harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{\Gamma}_{s}=\left\{z=z^{(s)}+\vartheta^{(s)}\varepsilon_{s}(z^{(s)})\,,\quad\vartheta=\vartheta^{(s)}\left(1+\frac{\varepsilon_{s}\varepsilon_{s}^{\prime}}2\right)+\varepsilon_{s}(z^{(s)})\right\}\,,
\kappa(\zeta)=\zeta\frac{(x^{4}z;x^{4})_{\infty}(x^{2}z^{-1};x^{4})_{\infty}}{(x^{4}z^{-1};x^{4})_{\infty}(x^{2}z;x^{4})_{\infty}},~~~~(a;p_{1},\cdots,p_{n})_{\infty}=\prod_{k_{i}\ge0}(1-ap_{1}^{k_{1}}\cdotsp_{n}^{k_{n}}),
\delta^{s}S\simeq\int_{t_{1}}^{t_{2}}dt\;\epsilon\;\left[U_{_{(\nu)}}^{s}\Omega_{_{(\nu)}}-y^{s}\Omega_{_{(1)}}-z_{_{(\nu)}}^{s}\left({\frac{\partial\Omega_{_{(\nu)}}}{\partial\pi}}\pi-\Omega_{_{(\nu)}}\right)\right]\;.
{\calQ}\equiv\left(\begin{array}{cc}{\frac{\delta}{2}r}&{\betaq}\\{\gammap}&{-\frac{\delta}{2}r}\\\end{array}\right)\equiv{\textstyle\frac{1}{2}}\oint_{S^{1}}{\calJ}=\oint_{S^{1}}d{\calM}{\calM}^{-1}\,,
\begin{array}{rcl}{{A^{\prime}}^{a}T^{a}}&{=}&{UA^{a}T^{a}U^{-1}\\\,\\\{\pi^{\prime}}^{a}T^{a}=U\pi^{a}T^{a}U^{-1}\\\,}\\{{\phi^{\prime}}^{a}T^{a}}&{=}&{U\phi^{a}T^{a}U^{-1}\,+\,\frac{i}{g}U\frac{d}{dt}U^{-1}\\\.}\\\end{array}
S_{w}=1+\sum_{n=1}^{\infty}\frac{i^{n}}{n!}\int_{-\infty}^{\infty}d^{4}x_{1}...\int_{-\infty}^{\infty}d^{4}x_{n}T:\mathcal{H}_{WI}\left(g_{,}m_{e,}x\right)...\mathcal{H}_{WI}\left(g_{,}m_{e,}x\right)\tag{2.5.2}
S_{worm}=-6\pi^{5}\intd\rho\big[{\frac{R^{7}}{N}}(\partial_{\rho}R)^{2}+NR^{7}-{\frac{1}{144}}{\frac{R^{9}}{N}}(\partial_{\rho}\phi)^{2}-{\frac{Q^{2}}{\pi^{10}}}{\frac{N}{R^{9}}}e^{-2\phi}\big].
H_{qqqq}={\frac{g^{2}}{8\pi}}\intd^{2}x^{\perp}\int_{-\infty}^{\infty}{\frac{dp^{+}}{(p^{+})^{2}}}{\tilde{j}^{+a}}_{q}(p^{+},x_{\perp}){\tilde{j}^{+a}}_{q}(-p^{+},x_{\perp}).
\left.\nabla_{\underline{{A}}}W_{\underline{{BCD}}}\right|=\frac{1}{8}{\calR}_{\mu\nu\rho\sigma}\sigma_{\underline{{AB}}}^{\mu\nu}\sigma_{\underline{{CD}}}^{\rho\sigma}+\cdots=-\frac{1}{2}{\calW}_{ABCD}+\cdots
G^{{\mu}{\nu}{\sigma}}(B)={\frac{1}{3!}}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\partial}_{[{\rho}}B_{{\eta}{\delta}]}\equiv{\frac{1}{3!}}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}F_{{\rho}{\eta}{\delta}}(B),
\frac{y^{n+p}}{(n+p)^{r}n^{k}}=\frac{1}{n^{k}}\int_{0}^{y}\Omega^{r-1}\Omega_{n+p}=\int_{0}^{y}\Omega^{r-1}\bigg(\frac{x^{n}}{n^{k}}\bigg)x^{p}\frac{dx}{x}.
\chi(r)=\frac{1}{N}\left(\begin{array}{c}{\sqrt{{\calE}+m}\,\,(\varepsilon_{l})^{l}J_{\varepsilon_{l}\nu}(kr)}\\{i\sqrt{{\calE}-m}\,\,(\varepsilon_{l})^{l+1}J_{\varepsilon_{l}(\nu+1)}(kr)e^{i\varphi}}\\\end{array}\right),
\deltaS_{lin}=\intd^{6}x\,\left(\frac1{12}H_{\mu\nu\rho}^{*}-\frac23h_{\mu\nu\rho}^{+}\right)\deltaH^{\mu\nu\rho}+\frac1{2\sqrt{u^{2}}}\epsilon^{\mu\nu\rho\sigma\tau\phi}H_{\mu\nu}^{-}H_{\rho\sigma}^{-}v_{\tau}\partial_{\phi}\deltaa\.
\begin{array}{c}{\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}D_{m}^{2}(k_{1})D_{m}(k_{2})D_{m}(k_{1}+k_{2})\overline{{D(k_{1}+p)}}K_{1}.}\\\end{array}
\frac{1}{2}\sqrt{-\gamma}\gamma^{ij}\gamma^{k\ell}\partial_{k}x^{\mu}\partial_{\ell}x^{\nu}\eta_{\mu\nu}-\sqrt{-\gamma}\partial_{k}x^{\mu}\partial_{\ell}x^{\nu}\gamma^{ik}\gamma^{j\ell}=\frac{1}{2}(d-2)\sqrt{-\gamma}\gamma^{ij}~~.
