=}&{2f^{\prime}p}\\{\{p^{3},f\}_{\theta}}&{=}&{3f^{\prime}p^{2}+{\theta}^{2}f^{\prime\prime\prime}}\\\end{array}
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}(\partial_{\mu}\phi)^{2}+e(g^{\mu\nu}-\epsilon^{\mu\nu})(\partial_{\mu}\phi)A_{\nu}+\frac{1}{2}e^{2}\alphaA_{\mu}^{2},
\begin{array}{ll}{\displaystyle{\frac{2\pii}{\beta}}\sum_{n=-\infty}^{n=\infty}f\left(\frac{2\piin}{\beta}\right)=}&{\displaystyle{\frac12}\int_{-\infty}^{\infty}\mathrm{d}z\left[f\left(iz\right)+f\left(-iz\right)\right]}\\{}&{+\displaystyle{\int}_{-\infty+\delta}^{\infty+\delta}\mathrm{d}z\left[\displaystyle{\frac{f\left(iz\right)+f\left(-iz\right)}{\mathrm{e}^{i\betaz}-1}}\right]}\\\end{array},
K=\frac{\sqrt{G^{2}M^{2}-Gq^{2}}}{G^{2}M^{2}(1+\sqrt{1-\frac{q^{2}}{GM^{2}}})^{2}}\approx\frac{1}{4GM}\Bigl(1-\frac{q^{2}}{2GM^{2}}\Bigr)\Bigl(1+\frac{q^{2}}{2GM^{2}}\Bigr)\approx\frac{1}{4GM}
\begin{array}{lcr}{D^{++}=u^{+\alpha}{\frac{\partial}{\partialu^{-\alpha}}};\qquadD^{--}=u^{-\alpha}{\frac{\partial}{\partialu^{+\alpha}}}}\\{2D^{++}=[D^{0},D^{++}];\qquad-2D^{--}=[D^{0},D^{--}]}\\{D^{0}=[D^{++},D^{--}]=u^{+\alpha}{\frac{\partial}{\partialu^{+\alpha}}}-u^{-\alpha}{\frac{\partial}{\partialu^{-\alpha}}}}\\\end{array}
\frac{d^{2}\tilde{F}_{0}}{d\tilde{z}^{2}}+\tilde{z}\frac{d\tilde{F}_{0}}{d\tilde{z}}+(1-\frac{1}{4\tilde{z}^{2}}+\frac{\mu^{2/(7-p)}}{(7-p)^{2}\tilde{z}^{2+2/(7-p)}})\tilde{F}_{0}=0
S_{M}=\intd^{D}x\Big[\overline{{\psi}}_{ik}i(\gamma^{\mu}\partial_{\mu}-\sum_{p=0}^{D}\frac{(-1)^{p}}{p!}A_{\mu_{1}\cdots\mu_{p}}^{(p)}(\gamma^{\mu_{1}}\cdots\gamma^{\mu_{p}})-m)_{kl}\psi_{li}\Big].
\beta_{\mu}^{(1,0)}\beta_{\nu}^{(1,0)}\beta_{\alpha}^{(1,0)}+\beta_{\alpha}^{(1,0)}\beta_{\nu}^{(1,0)}\beta_{\mu}^{(1,0)}=\delta_{\mu\nu}\beta_{\alpha}^{(1,0)}+\delta_{\alpha\nu}\beta_{\mu}^{(1,0)}
\begin{array}{lcr}{P(W_{2})=(y^{2}+x^{3}+z^{6}+\muxyz)+}\\{+w(a_{0}z^{2n}+\sum_{i=1}^{n-1}a_{i}z^{{\frac{(4(n-i)-1)}{2}}}y^{{\frac{1}{2}}}x^{{\frac{(2i-1)}{2}}})+a_{n}x^{n}.}\\\end{array}
\frac12\,\epsilon^{\mu\nu\lambda\sigma}\,e_{\lambda}^{a}\,e_{\sigma}^{b}\,\hat{\xi}_{ab}^{\pm}=\pm\Omega^{-1}\,\hat{\eta}_{\mu\nu}^{\pm}\,\Omega\qquad\mathrm{or}\qquad\Omega\,\hat{\xi}_{ab}^{\pm}=\pm\frac12\,\epsilon_{abcd}\,e^{\mu\,c}\,e^{\nu\,d}\,\hat{\eta}_{\mu\nu}^{\pm}\,\Omega.