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OleehyO
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latex-formulas

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\begin{align*} t^\ell_{0n} = \left(\frac{(2\ell)!} {n!{(2\ell-n)!}}\right)^{1/2}\, e^{-i\ell\phi} \frac{\tau^n}{(1+|\tau|^2)^\ell}\,.\end{align*}
\begin{align*}C^{-1}_{\mu\nu} = -\frac{1}{m^2} \left (\begin{array}{cccc} 0 & \partial^x_1 & \partial^x_2 & -1 \\ \partial^x_1 & 0 & -m & 0 \\ \partial^x_2 & m & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right ) \delta^2 (x-y) .\end{align*}
\begin{align*}\left\{ \phi( x ) , \phi( y )\right\} = \phi'(x)\phi'(y)\epsilon(x-y) +{1 \over \alpha}\left\{ \phi'(y) - \phi'(x) \right\}\delta(x-y) -{1 \over \alpha^2}\delta'(x-y)\end{align*}
\begin{align*}\tilde\partial_{i}(g_{0}, \dots, g_{n}) = \begin{cases}(g_{0}, \dots, g_{i-1}, g_{i}g_{i+1}, g_{i+2}, \dots, g_{p}) & i=0, \dots p-1, \\(g_{0}, \dots, g_{p-1}) & i=p.\end{cases}\end{align*}
\begin{align*} \rho = \theta_1 R_{12} + \theta_2 R_{34} + \theta_3 R_{56} + \theta_4 R_{78}~,\end{align*}
\begin{align*}\gamma_{ijkl \ldots }=\gamma_{i}\gamma_{j}\gamma_{k}\gamma_{l} \ldots\end{align*}
\begin{align*}S_{nm}=-\frac{1}{\sqrt{nm}} \oint\frac{dzdw}{(2\pi i)^2 z^{n}w^{m}}\frac{f'(-z)f'(-w)}{(f(-z)-f(-w))^2}\,,\end{align*}
\begin{align*}\chi(E) = \sum_{\{x_{k}\in M \vert s(x_{k})=0\}}(\pm 1), \end{align*}
\begin{align*}\delta^{[r]}(M)=\prod_{a=1}^{n_r}\delta\big({\rm tr}_K\,T^{(-r)}_aM\big)\ .\end{align*}
\begin{align*}J^{\alpha}{}_{\beta}(z)=J^{\mu}(z)(\gamma_{\mu})^{\alpha}{}_{\beta}\,,\end{align*}
\begin{align*}L^{f}_{gh}(x_{f})=J_{\psi}L_{gh}(x)=J_{\psi}<(e^{\mu}\partial_{\mu}C)^{+},e^{\nu}\partial_{\nu}C>=J_{\psi}S(B_{f})^{2}(D_{l}C_{f})^{+}D_{l}C_{f}.\end{align*}
\begin{align*}S = \int \left[-\frac{1}{4}tr F^2 -\frac{1}{2\xi }tr(\partial ^{\mu }A_\mu )^2 -i tr(\partial ^\mu \bar{C})(\nabla _\mu C)\right].\end{align*}
\begin{align*}I^{\pm}_{\omega \omega'}=\int_{-\infty}^{\infty}d\bar{v} e^{-i\omega v(\bar{v}) \pm i\omega' \bar{v}} .\end{align*}
\begin{align*}\mathcal{S}_{BF}=\int\limits_{M}\;X^i\;F_i\;,\end{align*}
\begin{align*}\Omega^q_\epsilon(W) := \bigwedge\nolimits_K^q(K^n)^\vee \otimes_K F_\epsilon(W).\end{align*}
\begin{align*}P_{-}\left( t,x,y\right) =\sum_{E_{n}<0}\zeta _{n}^{+}\left( t,x\right)\zeta _{n}\left( t,x\right) . \end{align*}
\begin{align*}U = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right)~.\end{align*}
\begin{align*}\mu_1=\sin\theta,\ \ \mu_2=\cos\theta\sin\psi_1,\ \ \mu_3=\cos\theta\cos\psi_1\sin\psi_2,\ \ \mu_4=\cos\theta\cos\psi_1\cos\psi_2\sin\psi_3.\end{align*}
\begin{align*}{\int}d^4x\,e\,\left(\,\sigma\,,\,{\bar{\psi}\psi}\,\right)\,\,,\end{align*}
\begin{align*}Q^2=H \end{align*}
\begin{align*}\mid B\rangle =\mid NS\rangle \otimes \mid NS\rangle +\mid R\rangle \otimes\mid R\rangle \end{align*}
\begin{align*}iG^{[s]}(x,x')_{mn}^{ab}=\int\limits_0^\infty\,dT\int\limits_{x(0)=x',x(T)=x}Dx(t) exp\left({-1\over 4}\int\limits_0^Tdt\,\dot{x}^2\right)\Phi^{[s]}_{mn}[C^{xx'}]U[C^{xx'}]^{ab},\end{align*}
\begin{align*}ds^2 = - \left( dt + \beta \sum_{i,j=1}^6 J_{ij} x^i dx^j \right)^2 + \sum_{i=1}^{\sharp} ( dx^i )^2,\end{align*}
\begin{align*}k^2(t)=\frac{1}{2\sqrt{2}}\ e^{\alpha (t)x}\left(\alpha (t)^2+2x^2\alpha '(t)^2+2x\alpha ''(t)\right)k^2_0\end{align*}