\hat{S}\[X^{\mu},c^{(0)},b_{\xi},\gamma]=-{\textstyle\frac{p}{2}}\intd\xi\\sqrt{|\gamma|}\\gamma^{-1}\left[e^{-\frac{2}{3}\phi}g_{\xi\xi}-(2\pi\alpha^{\prime})^{2}e^{\frac{4}{3}\phi}\left({\calG}^{(1)}{}_{\xi}\right)^{2}\right]\,,
{\bf\mathrm{P}_{1}}(n)=\left[\begin{array}{cccccc}{1}&{0}&{\cdots}&{0}&{|}&{1}\\{0}&{1}&{0}&{\vdots}&{|}&{2}\\{\vdots}&{0}&{\ddots}&{0}&{|}&{\vdots}\\{\vdots}&{\vdots}&{0}&{1}&{|}&{n-1}\\{0}&{\cdots}&{\cdots}&{0}&{|}&{n}\\\end{array}\right].
S_{\mathrm{\scriptsize~eff}}=\int\,d^{4}x\left({\bar{\psi}}\,(i\!\not{\!\partial}-m)\,\psi-{\frac{1}{2}}\phi(Z_{\phi}\partial^{2}+Z_{\mu}\mu^{2})\phi+\beta\,\phi({\bar{\psi}}\psi)-\frac{Z_{4}\beta^{4}}{4!}\phi^{4}\right)\,.
\begin{array}{ccccccc}{{\calL}_{\mathbf{I}}\Omega^{x}}&{=}&{\epsilon^{xyz}\Omega^{y}W_{\mathbf{I}}^{z}}&{;}&{{\calL}_{\mathbf{I}}\omega^{x}}&{=}&{\nablaW_{\mathbf{I}}^{x}}\\\end{array}
ds^{2}=\frac{1}{1-\xi^{0}\xi^{0}}d\xi^{0}d\xi^{0}-(1-\xi^{0}\xi^{0})\frac{d{\bfx}(I-{\bfx}^{\prime}{\bfx})^{-1}d{\bfx}^{\prime}}{1-{\bfx}{\bfx}^{\prime}}~,
\mathrm{det}\C\;=\;w_{N-3}\!\!\!\!\!\!\!\prod_{2\leqslantr<s\leqslantN-2}\!\!\!\!\!\!\!(z_{s}-z_{r})\prod_{2\leqslantI<J}\!\!(Z_{J}-Z_{I})\prod_{r,I=2}^{N-2}(Z_{I}-z_{r})^{-1}\,
\frac{d\tau}{d\alpha}\langle\int_{\tilde{c}}\frac{dz}{2\pii}(\frac{dz}{dw})J^{--}(z)~\tilde{G}_{0}^{+}\Phi_{D}(\alpha^{-1})~G_{0}^{+}\Phi_{A}(-\alpha^{-1})~G_{0}^{+}\Phi_{B}(-\alpha)~\tilde{G}_{0}^{+}\Phi_{C}(\alpha)\rangle
\theta_{g_{1}g_{2}h_{1}h_{2}}\left(\begin{array}{cc}{\tau_{1}}&{\epsilon}\\{\epsilon}&{\tau_{2}}\\\end{array}\right)=\sum_{n=0}^{\infty}\frac{2^{2n}}{(2n)!}\frac{d^{n}}{d\tau_{1}^{n}}\theta_{g_{1}h_{1}}(\tau_{1})\frac{d^{n}}{d\tau_{2}^{n}}\theta_{g_{2}h_{2}}(\tau_{2})\epsilon^{2n}
\partial_{\tau}^{2}X^{\mu}-c^{2}\partial_{\sigma}^{2}X^{\mu}+\Gamma_{\nu\rho}^{\mu}\left[\partial_{\tau}X^{\nu}\partial_{\tau}X^{\rho}-c^{2}\partial_{\sigma}X^{\nu}\partial_{\sigma}X^{\rho}\right]=0.
(r_{2n}\Psi_{1})\star\Psi_{2}=\Psi_{1}\star(l_{2n}\Psi_{2})\,\,,\quad\left(\frac{\partial}{\partialr_{2n}}\Psi_{1}\right)\star\Psi_{2}=-\Psi_{1}\star\left(\frac{\partial}{\partiall_{2n}}\Psi_{2}\right)\,\,.
\sum_{n=0}^{\infty}F(n)=\frac{1}{2}\Bigl(F(0)+F(\infty)\Bigr)+\int_{0}^{\infty}dxF(x)+\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}\Bigl(F^{(2k-1)}(\infty)-F^{(2k-1)}(0)\Bigr),
\rho(r)g(r)mg(r)^{-1}\rho(r)^{-1}=\left(\begin{array}{ccc}{-km_{11}k}&{-km_{12}k}&{km_{13}k}\\{-km_{12}k}&{-km_{22}k}&{km_{23}k}\\{km_{13}k}&{km_{23}k}&{-km_{33}k}\\\end{array}\right)
\tilde{G}_{\alpha\beta}(\sigma^{\alpha},X^{i}+\tilde{\Phi}^{i})=\eta_{\alpha\beta}+2kH_{\alpha\beta}+\partial_{\alpha}\tilde{\Phi}^{i}\partial_{\beta}\tilde{\Phi}^{i}+2kH_{ij}\partial_{\alpha}\tilde{\Phi}^{i}\partial_{\beta}\tilde{\Phi}^{j}
H\left(\begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\\end{array}\right)=\left(\begin{array}{c}{P_{-}P_{+}\qquad0}\\{0\qquadP_{+}P_{-}}\\\end{array}\right)\left(\begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\\end{array}\right).
