harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
*j_{F1}=\frac{\tilde{f}2\kappa_{10}T_{F1}}{2}\delta^{(6)}(x)\wedge_{i=1}^{6}dx_{i}\wedge\left(\delta(x_{9}-\hat{x}_{9}(\rho))\frac{d\psi}{2\pi}\wedgedx_{9}+\delta(\rho-\hat{\rho}(x_{9}))d\rho\wedge\frac{d\psi}{2\pi}\right),
{\Psi}_{BL}^{(w)}\simeqN_{w}[\Psi_{L}^{(w)}+e^{-\piw/\kappa}\Psi_{R}^{(-w)}],\qquad{\Psi}_{BR}^{(w)}\simeqN_{w}[\Psi_{R}^{(w)}+e^{-\piw/\kappa}\Psi_{L}^{(-w)}],
\begin{aligned}{\}&{{}(3+7+\ldots+(4k-1))(g(M_{1})-1)+(5+9+\ldots+(4k-3))(g(M_{2})-g(M_{1}))}\\{=}&{{}k(2k+1)(g(M_{1})-1)+\left(k(2k-1)-1\right)(g(M_{2})-g(M_{1})).}\\\end{aligned}
E_{1}(x)\Psi(y)\midE\rangle=\left({\varepsilon_{1}(x)+q\left({1+\frac{{e^{2}}}{\pi}\frac{1}{{\partial_{x}^{2}}}}\right)^{-1}\int_{0}^{1}{d\alphay_{1}\delta^{(1)}(\alphay_{1}-x_{1})}}\right)\Psi(y)\midE\rangle.
(q+q^{-1})\mathrm{[}\hat{E}_{-},\hat{E}_{+}\mathrm{]}-(q-q^{-1})\hat{E}_{0}^{\;2}=\hat{E}_{0}\quad,\quad\mathrm{[}\hat{E}_{\pm},\hat{E}_{0}\mathrm{]}_{(q^{\mp1},q^{\pm1})}=\pm\hat{E}_{\pm}\;.
S_{eff}=NS_{0}(\sigma^{0})+S_{1}(\sigma^{0})-{\frac{i}{2}}C_{2}+{\frac{1}{2}}D_{2}+\sum_{n=1}^{\infty}{\frac{1}{\sqrt{N}^{n}}}\Big[i(-1)^{n+1}[{\frac{C_{n+2}}{(n+2)}}+mL^{d}\delta^{d}(0){\frac{C_{n}}{n}}]+{\frac{1}{(n+2)!}}D_{n+2}\Big]
f^{ilk}\intd^{4}xd^{4}yA_{\mu}^{l}(x)A_{\mu}^{m}(y)\langleT[j_{A^{0}}^{\mu,k}(x)j_{A^{0}}^{\nu,m}(y)]\rangle+if^{ilk}\intd^{4}yA_{\nu}^{l}(y)\langlej^{\nu,k}(y)\rangle=0
{\frac{\partialS}{\partial\chi_{0}}}=-{\frac{2}{\kappa}}\dot{\chi}_{0}\,\,,\,\,\,{\frac{\partialS}{\partial\Omega_{0}}}={\frac{2}{\kappa}}\dot{\Omega}_{0}\,\,,\,\,\,{\frac{\partialS}{\partialf_{i0}}}=\dot{f}_{i0}\\,
\begin{array}{lcl}{1\starf(x)g(p)}&{=}&{f(x)g(p)}\\{f(x)p^{n}\starg(x)}&{=}&{f(p^{n}\starg)}\\{p^{n}\starf(p)g(x)}&{=}&{(p^{n}\starg)f}\\{f(x)\starg(x)h(p)}&{=}&{(f\starh)g}\\{f(x)p\starg(x)p}&{=}&{fgp^{2}+\theta(fg^{\prime}-f^{\prime}g)p-(\theta)^{2}f^{\prime}g}\\\end{array}
\int_{0}^{\infty}dx\x^{\mu-1}e^{-px}I_{\nu}(cx)=p^{-(\mu+\nu)}\left(\frac{c}{2}\right)^{\nu}\frac{\Gamma[\nu+\mu]}{\Gamma[\nu+1]}{}_{2}F_{1}\left[\frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;\frac{c^{2}}{p^{2}}\right],
\mathrm{det}\,T_{(0,\,r)}-\frac{\mathrm{det}\,T_{(0,\,k-1)}\,\mathrm{det}\,T_{(k+1,\,r)}}{\Omega_{(0,\,k-1)}\Omega_{(k+1,\,r)}}\Omega_{(0,\,r)}=0,
\underset{x_{1}\rightarrowx_{2}}{\mathrm{lim}}(x_{12}^{2})^{-\frac{1}{2}\eta}:\sum_{\substack{k_{1},k_{2}\,k_{1}+k_{2}=k}}A_{k_{1},k_{2}}\gamma_{[r,s,t],k_{1}k_{2}}^{J,I_{1}I_{2}}\prod_{i=1}^{2}O_{k_{i}}^{I_{i}}(x_{i}):
K_{I}^{\mu}(x)=\frac{1}{2\pi}\varepsilon^{\mu\nu\lambda}\partial_{\nu}\partial_{\lambda}\bar{\omega}_{I}=\sum_{p_{I}}\sigma_{p_{I}}\intd\tau\frac{dR_{p_{I}}^{\mu}(\tau)}{d\tau}\delta[x-R_{p_{I}}(\tau)]
e^{{\cal{G}}_{\mathrm{ac}}\{{\phi}\}}=\frac{\int{\cal{D}}\{\phi^{\prime}\}e^{-S_{0}\{\phi^{\prime}\}-S_{\mathrm{int}}\{\phi^{\prime}+{\phi}\}}}{\int{\cal{D}}\{\phi^{\prime}\}e^{-S_{0}\{\phi^{\prime}\}}}\;.