0\stackrel{0}{\longrightarrow}\Omega^{0}(M,\xi)\stackrel{\nabla_{0}}{\longrightarrow}\dots\stackrel{\nabla_{m-2}}{\longrightarrow}\Omega^{m-1}(M,\xi)\stackrel{\nabla_{m-1}}{\longrightarrow}ker(S_{\nabla})\stackrel{0}{\longrightarrow}0
\left[\widehat{G_{1}}[\phi_{1},\ldots,\phi_{m}],\,\widehat{G_{2}}[\pi_{1},\ldots,\pi_{m}]\right]=i\hbar\left(\sum_{k=1}^{m}\intd^{n}\vec{x}\;{\frac{\deltaG_{1}}{\delta\phi_{k}}}\,{\frac{\deltaG_{2}}{\delta\pi_{k}}}\right)\!\!{}_{\mathrm{{op}}},
ds_{WB}^{2}\approxr_{0}^{2}[(1+\frac{2}{D-2}y^{2})(-d\tau^{2}+cosh^{2}\taud\Omega_{D-2}^{2})+(\frac{2}{D-2})^{2}(1+2\frac{D-1}{D-2}y^{2})dy^{2}]\.
S_{\mathrm{graviphoton}}=\intd^{4}x\sqrt{-g}\left[-\frac{1}{8}g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}^{(\alpha)}F_{\rho\sigma}^{(\alpha)}\right]\int_{-\infty}^{\infty}dz\,e^{-2k|z|},
\delta\Phi=-\frac{i}{48}{\bar{D}}^{4}D^{{\mathbfi}{\mathbfj}}(L_{{\mathbfi}{\mathbfj}}\Phi)+\frac{i}{48}{\bar{D}}^{4}(L_{{\mathbfi}{\mathbfj}}D^{{\mathbfi}{\mathbfj}}\Phi)\.
v_{i}(p^{+})=\frac{iN_{c}g^{2}}{4\pi^{2}}\intdk^{+}dk^{-}\frac{\calP}{(p^{+}-k^{+})^{2}}\frac{1}{k^{--}\frac{m_{i}^{2}}{k^{+}}-v_{i}(k^{+})+i\epsilon~\mathrm{sgn}(k^{+})}.
\gamma_{1}=\frac{1}{\sqrt{2}}(\Gamma_{1}+i\Gamma_{2}),\gamma_{2}=\frac{1}{\sqrt{2}}(\Gamma_{3}+i\Gamma_{3}),\gamma_{3}=\frac{1}{\sqrt{2}}(\Gamma_{5}+i\Gamma_{6}),\gamma_{4}=\frac{1}{\sqrt{2}}(\Gamma_{7}+i\Gamma_{8})
\langle\phi|x^{ml}\rangle=N^{1/2}(n)(\beta/2\pi)^{n/4}\int_{I_{n}}d^{n}\rho~(1-\beta\rho^{2}/2)^{1/2+\sqrt{1/4+n/8}}e^{ix\cdot\rho/\hbar}\phi(\rho)
\zeta=\frac{a\hat{\zeta}+b}{-\overline{{b}}\hat{\zeta}+\overline{{a}}},\qquad\eta=\frac{\hat{\eta}}{(-\overline{{b}}\hat{\zeta}+\overline{{a}})^{2}},\qquad|a|^{2}+|b|^{2}=1,
A_{\mu}^{AB}=\tau_{AB}^{+}\,\frac{1}{2\sqrt{2}}\,{\calA}_{\mu}^{1}\,+\,\tau_{AB}^{-}\,\frac{1}{2\sqrt{2}}\,{\calA}_{\mu}^{2}\,+\,\mathrm{26~non~singlet~fields}
\zeta_{0}^{(p)}(\nu)={\frac{1}{\pi\Gamma(\nu)}}\int_{0}^{\infty}dx\int_{0}^{1}dtt^{\nu-1}e^{-tx^{2}}{\frac{1}{(4\pit)^{(D-1)/2}}}\sum_{n=0}^{\infty}\breve{a}_{n}(x)t^{n}~~.
<\psi_{i}(z_{1})\psi^{j\dagger}(z_{2})\psi^{k\dagger}(z_{3}^{*})\psi_{l}(z_{4}^{*})>\to{\frac{(-\delta_{i}^{j}\delta_{l}^{k}+\delta_{l}^{j}\delta_{i}^{k})}{(2\pi)^{2}[(z_{1}-z_{4}^{*})(z_{2}-z_{3}^{*})]^{2/N}|\tau_{1}-\tau_{2}|^{2-4/N}}}
S=-\frac{1}{\lambda_{s}^{2}}\intd^{4}x~\sqrt{-g}~e^{-\phi}\biggl[R+g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi-\frac{1}{2}e^{2\phi}g^{\alpha\beta}\partial_{\alpha}\psi\partial_{\beta}\psi\biggr].
a_{0}=(\frac{2k_{5}^{2}}{3\sqrt{2\pi}})^{\frac{1}{2}}\sigma^{-\frac{1}{2}}(T_{44}-\frac{V}{4k^{2}}+\sqrt{\frac{\pi}{2}}\frac{\sigma}{k^{2}}\Lambda)^{\frac{1}{2}},
\phi_{1GE}^{(s)}(x)={\frac{g^{2}m^{2}C_{F}\phi(x)}{4\pi{(P^{+})}^{2}}}{\frac{1}{({\calE}-\omega(x)-\omega(1-x)+i\epsilon)^{2}}}\[x^{-2}+(1-x)^{-2}].