\times\frac{[1+A{\Bigl(\frac{q}{|m|}\Bigr)}^{-2(1-F)}]K_{F}^{2}(qr)-[1+A^{-1}{\Bigl(\frac{q}{|m|}\Bigr)}^{-2F}]K_{1-F}^{2}(qr)}{A{\Bigl(\frac{q}{|m|}\Bigr)}^{2F}+2+A^{-1}{\Bigl(\frac{q}{|m|}\Bigr)}^{2(1-F)}}.
R_{1}=\frac{1}{\sqrt{\mu^{2}+4\left(M^{2}+k_{3}^{2}\right)}}\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{m_{1}^{2}-\left(M^{2}+k_{3}^{2}\right)}&{i\mum_{1}}&{0}\\{0}&{-i\mum_{1}}&{m_{1}^{2}-\left(M^{2}+k_{3}^{2}\right)}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right).
\frac{ds^{2}}{l_{s}^{2}}=\frac{u^{7-p}}{R^{7-p}}\Big[-fd\tilde{t}^{2}+(d\tilde{x}^{1})^{2}\Big]+(d\tilde{x}^{2})^{2}+\cdots+(d\tilde{x}^{p})^{2}+f^{-1}du^{2}+u^{2}d\Omega_{8-p}^{2}
\frac{1}{\rho}\frac{d}{d\rho}\left(\rho\frac{d\psi_{4}}{d\rho}\right)-\frac{a_{+}^{2}}{\rho^{2}}\psi_{4}-g_{1}\eta^{2}\psi_{4}(\psi_{4}^{2}+\psi_{6}^{2}-1)-g_{2}\eta^{2}\psi_{4}(\psi_{4}^{2}-\psi_{6}^{2}-c)=0,
\renewcommand{\arraystretch}{1.2}\begin{array}{rcl}{\chi_{\mathcal{H}_{\frac12,0}}(\tau)}&{=}&{|\Lambda_{0,2}(\tau)|^{2}=\chi_{\mathcal{V}_{-1/8,-1/8}}(\tau)\,,}\\{\chi_{\mathcal{H}_{\frac12,\frac12}}(\tau)}&{=}&{|\Lambda_{2,2}(\tau)|^{2}=\chi_{\mathcal{V}_{3/8,3/8}}(\tau)\,,}\\\end{array}
\begin{array}{cccc}{\mathrm{long~roots:}}&{\pm\,\beta_{kl}~=~\pm(2\epsilon_{jkl}e_{j}-e_{k}-e_{l})}&{}&{(1\leqk<l\leq3)}\\{\mathrm{short~roots:}}&{\pm\,\alpha_{kl}~=~\pm(e_{k}-e_{l})}&{}&{(1\leqk<l\leq3)}\\\end{array}~,
\delta=-\sigma\simeq\frac{\epsilonr_{s}}{\tilde{r}}\Theta(u^{+}-u_{0}^{+})+\frac{\kappa}{4\tilde{r}}(x_{H}^{-}e^{-u^{-}}-\frac{1}{4}{(x_{H}^{-})}^{2}e^{-2u^{-}}+c)
f_{D_{4}}=\frac{4\pi}{\sqrt{y^{2}+x_{0}^{2}}}\,K_{0}\left(\frac{\sqrt{(y^{2}+x_{0}^{2}+x_{1}^{2})(y^{2}+x_{0}^{2}+x_{2}^{2})(y^{2}+x_{0}^{2}+x_{3}^{2})}}{y^{2}+x_{0}^{2}}\right)e^{-i\,\frac{x_{0}x_{1}x_{2}x_{3}}{y(y^{2}+x_{0}^{2})}}\,.
\begin{array}{lll}{T(z)}&{=}&{\frac{1}{k+\check{g}}\{\frac{1}{2}(U_{a}U^{a}+U^{a}U_{a})+\frac{1}{2}(K_{+}K^{+}+K^{+}K_{+})+K_{3}^{2}}\\{}&{}&{}\\{}&{}&{-\frac{k+1}{2}(\psi_{a}\partial\psi^{a}+\psi^{a}\partial\psi_{a})+H_{a}^{b}\psi^{a}\psi_{b}+\frac{1}{4}\Omega_{ab}\psi^{a}\psi^{b}\Omega^{cd}\psi_{c}\psi_{d}\}(z)}\\\end{array}
\frac13\varepsilon^{ij}\left(\begin{array}{l}{2\bar{\varepsilon}\gamma_{\mu\rho}\partial_{i}\theta\bar{\theta}\gamma^{\mu\nu}\partial_{j}\theta}\\{-\bar{\varepsilon}\gamma_{\mu\rho}\theta\partial_{i}\bar{\theta}\gamma^{\mu\nu}\partial_{j}\theta}\\\end{array}\right)\lambda_{\nu}\lambda^{\rho},
(L_{3})_{ikms}^{~jlnp}=\delta_{i}^{~j}\delta_{k}^{~n}\delta_{s}^{~l}\delta_{m}^{~p}+\delta_{i}^{~n}\delta_{s}^{~j}\delta_{k}^{~l}\delta_{m}^{~p}-\delta_{i}^{~j}\delta_{k}^{~p}\delta_{m}^{~l}\delta_{s}^{~n}-\delta_{i}^{~p}\delta_{k}^{~l}\delta_{m}^{~j}\delta_{s}^{~n},
I(E)=\frac{4\pi^{d/2}\mu}{\Gamma(d/2)(2\pi)^{d}}P\int_{0}^{\infty}dk\,\frac{k^{d-1}}{2\muE-k^{2}}-i\frac{2\pi^{d/2+1}\mu}{\Gamma(d/2)(2\pi)^{d}}(2\muE)^{d/2-1}.