L_{\mu\nu}^{-1}(\tau,\tau^{\prime})-\frac{{\calF}_{\mu}^{\alpha}}{2}\int_{0}^{1}\epsilon(\tau-\tau^{\prime\prime})L_{\alpha\nu}^{-1}(\tau^{\prime\prime},\tau^{\prime})d\tau^{\prime\prime}=\eta_{\mu\nu}\delta(\tau-\tau^{\prime})
\begin{array}{l}{\Phi(x_{+},\theta)~\rightarrow~\mathrm{sdet}_{i}^{\frac{1}{3}}\overline{{\Phi}}(x_{-}^{\prime},\bar{\theta}^{\prime})}\\{{}}\\{\overline{{\Phi}}(x_{-},\bar{\theta})~\rightarrow~\overline{{\mathrm{sdet}}}_{i}^{\frac{1}{3}}\Phi(x_{+}^{\prime},\theta^{\prime})}\\\end{array}
(ds)^{2}=-\frac{1}{\sqrt{\Lambda}}\left(1+\frac{2}{3}\delta\mathrm{cos}R\right)\mathrm{sin}^{2}Rd^{2}T+\frac{1}{\Lambda}\left(1-\frac{2}{3}\delta\mathrm{cos}R\right)d^{2}R+\frac{1}{\Lambda}\left(1-2\delta\mathrm{cos}\right)d^{2}\Omega_{(2)}
\Sigma_{(daisy)}=\sum_{n=0}^{\infty}{\frac{1}{n!}}(m^{2}-M^{2}+\bar{G}(M^{2}))^{n}({\frac{\partial}{\partial{M^{2}}}})^{n}{\bar{G}(M^{2})}=\bar{G}(m^{2}+\bar{G}(M^{2}))
\left.\;J^{(N)}\left(n;\nu_{1},\ldots,\nu_{N}\right)\right|_{\Sigma\nu_{i}=n,\;\;m_{i}=m}=\frac{1}{m^{\Sigma\nu_{i}}}\;\Psi\left(\left\{\overline{{\overline{{c}}}}_{jl}\right\}\right).
h_{+p\dot{q}}^{1-}=\gamma_{p\dot{q}}^{i}h_{+}^{1-i}+\gamma_{p\dot{q}}^{ijk}h_{+}^{1-ijk},\h_{-p\dot{q}}^{2+}=\gamma_{p\dot{q}}^{i}h_{+}^{2+i}+\gamma_{p\dot{q}}^{ijk}h_{+}^{2+ijk},\
\{\widetilde{{\cal{Q}}}_{r},\overline{{\widetilde{{\cal{Q}}}}}_{s}\}={\cal{M}}_{rs}=(C\Gamma_{\widetilde{\mu}_{1}\widetilde{\mu}_{2}}^{(12)})_{rs}\,M^{[\widetilde{\mu}_{1}\widetilde{\mu}_{2}]}\,,
{\calA}(x)=\frac{1}{8\pi^{2}}[R^{ab}\mathrm{\tiny~\wedge~}R_{ab}+2(T^{a}\mathrm{\tiny~\wedge~}T_{a}-R_{ab}\mathrm{\tiny~\wedge~}e^{a}\mathrm{\tiny~\wedge~}e^{b})],
\tilde{D}_{\mu\nu}\approxD_{\mu\nu}+iD_{\mu\sigma}\Pi^{\sigma\rho}D_{\rho\nu}\approx\frac{z_{\Phi}}{p^{2}-p_{3}^{2}-z_{\Phi}M^{2}+i\epsilon}\left[-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p_{3}^{2}+z_{\Phi}M^{2}}\right].
\sum_{\epsilon_{1}^{\prime},\epsilon_{2}^{\prime}}R_{\epsilon_{1}\epsilon_{2}}^{\epsilon_{1}^{\prime}\epsilon_{2}^{\prime}}(\zeta_{1}/\zeta_{2})\Phi_{\epsilon_{1}^{\prime}}(\zeta_{1})\Phi_{\epsilon_{2}^{\prime}}(\zeta_{2})=\Phi_{\epsilon_{2}}(\zeta_{2})\Phi_{\epsilon_{1}}(\zeta_{1}),
\triangle(\tau^{\prime},\tau)^{ab}=\theta^{ab}\frac{1}{2\pi}\sum_{n}ne^{in(\tau^{\prime}-\tau)}\Rightarrow\triangle(\tau,\tau^{\prime})_{ab}^{-1}=\theta_{ab}\frac{1}{2\pi}\sum_{n}\frac{1}{n}e^{in(\tau-\tau^{\prime})}\.
S=\int\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}\left(B_{\mu\nu}^{a}(x)R_{\lambda\rho}^{a}(x)+\underbrace{\phi^{ab}(x)B_{\mu\nu}^{a}(x)B_{\lambda\rho}^{b}(x)}_{constraint\:term}\right),
{\calL}=-\frac{{pe^{2}}}{4}F_{\mu\nu}^{2}+\frac{{qe^{2}}}{2}\varepsilon_{\mu\nu\lambda}A^{\mu}F^{\nu\lambda}+\overline{\psi}i\gamma^{\mu}\left({\partial_{\mu}-ieA_{\mu}-iB_{\mu}}\right)\psi\,-\frac{1}{{2g}}B_{\mu}^{2}-m\overline{\psi}\psi+J^{0}A_{0}\,.
\begin{array}{c}{\Omega_{i_{1}i_{2}i_{3}i_{4}p}\Omega_{i_{1}i_{2}i_{3}i_{4}q}=5\delta_{pq}\quad,}\\{\displaystyle\Omega_{i_{1}i_{2}i_{3}ab}\Omega_{i_{1}i_{2}i_{3}pq}={\frac{1}{2}}(\delta_{ap}\delta_{bq}-\delta_{aq}\delta_{bp}+f_{abi}f_{pqi})\quad,}\\\end{array}
\times\lambda_{1}^{l_{2}}\lambda_{2}^{r_{1}-l_{1}}\prod_{j=0}^{r_{2}-l_{2}-1}(e^{u}\lambda_{2}q^{-j}-e^{-u}\lambda_{1}q^{j})\prod_{j=0}^{l_{1}-1}(e^{u}\lambda_{1}q^{-j+r_{2}-l_{2}}-e^{-u}\lambda_{2}q^{j+l_{2}-r_{2}})
dJ^{-}=\Bigl(+e\Bigr)\times\Bigl(e^{-2\pi\lambda(k^{+},k_{\bot})}\Bigr)\times\Bigl(2\Bigr)\times\Biggl({\frac{-eE(x^{+})dx^{-}}{2\pi}}\Biggr)\times\Biggl({\frac{d^{2}p_{\bot}}{(2\pi)^{2}}}\Biggr)\;.