\hat{\theta}_{e}\,=\,\theta\,({\frac{P_{0}}{2}}+\mu_{e})\,\theta\,({\frac{P_{0}}{2}}-\mu_{e})\,\theta\,(\,\big[{\frac{P_{0}}{2}}+\mu_{e}\big]^{2}-h_{1e}^{2}+m_{1}^{2}\,)\,,
C=\Gamma_{0}=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\,,\qquad\Gamma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\qquad\Gamma_{2}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,.
\begin{array}{cl}{j_{L}=}&{|A|^{2}\frac{\displaystylek}{\displaystylem^{2}}(E-k)-|B|^{2}\frac{\displaystylek}{\displaystylem^{2}}(E+k)}\\{j_{R}=}&{|d_{1}|^{2}\frac{\displaystylek}{\displaystylem^{2}}(E-k)}\\\end{array}
g_{ij,kl}=\frac{-q^{-1}}{\left[2\right]}D_{si}\hat{R}_{js,kl}^{\epsilon}=\frac{-q^{-1}}{\left[2\right]}\epsilon_{im}\hat{R}_{jm,kt}\epsilon_{lt}^{-1}\quad,\quad(g=\frac{-q^{-1}}{\left[2\right]}D_{1}{\calP}\hat{R}_{12}^{\epsilon})\quad,
\Theta_{\mu\nu}(x)=\frac{1}{3}\left[\partial_{\mu}\phi\,\partial_{\nu}\phi+\partial_{\nu}\phi\,\partial_{\mu}\phi-\frac1{2}\left(\phi\partial_{\mu}\,\partial_{\nu}\phi+\partial_{\mu}\partial_{\nu}\phi.\phi+\eta_{\mu\nu}\partial_{\alpha}\phi\partial^{\alpha}\phi\right)\right],
[\,2i\sqrt{\pi}\,(\xim_{1}a+i\frac{m_{2}}{\xi}a^{\dagger}),\,2i\sqrt{\pi}\,(\xin_{1}a+i\frac{n_{2}}{\xi}a^{\dagger})\,]=-4\pii\,(m_{1}n_{2}-m_{2}n_{1})\in4\pii\,{\mathbbZ},
\tilde{\theta}_{\kappa(\mu\nu)}~=~g_{ij}(\varphi)\,\partial_{\kappa}\varphi^{i}\,\nabla_{\mu}\partial_{\nu}\varphi^{j}~=~g_{ij}(\varphi)\,\partial_{\kappa}\varphi^{i}\left(\partial_{\mu}\partial_{\nu}\varphi^{j}\,+\,{\mit\Gamma}_{kl}^{j}(\varphi)\,\partial_{\mu}\varphi^{k}\,\partial_{\nu}\varphi^{l}\right)~~.
\beta_{n}^{\prime}\left(t,z;x\right)\,=\,\sum_{n=1}^{N}\beta_{n_{0}}^{\prime}e^{\theta_{n}\left(mz\right)}\left[e^{\theta_{n}\left(\left(m+1\right)t-i\theta_{n}\left(2m+1\right)^{2}x\right)}\right],\tag{3.2b}
\mathrm{Im}~S=\mathrm{Im}\int_{r_{\mathrm{in}}}^{r_{\mathrm{out}}}p_{r}\,dr=\mathrm{Im}\int_{r_{\mathrm{in}}}^{r_{\mathrm{out}}}\!\!\int_{0}^{p_{r}}dp_{r}^{\prime}\,dr\;.
\acute{o}^{A}\bar{o}^{B^{\prime}}+o^{A}\acute{\bar{o}}^{B^{\prime}}=o^{A}\dot{\bar{\iota}}^{B^{\prime}}+\dot{o}^{A}\bar{\iota}^{B^{\prime}}+\iota^{A}\dot{\bar{o}}^{B^{\prime}}+\dot{\iota}^{A}\bar{o}^{B^{\prime}}.
\eta_{\mathrm{a}}^{N=2}=\frac{64NG}{\Gamma(3N^{2}\!-2)\pi^{2}}\int_{0}^{\infty}dt\,L(t)\int_{0}^{\infty}ds_{1}\,\widetilde{K}(s_{1})\,\left[S\left(\frac{cGs_{1}t^{2}}{2\pi^{4}}\right)-\frac{1}{2}\,S\left(\frac{cGs_{1}t^{2}}{8\pi^{4}}\right)\!\right].
Tr\left(Q\mathcal{M}\right)\;=\;2i\vec{V}_{-}^{T}\varepsilonM\vec{V}_{+}\,=\,\frac{e^{\phi}}{2}\left(m_{3}+m_{2}+2m_{1}C_{(0)}+(m_{3}-m_{2})\bar{\lambda}\lambda\right)\;.
\left(\begin{matrix}{\Gamma^{\phi_{S}^{0}11}(p)}&{\Gamma^{\phi_{S}^{0}12}(p)}\\{\Gamma^{\phi_{S}^{0}21}(p)}&{\Gamma^{\phi_{S}^{0}22}(p)}\\\end{matrix}\right)=M_{S}^{-1}\left(\begin{matrix}{\Gamma_{R}^{\phi_{S}^{0}}(p)}&{0}\\{0}&{\Gamma_{R}^{\phi_{S}^{0}*}(p)}\\\end{matrix}\right)M_{S}^{-1}
\begin{array}{c}{\{\Theta_{i}^{+},\Theta_{j}^{-}\}=\delta_{ij}{\calH}+{\calZ}\sigma_{i{j}}^{3},\quad\{\Theta_{i}^{\pm},\Theta_{j}^{\pm}\}=0,}\\{\{{\calZ},{\calH}\}=\{{\calZ},\Theta_{k}^{\pm}\}=0.}\\\end{array}
P=\sum_{n=0}^{\infty}\sum_{\mu_{1},\cdots,\mu_{n}=1}^{N}\int{\frac{{d\theta_{1}}\cdots{d\theta_{n}}}{{n!}(2\pi)^{n}}|V_{\mu_{1}}(\theta_{1})\cdotsV_{\mu_{n}}(\theta_{n})\rangle\langleV_{\mu_{1}}(\theta_{1})\cdotsV_{\mu_{n}}(\theta_{n})|}
{\frac{\sqrt{\pi\sigma}}{4\pi\kappa}}\left[{\frac{\kappa^{2}}{E^{2}}}+\sum_{n=1}^{\infty}{\frac{2}{n^{2}+\left({\frac{E}{\kappa}}\right)^{2}}}\right]-{\frac{\sqrt{\pi\sigma}}{4E}}.
\langle\vec{q}|n_{1}n_{2}\rangle=\Phi_{0}(\vec{q},t)\{\bfH}_{n_{1},n_{2}}^{\biggl\{-{\frac{\mu_{p}^{\ast}}{\mu_{p}}}{\bfI_{2}}\biggr\}}\biggl(-{\frac{i}{\hbar\mu_{p}^{\ast}}}{\bfR}\vec{q}\biggr)\.
\mathrm{~e}^{-S_{\mathrm{eff}}[A_{0}]}\equiv\int[d\vec{a}][d\vec{\pi}]\mathrm{~e}^{\int_{0}^{1/T}d\tau\int_{S^{2}}\left\{i\vec{\pi}\cdot\dot{\vec{a}}-\frac{1}{2}(\pi_{i}-\frac{\kappa}{2}\epsilon_{ij}a_{j})^{2}-\frac{1}{2}(B_{M}^{2}+b^{2})\right\}+i\frac{\kappa}{T}\int_{S^{2}}A_{0}\left(B_{M}+b/2\right)}
\delta\mathrm{A}=\mathrm{A}^{\prime}\xi_{\mathrm{B}}+\xi_{\mathrm{B}}^{\prime}\mathrm{A}-[\Lambda_{\mathrm{B}}\,\mathrm{A}-\mathrm{A\,}\Lambda_{\mathrm{B}}]+k\,\mu\,\Lambda_{\mathrm{B}}^{\prime}=0,
\Gamma_{2}[\phi_{0}]=\int_{0}^{\infty}\frac{4q^{2}dq}{(4\pi)^{4}}\left[\left(2\frac{\sqrt{q^{2}+4M^{2}}}{q}\mathrm{arctanh}\frac{q}{\sqrt{q^{2}+4M^{2}}}-\frac{5\pi}{3\sqrt{3}}+1\right)|\tilde{\chi}(q)|^{2}-4q^{2}\left(\frac{2\pi}{3\sqrt{3}}-1\right)|\tilde{\phi}(q)|^{2}\right]
\vec{\tilde{F}}(\varphi(u))=\big(\varphi_{*}\vec{F}(u)\big)+\sum\partial_{i}(\big(\varphi_{*}^{i}\vec{F}(u)\big))+\sum\partial_{i}\partial_{j}(\big(\varphi_{*}^{ij}\vec{F}(u)\big))+...,
\biggl(L\biggl(x-1,{\frac{\partial}{\partialx}}\biggr)-z^{p}\biggr)\Psi(x,0,z)=0\,\\mathrm{and}\\,\biggl(L\biggl(z,{\frac{\partial}{\partialz}}-1\biggr)-(x-1)^{p}\biggr)\Psi(x,0,z)=0\,,
{\calL}_{F}=i\psi_{i}\sigma^{\mu}D_{\mu}\bar{\psi}_{i}+i\chi_{I}\sigma^{\mu}D_{\mu}\bar{\chi}_{I}-i\tilde{\xi}^{\alpha}\sigma_{\alpha\dot{\alpha}}^{\mu}\partial_{\mu}{\bar{\tilde{\xi}^{\dot{\alpha}}}}-i{\tilde{\lambda}^{\alpha}}\sigma_{\alpha\dot{\alpha}}^{\mu}\partial_{\mu}\bar{\tilde{\lambda}^{\dot{\alpha}}},
\begin{array}{rrrr}{\mathrm{{Previous\,\,sections:}}}&{X^{-}(z),}&{X_{2}(z),}&{X^{+}(z)}\\{\mathrm{{Appendix:}}}&{-X^{-}(z),}&{-\frac2ku(z),}&{-X^{+}(z)}\\\end{array}
K_{\beta}(\alpha)\equiv\int_{0}^{\infty}\!dk\left(\frac{1}{\sqrt{k^{2}+\alpha}}\,\,\frac{1-e^{-2\beta\sqrt{k^{2}+\alpha}}}{f\!\left(\sqrt{k^{2}+\alpha}\right)}-\frac{1}{\sqrt{k^{2}+m^{2}}}\right).
\begin{array}{l}{\{n_{a}(x),~n_{b}(y)\}=0}\\{\{n_{a}(x),~\pi_{b}(y)\}=(\delta_{ab}-n_{a}n_{b})\delta(x-y)}\\{\{\pi_{a}(x),~\pi_{b}(y)\}=(\pi_{a}n_{b}-\pi_{b}n_{a})\delta(x-y)}\\\end{array}
S(\gamma,X,A)=-\frac{1}{2}\intd^{3}\xi\sqrt{-\gamma}\left(\gamma^{ij}\partial_{i}X^{m}\partial_{j}X^{n}\eta_{mn}+\frac{1}{2}\gamma^{ij}\gamma^{kl}F_{ik}^{r}F_{jl}^{r}-1\right)
\biggl\langleF[A]\biggr\rangle_{x}\;=\;\frac{1}{\calN}\;\int{\calD}x_{1\ldotsd}\;F[A]\;e^{-\frac{1}{4}\int_{0}^{T}d\tau\;\Bigl(\dot{x}_{1}^{2}+\ldots+\dot{x}_{d}^{2}\Bigr)}\;,
\langle\dot{y}_{a}\cdot\dot{y}_{b}\rangle=\mathrm{tr}\biggl[\ddot{\calG}_{Bab}+{\frac{1}{2}}{\frac{(\dot{\calG}_{Baa}-\dot{\calG}_{Bab})(\dot{\calG}_{Bab}-\dot{\calG}_{Bbb})}{\bar{T}-{\frac{1}{2}}{\calC}_{ab}}}\biggr]\;
\ddot{Z}={\frac{\dot{Z}^{2}}{Z}}-{\frac{1}{\dot{\eta}}}\bigl({\frac{\ddot{\eta}}{\dot{\eta}}}-{\frac{\dot{g}}{g}}\bigr)\dot{Z}+{\frac{1}{\dot{\eta}^{2}\mu}}\bigl(\ddot{\mu}-{\frac{\dot{\mu}^{2}}{\mu}}-{\frac{\dot{g}}{g}}\dot{\mu}\bigr)Z+{\frac{h^{2}-l}{4\dot{\eta}^{2}\mu^{2}}}{\frac{Z^{4}-\mu^{4}}{Z}}.