\frac{\bar{b}(\pmp)}{\bar{a}(\pmp)}=e^{\pm\eta(p^{0})\bar{\Theta}},\\frac{\bar{d}(\pmp)}{\bar{c}(\pmp)}=e^{\mp\eta(p^{0})\bar{\Theta}},\\eta(p^{0})=\frac{p^{0}}{|p^{0}|},
\frac{\rho_{\mu}(x)}{\rho_{\nu}(x)}=\frac{q^{I(\Phi_{1}+\Phi_{2}+\mu)}+(-1)^{\mu+\nu}xq^{I(\Phi_{1}+\Phi_{2}+\nu)}}{xq^{I(\Phi_{1}+\Phi_{2}+\mu)}+(-1)^{\mu+\nu}q^{I(\Phi_{1}+\Phi_{2}+\nu)}}\,,~~~~\mu=\nu\pm1\,.
G_{r}^{(2)}=x^{\frac{r-1}{2}}\cdot\left(\frac{U_{\nu-r}U_{1}}{U_{\nu-r+1}}\right)^{r}\cdot\left[\left(\frac{U_{\nu-r+1}}{U_{\nu-r}}\right)^{r}\cdot\frac{U_{\nu-r}}{U_{\nu}}\right]=U_{1}\cdot\frac{U_{\nu-r}}{U_{\nu}}.
\hbar^{2}\frac{\delta^{2}}{\delta\phi\left(w\right)\delta\phi\left(z\right)}\Psi[\phi]=\left(\sqrt{m^{2}-\nabla_{w}^{2}}\,\phi\left(w\right)\right)\left(\sqrt{m^{2}-\nabla_{z}^{2}}\,\phi\left(z\right)\right)\,\Psi[\phi]-\hbar\sqrt{m^{2}-\nabla_{z}^{2}}\,\delta\left(w-z\right)\,\Psi[\phi].
Y=\left(\begin{matrix}{0}&{y_{1}}&{0}&{\dots}\\{0}&{0}&{y_{2}}&{\dots}\\{\vdots}&{\vdots}&{\ddots}&{\ddots}\\\end{matrix}\right)\quadZ=\left(\begin{matrix}{0}&{0}&{0}&{\dots}\\{\zeta_{1}}&{0}&{0}&{\dots}\\{0}&{\zeta_{2}}&{0}&{\dots}\\{\vdots}&{\vdots}&{\ddots}&{\ddots}\\\end{matrix}\right)
\begin{aligned}{a}&{{}=a_{4}-\frac{1}{3}a_{2}^{2},}\\{b}&{{}=\frac{1}{27}a_{2}\left(2a_{2}^{2}-9a_{4}\right),}\\{\delta}&{{}=a_{4}^{2}\left(4a_{4}-a_{2}^{2}\right)}\\\end{aligned}
E_{H.F.}^{(p)}(\omega)=\sum_{n=0}^{p}{\frac{C_{n}}{4^{n}}}{\frac{\Gamma(p+n/2+1/2)}{\Gamma(3n/2+1/2)\Gamma(p-n+1)}}\omega({\frac{1}{\omega^{3}}})^{n}\,,
\begin{array}{c}{\mathcal{E}(x)\rightarrow\mathcal{E}^{V}(x)=\Lambda\mathcal{E}(\Lambda^{-1}(x-a)),}\\{\mathcal{E}(x)\rightarrow\mathcal{E}^{TS}(x)=F(\Lambda)\mathcal{E}(\Lambda^{-1}(x-a)),}\\{\mathcal{E}(x)\rightarrow\mathcal{E}^{S}(x)=S(\Lambda)\mathcal{E}(\Lambda^{-1}(x-a)),}\\\end{array}
\sum_{p=0}^{\infty}p^{2k}\alpha_{p}^{-2k-l}\sim{\frac{\Gamma\left(k+{\frac{1}{2}}\right)\Gamma\left({\frac{l}{2}}-{\frac{1}{2}}\right)}{2\Gamma\left(k+{\frac{l}{2}}\right)}}x^{1-l},\;k=1,2,...,
\left(\partial_{(--)}U_{{\calL}}^{kj}\right)U_{{\calL}}^{ki}=\left(\partial_{(--)}G_{{\calR}}^{jk}\right)G_{{\calR}}^{ik}~,\qquad\left(\partial_{(++)}U_{{\calR}}^{kj}\right)U_{{\calR}}^{ki}=\left(\partial_{(++)}G_{{\calL}}^{jk}\right)G_{{\calL}}^{ik}~.
{}\{\bfg}_{Y}\left(t^{\mu}\frac{\partial}{\partialx^{\mu}},\frac{\partial}{\partialx^{a}}\right)=0,\\foralla\Rightarrow\{\bfg}_{Y}\left(t^{\mu}\frac{\partial}{\partialx^{\mu}},t^{\mu}\frac{\partial}{\partialx^{\mu}}\right)<0,
P_{b}^{(1,2)}=P_{b}(x_{1,2})=(\frac{g}{\tilde{g}})^{\frac{1}{2}}(\frac{-3\pi^{2}}{1440a^{4}})=\frac{\eta^{4}}{\alpha^{4}}(\frac{-3\pi^{2}}{1440a^{4}})=\frac{-\eta^{4}\Lambda^{2}}{3}\frac{\pi^{2}}{1440a^{4}},
\tilde{T}_{\cdot\,\nu}^{\mu\,\cdot}=\frac{i}{2}\left[\overline{{\psi}}\gamma^{\hat{\alpha}}e_{\hat{\alpha}}^{\mu}\partial_{\nu}\psi-(\overline{{\partial_{\nu}\psi}})\gamma^{\hat{\alpha}}e_{\hat{\alpha}}^{\mu}\psi\right]
{\mathcalG}_{l+\nu}(r^{\prime\prime},r^{\prime};E)=\frac{u_{l,\nu}^{(<)}(r_{<})u_{l,\nu}^{(>)}(r_{>})}{p(r^{\prime})\;W\left\{u_{l,\nu}^{(<)},u_{l,\nu}^{(>)}\right\}(r^{\prime})}\;,
\int\!\!d^{4}x_{_{E}}{\overline{F}}_{\mu\nu}{\overline{F}}_{\mu\nu}=2\int\!\!d^{4}x_{_{E}}\left\{\sum_{\alpha=1}^{2}\left(\left[{\frac{1}{\vert{\boldsymbol{\nabla}}\vert}}\dot{\beta}_{\alpha}\right]^{2}+\beta_{\alpha}^{2}\right)+\rho^{2}\right\}.