\begin{array}{ccc}{{\calK}\equiv\left(\begin{array}{c}{a^{\mu}}\\{b^{\mu}}\\{\varepsilon^{a}}\\{\rho^{a}}\\{\lambda}\\{w^{\mu}{}_{\nu}}\\{t^{a}{}_{b}}\\\end{array}\right)~~~~}&{\longrightarrow}&{~~~~\left(\begin{array}{c}{b^{\mu}}\\{a^{\mu}}\\{\rho^{a}}\\{\varepsilon^{a}}\\{-\lambda}\\{w^{\mu}{}_{\nu}}\\{t^{a}{}_{b}}\\\end{array}\right)\,.}\\\end{array}
\begin{array}{llc}{}&{R=+}&{SO(4)\mathrm{~rep.}}\\{|s_{3}s_{4},ij\rangle\lambda_{ji},\s_{3}=-s_{4}\quad}&{\lambda=\gamma_{R,I}\lambda\gamma_{R,9}^{-1}\}&{2({\bf1},{\bf1})}\\\end{array}
S_{2}({\bf{k}})=\sum_{{\bf{q}},i}\frac{{\bar{n}}_{{\bf{k}}-{\bf{q}}}}{({\tilde{\omega}}_{i}(-{\bf{q}})+\frac{{\bf{k.q}}}{m}-\epsilon_{{\bf{q}}})^{2}}g_{i}^{2}(-{\bf{q}})
{\calE_{C}}[\phi_{0}]=\Gamma^{(2)}[\phi_{0}]+\sum_{\ell=0}^{\infty}(2\ell+1)\left[\int_{0}^{\infty}\frac{dk}{\pi}E\frac{d}{dk}\left(\delta_{\ell}(k)-\delta_{\ell}^{(1)}(k)-\delta_{\ell}^{(2)}(k)\right)+\sum_{j}\mathrm{max}(\omega,E_{j,\ell})\right]\,.
\frac{p_{r}^{2}}{g_{rr}}+\frac{p_{\theta}^{2}}{g_{\theta\theta}}-\frac{{g^{\prime}}_{tt}}{-\calD}\left(p_{\phi}+\frac{g_{t\phi}+\Omega_{0}g_{\phi\phi}}{{g^{\prime}}_{tt}}E\right)^{2}=-\left(\frac{E^{2}}{{g^{\prime}}_{tt}}+\mu^{2}\right),
V^{\prime\prime}(A)=\frac{2}{\ell^{2}}-\frac{32\,\pi^{2}\{[(d-2)\,\rho+(d-1)\,p]^{2}+(d-1)\,\rho\,(\rho+p)[(d-2)+(d-1)\,v_{s}^{2}]\}}{(d-1)^{2}\,L_{p}^{2(3-d)}}
\langle\partial_{\mu}j_{R}^{\mu}\rangle=\frac{1}{16\pi}\mathrm{Re}\intd^{2}\theta_{1}d^{2}\theta_{2}\Big(F+F_{D}\Big)=\frac{1}{8\pi^{2}}\mathrm{Im}\intd^{2}\theta_{1}d^{2}\theta_{2}u.
d\hat{\calE}=0\;,\;\hat{\calE}\wedge\hat{\calE}=0\;,\;(\hat{\calE}\wedge\hat{\vec{\calN}}){}^{T}=-\hat{\vec{\calN}}{}^{T}\wedge\hat{\calE}^{\T}\;,\;\hat{\calE}^{\T}\eta\,=\,-\eta\hat{\calE}\;,
\Box^{(4)}{h_{\mu\nu}}=-\frac{\kappa_{5}^{2}}{\int_{0}^{r_{c}}dza^{2}}(T_{\mu\nu}^{+}-\frac{1}{3}T^{+}\eta_{\mu\nu})+\frac{U_{B}}{12}\vert_{0}\kappa_{5}^{2}T^{+}\eta_{\mu\nu},
{\calL}(w)=\left(\begin{array}{ccccc}{-p_{1}}&{e^{{\frac{1}{2}}(q_{2}-q_{1})}}&{0}&{}&{{\frac{1}{w}}e^{{\frac{1}{2}}(q_{N_{c}}-q_{1})}}\\{e^{{\frac{1}{2}}(q_{2}-q_{1})}}&{-p_{2}}&{e^{{\frac{1}{2}}(q_{3}-q_{2})}}&{\ldots}&{0}\\{0}&{e^{{\frac{1}{2}}(q_{3}-q_{2})}}&{-p_{3}}&{}&{0}\\{}&{}&{\ldots}&{}&{}\\{{w}e^{{\frac{1}{2}}(q_{N_{c}}-q_{1})}}&{0}&{0}&{}&{-p_{N_{c}}}\\\end{array}\right)
J_{ij}^{N}=\intd^{2}\vec{x}(x_{i}\theta_{\circj}^{N}-x_{j}\theta_{\circi}^{N}+\frac{\partial{\calL}}{\partial(\partial^{\circ}A^{k})}(g_{i}^{k}g_{j}^{l}-g_{j}^{k}g_{i}^{l})A_{l})
:\prod_{\substack{a=1\,a\neqc}}^{r}O_{2}^{J_{a}}(x_{4})[O_{2}^{J_{c}}(x_{4})\partial_{\nu}O_{2}^{J_{r+1}}(x_{4})-O_{2}^{J_{r+1}}(x_{4})\partial_{\nu}O_{2}^{J_{c}}(x_{4})]:=W_{\nu}^{J_{1}J_{2}...(J_{c})...J_{r}J_{r+1}}(x_{4})
{\calB}_{N_{1},\ldots,N_{10}}:={S}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}}\{S}_{\alpha_{4}\alpha_{5}\alpha_{6}\alpha_{7}}^{N_{4}N_{5}N_{6}N_{7}}\{S}_{\alpha_{7}\alpha_{3}\alpha_{8}\alpha_{9}}^{N_{7}N_{3}N_{8}N_{9}}\{S}_{\alpha_{9}\alpha_{6}\alpha_{2}\alpha_{10}}^{N_{9}N_{6}N_{2}N_{10}}\{S}_{\alpha_{10}\alpha_{8}\alpha_{5}\alpha_{1}}^{N_{10}N_{8}N_{5}N_{1}}.
ds_{11}^{2}=\frac{1}{({\frac{1}{6}}C_{ABC}H^{A}H^{B}H^{C})^{\frac{1}{3}}}\Big[du\,dv+H_{0}\,du^{2}+{\textstyle{\frac{1}{6}}}C_{ABC}H^{A}H^{B}H^{C}\,d\vec{x}^{2}+H^{A}\omega_{A}\Big]\;,
v=\frac{1}{detD}\sqrt{(e^{\phi}\vec{\pi}.\vec{\pi}+1)(e^{-\phi}\vec{B}.\vec{B}+1)-[(\vec{\pi}.\vec{B})^{2}+2\psie^{\phi}(\vec{\pi}.\vec{B})-\psi^{2}e^{\phi}\vec{B}.\vec{B}]}.
{\calH}(p)=\frac{\;\Gamma\left(2-\frac{d}{2}\right)\mu^{\epsilon}}{2^{d+1}\pi^{\frac{d}{2}}}\;\varepsilon^{\mu\nu\rho}\;\varepsilon^{\alpha\beta\lambda}\;g_{\nu\alpha}\int_{0}^{1}dyy\int_{0}^{1}dx\frac{1}{[-a^{\prime}]^{2-\frac{d}{2}}}\frac{\mu^{\epsilon}}{(2\pi)^{d}}I_{DivPar}(p)+\mathrm{fin~parts}
\vec{y}(\omega)={\frac{-e\vec{E}^{e}(\omega)}{\omega^{2}\left[m_{0}+{\frac{2}{3}}e^{2}\int{\frac{d\vec{k}}{(2\pi)^{3}}}\,{\frac{|f(\vec{k})|^{2}}{k^{2}-\omega^{2}}}\right]}}~~,
\int_{-w\prime}^{w\prime}\int_{-w\prime^{\ast}}^{w\prime^{\ast}}d_{q}w^{\ast}d_{q}w\,S(\kappaw^{\ast},\kappa^{\ast}w)S(\kappa^{\prime}w^{\ast},\kappa^{\ast}w)=2M(\kappa,w\prime^{\ast})N(\kappa^{\ast},w\prime)\delta_{\kappa,\kappa^{\prime}},
\displaystyle\frac{{\calJ}_{-}}{{\calJ}_{+}}=2\displaystyle\frac{J_{l-\xi}(ka)J_{l+\Phi}(ka)-J_{l}(ka)J_{l+\Phi-\xi}(ka)}{J_{l-\xi}(ka)H_{l+\Phi}^{+}(ka)-J_{l}(ka)H_{l+\Phi-\xi}^{+}(ka)}-1.
Q_{1}(Q_{2})={\frac{1}{{\sqrt{-2\nabla^{2}}}}}[\epsilon_{ij}\partial_{i}A_{j}\pm{\frac{1}{m}}\Pi_{\theta}];~~P_{1}(P_{2})=[{\frac{1}{{\sqrt{-2\nabla^{2}}}}}\epsilon_{ij}\partial_{i}\Pi_{j}\mp{\frac{m}{2}}{\sqrt{-2\nabla^{2}}}\theta],
\Pi_{\mu}\Pi_{\nu}^{*}a_{\mu}b_{\nu}=\frac{1-\gamma_{5}}{2}\gamma_{\mu}\frac{1+\gamma_{5}}{2}\gamma_{\nu}a_{\mu}b_{\nu}=\frac{1-\gamma_{5}}{2}\gamma_{\mu}\gamma_{\nu}a_{\mu}b_{\nu}
\begin{array}{ll}{\Gamma^{+}\theta=0,\qquad}&{x^{+}=\taup^{+},}\\{e=1,\qquad}&{\Lambda^{i}=0,}\\{A^{2}=1,\qquad}&{A^{i}=0,}\\{B^{2}=1,\qquad}&{B^{i}=0,}\\\end{array}
T\sum_{n=-\infty}^{\infty}f(p_{3}=2\piTn)=\frac{1}{2}\int_{-\infty}^{\infty}\frac{dz}{2\pi}[f(z)+f(-z)]+\int_{-\infty+i\epsilon}^{\infty+i\epsilon}\frac{dz}{2\pi}[f(z)+f(-z)]\frac{1}{e^{-i\betaz}-1}\;.
{\calL}=-\,\frac{1}{4}\,\pi_{ij}\pi^{ij}+\frac{1}{2}\,\Bigl(\dot{A}_{ij}+\partial_{i}A_{j0}-\partial_{j}A_{i0}\Bigr)\,\pi^{ij}+\frac{1}{4}\,\partial_{i}A_{jk}\,\partial^{i}A^{jk}+\frac{1}{2}\,\partial_{i}A_{jk}\,\partial^{k}A^{ji}
\frac{1}{m}E_{\mathrm{cl}}^{(3,4)}=\frac{\pi\mu_{H}^{2}}{2g^{2}}\int_{0}^{\infty}dxx^{2}\left\{\left[2(s-1)+(s-1)^{2}+\vec{p}\,^{2}\right]^{2}-4(s-1)^{2}\right\}\.
Im\left({{\frac{\Pi(P^{2})}{(P^{2}-M^{2}+i\varepsilon)^{2}}}}\right)={\frac{Im\Pi(P^{2})}{(P^{2}-M^{2})^{2}}}-2\piRe\Pi^{\prime}(M^{2})\delta(P^{2}-M^{2})
\Gamma^{\alpha}{}_{\beta}=\frac{(-1)^{d(d-3)/4}}{d!{\sqrt-\gamma}}\epsilon^{i_{1}..i_{d}}E_{i_{1}}{}^{a_{1}}E_{i_{2}}{}^{a_{2}}\ldotsE_{i_{d}}{}^{a_{d}}\Gamma_{a_{1..a{_d}}}~~.