\|\widetilde{c}_{jl}\|=\left(\begin{array}{cccc}{1}&{0}&{\widetilde{c}^{\prime}}&{0}\\{0}&{1}&{0}&{\widetilde{c}^{\prime}}\\{\widetilde{c}^{\prime}}&{0}&{1}&{\widetilde{c}}\\{0}&{\widetilde{c}^{\prime}}&{\widetilde{c}}&{1}\\\end{array}\right),
Tr_{\{\eta,\overline{{\eta}}\}}(I)=Tr_{\{\eta,\overline{{\eta}}\}}(\theta)=Tr_{\{\eta,\overline{{\eta}}\}}(\overline{{\theta}})=0\;\;,Tr_{\{\eta,\overline{{\eta}}\}}(\overline{{\theta}}\theta)=q\;.
\Gamma=\frac{1}{2}\intd^{3}xd^{3}y\,A_{i}(x)\left(K(x-y)+\frac{\lambda}{2}\,K^{2}(x-y)\right)A_{i}(y)+(A_{i}\longleftrightarrowA_{i}+\partial_{i}\varphi)-\frac{\lambda}{4}\intd^{3}x\,F_{ij}F_{ij}.
\intd^{4}q{\frac{1}{[q^{2}]^{1+n\epsilon}(q-p)^{2}}}={\frac{\Gamma(1-\epsilon)\Gamma((n+1)\epsilon)\Gamma(1-(n+1)\epsilon)}{\Gamma(1+n\epsilon)\Gamma(2-(n+2)\epsilon)}}{\frac{1}{[p^{2}]^{(n+1)\epsilon}}}
e^{-4U}=\,1-2\Bigl(\frac{q^{I}}{|\vec{x}-\vec{x}_{1}|}-\frac{q^{I}}{|\vec{x}-\vec{x}_{2}|}\Bigr)^{2}=e^{4\phi}\,\qquadH_{i}^{I}={\frac{(x-x_{2})^{i}q^{I}}{|\vec{x}-\vec{x}_{2}|^{3}}}-{\frac{(x-x_{1})^{i}q^{I}}{|\vec{x}-\vec{x}_{1}|^{3}}}\,
\varepsilon^{ijk}{^{\pm}}A_{iA}^{\\B}{^{\pm}}A_{jB}^{\\C}{^{\pm}}A_{kC}^{\\A}={\frac{1}{2}}\varepsilon^{ijk}\big(A_{iA}^{\\B}A_{jB}^{\\C}A_{kC}^{\\A}\pm{\frac{i}{2}}\varepsilon^{ABCD}A_{iA}^{\\E}A_{jEB}A_{kCD}\big).
W^{(m,m^{\prime})}\left(\left.\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right|v\right)=\sum_{\sigma\in\mathfrak{S}_{m}}\mathrm{sgn}\,\sigma\prod_{j=1}^{m}W_{II}^{(1,m^{\prime})}\left(\left.\begin{array}{cc}{b_{j}^{\sigma}}&{a_{j}}\\{b_{j-1}^{\sigma}}&{a_{j-1}}\\\end{array}\right|z-\frac{m+1}{2}+j\right),
\frac{ds^{2}}{\alpha^{\prime}}\approx\frac{R^{3/2}}{3U_{T}^{1/2}}\frac{dU^{2}}{U-U_{T}}+\frac{3U_{T}^{1/2}}{R^{3/2}}(U-U_{T})dt^{2}\equivdr^{2}+r^{2}d\theta^{2}\equivdX^{2}+dY^{2},
{\calL}_{\mathrm{GF}}=-\frac{1}{2}\Bigl(\partial_{\mu}A^{\mu}+\sqrt{N}\frac{\xi(\partial^{2})}{G}\phi\Bigr){\frac{1}{\xi(\partial^{2})}}\Bigl(\partial_{\nu}A^{\nu}+\sqrt{N}\frac{\xi(\partial^{2})}{G}\phi\Bigr),
S=\intd\taud\sigma\left[\frac{\mathrm{det~}(\partial_{\mu}{\tilde{x}}^{M}G_{MN}(\tilde{x})\partial_{\nu}{\tilde{x}}^{N})}{\tilde{E}}-\left(\frac{\gamma}{\alpha^{\prime}}\right)^{2}\tilde{E}\right]
\frac{\partialP_{c}(\phi,t|\chi)}{\partialt}=\frac{1}{2}\frac{\partial}{\partial\phi}\left(\frac{H^{3/2}(\phi)}{2\pi}\,\frac{\partial}{\partial\phi}\Bigl(\frac{H^{3/2}(\phi)}{2\pi}P_{c}(\phi,t|\chi)\Bigr)+\frac{V^{\prime}(\phi)}{3H(\phi)}\,P_{c}(\phi,t|\chi)\right)\.