\sigma_{\underline{\alpha}\underline{\beta}}^{\underline{m}}\sigma_{\underline{m}\underline{\gamma}\underline{\delta}}+\sigma_{\underline{\alpha}\underline{\delta}}^{\underline{m}}\sigma_{\underline{m}\underline{\beta}\underline{\gamma}}+\sigma_{\underline{\alpha}\underline{\gamma}}^{\underline{m}}\sigma_{\underline{m}\underline{\delta}\underline{\beta}}=0.
\begin{array}{rcl}{g(s;y,z)}&{=}&{\frac{2^{1-s}}{\Gamma(s)}\sum_{k=0}^{N}\frac{{\bfB}_{2k}}{(2k)!}y^{2k}\Gamma(s-1+2k)\Bigl\{2\zeta_{R}\left(s-1+2k;1+\frac{z}{2}\right)+}\\{}&{+}&{\left(\frac{z}{2}\right)^{1-2k-s}\Bigr\}+O\left(y^{2N+2}\right),}\\\end{array}
S=\int_{0}^{1}Ld\tau\,,\;\L=-\frac{\left(\dot{x}^{\mu}-i\xi^{\mu}\chi\right)^{2}}{2e}-q\dot{x}^{\mu}A_{\mu}+iqeF_{\mu\nu}\xi^{\mu}\xi^{\nu}-im\xi^{4}\chi-\frac{e}{2}m^{2}-i\xi_{n}\dot{\xi}^{n}\;,
\partial_{\mu}R^{\mu}=-\frac{4}{3}\left\{\left.\left[3{\calW}-\sum_{i}S_{i}\,\frac{\partial{\calW}}{\partialS_{i}}\left(1+\frac{c_{i}+\gamma_{i}}{2}\right)\right]\right|_{G}-\frac{\mathrm{Tr}\,W^{2}|_{G}}{16\pi^{2}}\left[b+\sum_{i}T_{i}\,(c_{i}+\gamma_{i})\right]\right\}\;.
V_{F}^{(div)}(H)=-\frac{N}{64\pi^{2}\epsilon}\left(m_{t}(H)^{4}+\sum_{k=1}^{\infty}\left(\frac{2k}{R}+m_{t}(H)\right)^{4}+\sum_{k=1}^{\infty}\left(-\frac{2k}{R}+m_{t}(H)\right)^{4}\right)
\stackrel{(J)}{\omega}_{1}=a_{J}\left(\left[\Phi^{\alpha}\right],\left[\mathcal{P}_{a_{k}}\right]_{k=0,\cdots,n-2}\right)e^{J+1}\left(\eta^{a},\eta_{a}^{i_{1}\cdotsi_{n-1}},C_{i_{1}\cdotsi_{n-1}}^{a}\right),
E_{a}^{\underline{\beta}}(\eta^{ab}E_{b}^{\underline{a}}\Gamma_{\underline{a}})_{\underline{\beta}\underline{\gamma}}v_{\alphaq^{\prime}}^{~~~\underline{\gamma}}v_{\underline{\alpha}}^{~\alphaq^{\prime}}\equiv{\frac{1}{2}}\eta^{ab}E_{a}^{\underline{\beta}}\left[E_{b}^{\underline{a}}\Gamma_{\underline{a}}(1-\bar{\Gamma})\right]_{\underline{\beta}\underline{\alpha}}=0
s=-\sum_{i=1}^{N}\frac{\alpha_{i}}{\alpha}-\frac{Q}{\alpha}~,~\alpha_{i}=-\frac{Q}{2}+\frac{1}{2}\sqrt{Q^{2}+4(h_{i}-2)},~\alpha=-\frac{Q}{2}+\frac{1}{2}\sqrt{Q^{2}+8}
q=-\frac{{\dot{b}}\left(-{\dot{\Phi}}+\frac{{\dot{\sigma}}_{1}+5{\dot{\sigma}}_{2}}{2}+{\dot{b}}\right)-{\ddot{\Phi}}+\frac{{\ddot{\sigma}}_{1}+5{\ddot{\sigma}}_{2}}{2}+{\ddot{b}}}{\left(-{\dot{\Phi}}+\frac{{\dot{\sigma}}_{1}+5{\dot{\sigma}}_{2}}{2}+{\dot{b}}\right)^{2}}
R:~~~~~~~~~`L_{m}=\frac{1}{2}\sum_{-\infty}^{\infty}:\alpha_{-n}\alpha_{m+n}:+\frac{1}{2}\sum_{-\infty}^{\infty}(n+\frac{1}{2}m):(d_{-n}\cdotd_{m+n}-d_{-n}^{\prime}\cdotd_{m+n}^{\prime}):
X^{\mu}=X_{0}^{\mu}+{\frac{1}{\piT}}P^{\mu}\tau+{\frac{i}{2\sqrt{\piT}}}\sum_{n\neq0}{\frac{1}{n}}\bigg\{\alpha_{n}^{\mu}exp\bigg(-i2n(\tau-\sigma)\bigg)+\tilde{\alpha}_{n}^{\mu}exp\bigg(-i2n(\tau+\sigma)\bigg)\bigg\}
F=VkT{\frac{1}{2a}}{\frac{d}{ds}}\int{\frac{d^{d}k}{(2\pi)^{d}}}\sum_{n=-\infty}^{\infty}{\frac{1}{2}}\beta^{2s}\left({\frac{n^{2}\pi^{2}}{a^{2}}}+k^{2}\right)^{s}\bigg|_{s=0},
(L^{+})^{n}\vert1/2\rangle=\frac{q^{n}-q^{-n}}{q-q^{-1}}\times\frac{q^{n-1}-q^{-n+1}}{q-q^{-1}}\times\ldots\times\frac{q-q^{-1}}{q-q^{-1}}\vertn+1/2\rangle.
\frac{dT}{dx}=-\frac{4c\varepsilon^{2}HZ^{4}n}{\nu_{0}^{2}v^{2}}\int_{0}^{\infty}\xid\xi\left\{1-e^{-\alpha\xi\int_{\xi}^{\infty}\frac{du}{u^{2}}B(u)}\right\}\frac{\int_{\xi}^{\infty}\frac{du}{u}B(u)}{\alpha\xi\int_{\xi}^{\infty}\frac{du}{u^{2}}B(u)}~~.
\Bigl(3\lambda_{1}^{2}+{\frac{1}{12}}\lambda_{3}^{2}\Bigr)\;{\frac{\partial\lambda_{3}}{\partial\lambda_{1}}}\;+\;\Bigl(3\lambda_{2}^{2}+{\frac{1}{12}}\lambda_{3}^{2}\Bigr)\;{\frac{\partial\lambda_{3}}{\partial\lambda_{2}}}={\frac{2}{3}}\lambda_{3}^{2}+\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{3}\;.
i\Pi_{S\mu\nu}^{\underline{{a}}\underline{{b}}}(q)=Ng^{2}\delta_{\underline{{a}}\underline{{b}}}\mu^{2\epsilon}\int\frac{d^{n}\ell}{(2\pi)^{n}}\left\{\frac{\ell_{\mu}\ell_{\nu}-m_{G}^{2}g_{\mu\nu}}{(\ell+q)^{2}(\ell^{2}-m_{G}^{2})}-\frac{1}{8}\frac{(2\ell+q)_{\mu}(2\ell+q)_{\nu}}{\ell^{2}(\ell+q)^{2}}\right\}
\chi^{\prime\prime}+{\frac{3}{2}}{\frac{B^{\prime}}{B}}\chi^{\prime}={\frac{2\sqrt{B}}{\eta^{2}}}\left[{\frac{\partial\sigma_{I}}{\partial\chi}}\delta(y)+{\frac{\partial\sigma_{II}}{\partial\chi}}\delta(y-y_{*})\right]\,.
\Theta_{f}(x_{f})=g\int_{0}^{x_{f}}{\bfB}_{l}(x_{f})dx_{f}^{l},\quad\Theta_{g}(x)=\int_{0}^{x}\left[R_{f}^{+}\Gamma_{\mu}R_{f}+\psi^{-1}\partial_{\mu}\psi\right]dx^{\mu},\quad\psi\equiv\left(\psi_{l}^{\mu}\right),
{\frac{2p^{-}}{\mu}}=\sum_{n}\left[e^{U_{0}}\,{\frac{s}{J_{V}}}N_{n}^{r,w}|n|+\sqrt{1+e^{2U_{0}}\,{\frac{s^{2}}{J_{V}^{2}}}\,n^{2}}\,\,N_{n}^{y}\right]\.
\frac{v^{2}}{\lambda}=\frac{(a\Lambda)^{2}}{8\pi^{2}}+\frac{1}{2}\hat{m}^{2}a^{2}\left(1+\frac{\lambda_{R}}{16\pi^{2}}\bar{c}\right)\bar{I}_{2}+Z_{\phi^{2}}\left(\frac{v_{\mathrm{R}}^{2}}{\lambda_{\mathrm{R}}}\right),
F_{t_{i}t_{j}t_{p}}:=\partial_{t_{i}}\partial_{t_{j}}\partial_{t_{p}}{\calF}_{mirror}=J_{i}\cdotJ_{j}\cdotJ_{p}+\sum_{n_{1},\cdots,n_{k}}d_{\vec{n}}{\frac{e^{n_{1}t_{1}+\cdots+n_{k}t_{k}}}{1-e^{n_{1}t_{1}+\cdotsn_{k}t_{k}}}}.
S_{(\xi>0)}=\intd^{4}x\sqrt{-g}\left[-{\frac{1}{2}}\xi\hat{\phi}^{2}\tilde{R}-{\frac{1}{2}}{\calD}_{\mu}\hat{\phi}{\calD}^{\mu}\hat{\phi}-V(\hat{\phi})-{\frac{1}{4}}H_{\mu\nu}H^{\mu\nu}\right],
I_{1}=\frac{1}{16\pi^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{d^{4}l}{(2\pi)^{4}}\,\frac{1}{\Big(1+k^{2n}/\Lambda^{2n}\Big)\,\Big(1+l^{2n}/\Lambda^{2n}\Big)\,k^{4}l^{4}}.
e_{\hat{0}}^{\;\;\mu}e_{\alpha}^{\;\;\nu}\left.\frac{\partial\Gamma_{R}}{\partialb^{\mu\nu}}\right|_{g_{\mu\nu}}={\bar{e}}_{\alpha}^{\;\;i}{e}_{\hat{0}}^{\;\;0}a\frac{\partial\Gamma_{R}}{\partialw^{i}},
{\calG}({\bfx})\approx{\frac{\betam}{4\pi^{2}\vert{\bfx}\vert}}\;K_{1}(m\vert{\bfx}\vert)\approx{\frac{\betam^{1/2}}{2^{5/2}\pi^{3/2}\vert{\bfx}\vert^{3/2}}}\;e^{-m\vert{\bfx}\vert}\;,
Q=+{\frac{\pi}{2}}+\rho\left[\int_{1}^{n^{2}}{\frac{du^{\prime}}{1-u^{\prime}}}{\frac{\sqrt{n^{2}-u^{\prime}}\sqrt{u^{\prime}-1}}{(n^{2}-u^{\prime})+\rho^{2}(u^{\prime}-1)}}\right].
\left((D^{2}({\Phi}))^{ab}+gf^{adc}{\tilde{A}}_{i}^{d}D_{i}^{cb}({\Phi})\right)\tilde{\sigma}^{b}=-j_{\mathrm{tot},0}^{b};~~~~j_{\mathrm{tot},0}^{b}=gf^{abc}{\tilde{A}}_{i}^{a}E_{i}^{c}+j_{0}^{b}~.