{\calF}^{(n+1)}\=\{\calF}^{(n)}\+\\frac{1}{(n+1)(2n+1)}\\frac{\partial^{\;2}}{\Box}\,{{\calF}^{(n)}}\;^{\prime}\-\\frac{1}{n+1}\\frac{\partial}{\Box}\\partial\cdot{\calF}^{(n)}\,
H={\frac{1}{2m_{(+)}}}\intd^{2}x|i\hbarD_{i(+)}^{(1)}\psi_{(+)}^{(1)}(x)|^{2}+{\frac{1}{2m_{(-)}}}\intd^{2}x|i\hbarD_{i(-)}^{(1)}\psi_{(-)}^{(1)}(x)|^{2}
\{A_{i}^{1},B_{i}^{2}\}=-r_{+}A_{i}^{1}B_{i}^{2}-A_{i}^{1}B_{i}^{2}r_{+}+A_{i}^{1}r_{-}B_{i}^{2}+B_{i}^{2}r_{+}A_{i}^{1}.
C\Big[\begin{array}{c}{b_{3}+n_{1}b_{1}}\\{\tilde{n}_{1}b_{1}}\\\end{array}\Big]=(-1)^{n_{1}},\qquadC\Big[\begin{array}{c}{b_{4}+m_{2}b_{2}}\\{\tilde{m}_{2}b_{2}}\\\end{array}\Big]=(-1)^{m_{2}}.
u^{\mu}(p_{3},0)\mid_{m\rightarrow0}=\left(\begin{matrix}{p_{3}}\\{0}\\{0}\\{{\frac{p_{3}^{2}}{E_{p}}}}\\\end{matrix}\right)\equiv\left(\begin{matrix}{E_{p}}\\{0}\\{0}\\{E_{p}}\\\end{matrix}\right)\quad,
\Omega(p)=\frac{4\Lambda^{2}c^{2}}{\pi}\left[2\lambda_{0}+8\pi\frac{\epsilon}{\Lambda^{2}}\partial_{1}^{2}I_{1}(0)\right]e^{-8\piI_{1}(0)}+\frac{4c^{2}\epsilon}{\pi}e^{-8\piI_{1}(0)}p_{1}^{2}
\Phi_{TT}^{(l=1)}\left[\Gamma^{[r,0]}\right]=I_{0}^{(r)}(k^{2})+\sum_{s=1}^{r}\,I_{s}^{(r)}(k^{2})\left(\frac{\Lambda^{2}}{k^{2}+u_{r,2s}^{\prime}\Lambda^{2}}\right)\,.
0=\biggl|\begin{matrix}{A_{jm}(x)}&{A_{jm}(y)}\\{A_{mk}(x)}&{A_{mk}(y)}\\\end{matrix}\biggr|-\biggl|\begin{matrix}{\psi_{jm}(x)}&{\psi_{jm}(y)}\\{\psi_{mk}(x)}&{\psi_{mk}(y)}\\\end{matrix}\biggr|A_{jmk}(x+y).
{\bfE}(x,y,z,t)=q\frac{({\bfR}-R\frac{{\bfV}}{c})(1-\frac{V^{2}}{c^{2}})}{(R-{\bfR}\frac{{\bfV}}{c})^{3}}+q\frac{[{\bfR},[({\bfR}-R\frac{{\bfV}}{c}),\frac{{\bf{\dot{V}}}}{c^{2}}]]}{(R-{\bfR}\frac{{\bfV}}{c})^{3}}.
q^{(m-n)(m-j)}\left[\begin{array}{l}{j+m}\\{m-n}\\\end{array}\right]_{q^{2}}^{1/2}\left[\begin{array}{l}{j-n}\\{m-n}\\\end{array}\right]_{q^{2}}^{1/2}P_{j+n}^{n-m,-m-n}(c^{*}c;q^{2})c^{m-n}(a^{*})^{n+m}.
V\=\V_{1}+\gamma_{15}\gamma_{25}V_{3}+\gamma_{1}^{\mu}\gamma_{2}^{\nu}(g_{\mu\nu}^{LL}V_{2}+g_{\mu\nu}^{TT}U_{4}+\frac{x_{\mu}^{T}x_{\nu}^{T}}{x^{T2}}T_{4})\,
\left(\varepsilon_{\bar{f}}^{f}\right)_{\alpha}\to\frac{i}{\sqrt{2}}\left[\delta_{1}^{f}\left(\delta_{\alpha}^{1}+i\delta_{\alpha}^{2}\right)\bar{\varepsilon}_{\bar{f}}-\delta_{2}^{f}\left(\delta_{\alpha}^{1}-i\delta_{\alpha}^{2}\right)\varepsilon_{\bar{f}}\right]\;.
\left(\begin{array}{ccc}{0}&{\kappa}&{0}\\{-\kappa}&{0}&{\tau}\\{0}&{-\tau}&{0}\\\end{array}\right)\Longrightarrow\left(\begin{array}{cccc}{0}&{-\kappa(s)}&{0}&{0}\\{\kappa(s)}&{0}&{-\tau_{1}(s)}&{0}\\{0}&{\tau_{1}(s)}&{0}&{-\tau_{2}(s)}\\{0}&{0}&{\tau_{2}(s)}&{0}\\\end{array}\right)~,
\delta_{a}\pi_{c\alpha_{1}}^{*(b)}=\left.\left(\pi_{c\alpha_{1}}^{*(b)},S\right)_{a}\right|_{\mathrm{ghosts}=0}=\left(-\right)^{\epsilon_{\alpha_{1}}+1}\delta_{ab}\varepsilon_{cde}\eta_{e\alpha_{1}}^{*(d)},
\int_{0}^{\infty}\frac{d\tau}{\tau^{2}}\,\mathrm{exp}\left(-\frac{p}{2\tau}\right)\Gamma\left(q,\frac{c\tau}2\right)=\left(\fracp4\right)^{\frac{q-2}2}c^{\fracq2}K_{q}(\sqrt{pc})\,.
\int\prod_{j=1}^{g}d\nu_{j}d\bar{\nu}_{j}d\hat{\nu}_{j}d\hat{\bar{\nu}}_{j}\,\,\mathrm{e}^{\int_{\Sigma_{g}}{\calL}_{5}}=\left|\prod_{j=1}^{g}\delta(a_{j})\,da_{j}\right|^{2},
exp~({\frac{i\theta}{4\pi^{2}}}\int\epsilon_{\mu\nu\lambda}L_{\mu}^{3}\partial_{\lambda}^{3}){}~=~\int_{a_{\mu}}exp~({\frac{i\theta}{4\pi^{2}}}(2\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\lambda}^{3}~-~\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}))
LF_{\beta}(\phi)_{R}=-\,{\frac{{M_{\beta}(\phi)}^{3}}{6\pi}}-{\frac{M_{\beta}(\phi)Li_{2}({e^{-M_{\beta}(\phi)\beta}})}{\pi\,{\beta}^{2}}}-{\frac{Li_{3}({e^{-M_{\beta}(\phi)\beta}})}{\pi\,{\beta}^{3}}}.
\hat{f}_{x}\equiv\sum_{j=1}^{m}(\frac{\partialf}{\partialx_{j+m}}\frac{\partial}{\partialx_{j}}-\frac{\partialf}{\partialx_{j}}\frac{\partial}{\partialx_{j+m}});\quad\hat{D}\equiv\sum_{A=1}^{2m}x_{A}\frac{\partial}{\partialx_{A}}.
\{Q(f),Q(g)\}=Q([f,g])-\frac{is}{\pi}\left<A_{\rho}A_{\rho}\right>\oint_{\partialD_{2}}d\varphi(f^{\rho}\partial_{\varphi}g^{\rho}-f^{\varphi}\partial_{\varphi}g^{\varphi}),
\mathrm{\bf~W}^{\prime}(\rho)+\mathrm{\bf~W}^{2}(\rho)+\mathrm{\bf~1}{\tilde{E}}_{1}^{(0)}=\left(\begin{array}{llll}{\frac{m^{2}-\frac{1}{4}}{\rho^{2}}}&{\;\;\;\frac{-2F}{\rho}}\\{\frac{-2F}{\rho}}&{\frac{(m+1)^{2}-\frac{1}{4}}{\rho^{2}}}\\\end{array}\right),
\eta_{\mathrm{j}}^{2}(n)\simeq\frac{n^{2}}{n_{\mathrm{b}}^{2}}\biggl[\biggl(\frac{n}{n_{\mathrm{inf}}}\biggr)^{2}+\biggl(\frac{n_{\mathrm{b}}}{n}\biggr)^{2}-1\biggr]\biggl[n^{2}\biggl(\frac{n}{n_{\mathrm{inf}}}\biggr)^{2}+\frac{6}{\eta_{0}^{2}}\biggr]^{-1}.
\begin{array}{ll}{g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},}\\{\Lambda(\phi)=\Lambda(\phi_{0})+\Lambda^{\prime}(\phi_{0})\phi_{(1)},}\\{T_{\mu\nu}=T_{(0)\mu\nu}+T_{(1)\mu\nu},}\\{\phi=\phi_{0}+\phi_{(1)},}\\\end{array}
U=\frac{\hat{\omega}+m+\Gamma_{(2n)}^{i}\hat{p}{}_{i}}{\sqrt{2\hat{\omega}(\hat{\omega}+m)}}=\frac{1}{\sqrt{2\hat{\omega}(\hat{\omega}+m)}}\left(\begin{array}{cc}{(\hat{\omega}+m)}&{\sigma_{(2n)}^{i}p_{i}}\\{-\sigma_{(2n)}^{i}p_{i}}&{(\hat{\omega}+m)}\\\end{array}\right)
k^{2}(r,l,E)=(1-{\frac{2M}{r}}+{\frac{Q^{2}}{r^{2}}})^{-1}[(1-{\frac{2M}{r}}+{\frac{Q^{2}}{r^{2}}})^{-1}E^{2}-{\frac{l(l+1)}{r^{2}}}-m^{2}].
D[\varphi]=-\hat{c}_{\alpha_{+}}\intd^{d}x\intd^{d}x^{\prime}\frac{\varphi_{0}(\mathbf{x})\varphi_{0}(\mathbf{x}^{\prime})}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{2\alpha_{+}}}\mathrm{~.}\tag{4.16}
S~=~\oint~d\tau\int~d^{2}r_{\perp}~\left({\frac{1}{2}}{\Omega}^{-}~{\vec{\nabla}}^{2}{\dot{\Omega}}^{+}-{\frac{1}{2}}{\Omega}^{+}~{\vec{\nabla}}^{2}{\dot{\Omega}}^{-}+{\dot{k}}^{+}{\Omega}^{-}-{\dot{k}}^{-}{\Omega}^{+}\right).
\gamma_{1}=-\tau+\sum_{a=1}^{N}\frac{1}{2}\mathrm{sign}(\pi_{a})\int\frac{\lambda_{a}^{N-1}d\lambda_{a}}{\left|A(\lambda_{a})\right|}\sqrt{\left|\frac{A(\lambda_{a})}{\lambda_{a}^{N+1}-(\alpha-1)\lambda_{a}^{N}+(1-\alpha+\beta)\lambda_{a}^{N-1}}\right|}
\overline{{\psi}}D\!\!\!/\psi=\overline{{\psi}}_{\delta}^{\prime}[\partial\!\!\!/+i\sqrt\pi\partial\!\!\!/(\widetilde\eta-\delta\widetilde\eta)+i\sqrt\pi\gamma_{5}\partial\!\!\!/(\sigma-\delta\sigma)]\psi_{\delta}^{\prime}.
I=\intdt\left(\frac12q^{2}-\frac{1}{2l}\varsigma^{1}{\varsigma^{1}}^{\prime}-\frac{1}{2l}\varsigma^{2}{\varsigma^{2}}^{\prime}-\frac{g}{2q^{2}}+\frac{\sqrtg}{2lq^{2}}[\varsigma^{1},\varsigma^{2}]+\frac{3}{cl^{2}}T_{B}q^{2}-\frac{3}{cl^{2}}T_{F}\varsigma^{1}q\right)\,.