\begin{align*}ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu} = -f(r) dT^2 + \frac{dr^2}{f(r)} + r^2 d\Sigma^2_K,\end{align*}
\begin{align*}\Omega^q_\epsilon(W) = \bigoplus_{1\leq k_{1}< \dots < k_{q}\leq n} F_\epsilon(W) dx_{k_1}\dots dx_{k_q}.\end{align*}
\begin{align*}A_{\mu}\equiv 2\sqrt{\omega}\tilde{A}_{\mu}; \qquad e=\frac{\tilde{e}}{2\sqrt{\omega}} \end{align*}
\begin{align*}\Big\langle : e^{i\varphi(f)} :_M e^{i\varphi(g)} :_M \Big\rangle_{C_{m=0}}\; := \;\lim_{m \rightarrow 0}\Big\langle : e^{i\varphi(f)} :_M e^{i\varphi(g)} :_M \Big\rangle_{C_{m}} \; .\end{align*}
\begin{align*}y_{1,2} =(- z)^{1/2} (1 + z)^{1/4} U_{1,2}(z)\ ,\end{align*}
\begin{align*}{\displaystyle{V=(\frac{1}{k} \partial_{\phi}-\frac{1}{a}\partial_{z})}}\end{align*}
\begin{align*}K(\varphi )=3\left(\frac{B'(\varphi)}{B(\varphi)}\right)^2-2\end{align*}
\begin{align*}V_l{}^{ij} = e^{2U} \delta_{l\, [i} \partial _j\left ( \ln \chi^{(1)}\chi^{(2)}\right ) \qquad V_l{}^{i4} = {1\over 2} e^{2U} \epsilon_{lim}\partial _m \left ( \ln {\chi^{(1)}\over\chi^{(2)}}\right )\end{align*}
\begin{align*}\nu =\frac{2}{5}\equiv\frac{1}{3}+\frac{1}{15}.\end{align*}
\begin{align*}{\tilde{g}}=g {\frac{{\tilde{V}}_p}{\ell_s^p}}=g{\frac{\ell_s^p}{V_p}}=\ell_s^{3-p}\left(E_f^\prime\right)^3{\tilde{V}}_p \quad .\end{align*}
\begin{align*}Z_\Psi=\int dZ \exp i\int_{\tau_1}^{\tau_2}d\tau [P_A\dot Q^A-H+\{\Psi,\Omega\}]\end{align*}
\begin{align*}(\Phi, H_{\rm coul} \Psi) = \int_{\Lambda} dA^{tr} \sigma \int d^3x \, 2^{-1}[(E_i^a\Phi)^{*} E_i^a\Psi + \Phi^{*} B_i^a B_i^a\Psi]\;\mbox{,}\end{align*}
\begin{align*}h(x^I dx_{k_1}\dots dx_{k_q}) := \frac{1}{|I|+q}\sum_{\alpha=1}^q(-1)^{\alpha-1}x^{I+e_{k_\alpha}} dx_{k_1}\dots \widehat{dx_{k_\alpha}} \dots dx_{k_q}.\end{align*}
\begin{align*}\left( \Gamma ^{M}\right) _{ab}\nabla _{M}\Psi ^{a}=0,\;\;\;\;\;\forall b\end{align*}
\begin{align*}\lim_{\tau\to \pm \infty} \frac{dq_a}{d\tau}=0,\qquad\lim_{\tau\to \pm \infty}q_a(\tau)=\pm \delta_{a1}\end{align*}
\begin{align*}U\left( a,b:z\right) \approx \frac{\Gamma \left( b-1\right) }{\Gamma \left( a\right) }z^{1-b},\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, Re\, b\geq 2,\, b\neq 2.\end{align*}
\begin{align*}(\mu, \nu, \lambda) = (\nu, \lambda, \mu)\\etc.\end{align*}
\begin{align*}\hat{h}\psi ({\bf x})=\epsilon \psi ({\bf x})\;,\;\;\psi ({\bf x})=\left( \begin{array}{c}\chi ({\bf x}) \\ \varphi ({\bf x})\end{array}\right) \;. \end{align*}
\begin{align*}{\cal L}\,=\, \partial_{+}\Phi\,\partial_{-}\Phi \,-\,{\frac{\partial_{+}\Lambda }{\partial_{-} \Lambda}}\,(\partial_{-}\Phi)^2 \;\;,\end{align*}
\begin{align*}e^{-\varphi J^2} e^{\sigma J^0} e^{\psi J^2}, \end{align*}
\begin{align*}\tilde{R}^{ab}=R^{ab}-K^a_c\wedge K^{cb}+\frac{1}{2}C^{ab}_{\;\cdot c}\tilde {T}^c+\frac{1}{4}\gamma^I_J \gamma^K_L\theta^a_I \theta^b_K A^J\wedge A^L+i\left(DK^{ab}-\frac{1}{4}\gamma^I_J(\theta^a_Ie^b+\theta^b_Ie^a)\wedge A^J\right). \end{align*}
\begin{align*}\left \{\;\int_{\cal R} d^np' \,(\tilde{\omega}_{p'}|e_{p'p}|\eta_p)\;,\;\int_{\cal R'} d^nq'\, (\tilde{\omega}_{q'}|f_{q'q}|\eta_q)\;\right \} =\end{align*}
\begin{align*}S(A+\lambda E) - S(A-\lambda E) = {k\lambda\over\pi} \int_{\cal M} {\rm Tr}\left( E\wedge F + {\lambda^2\over 3} E\wedge E\wedge E\right),\end{align*}
\begin{align*} (\mu_k^{N,M},z_k^{N,M})\in F_M, \ \ \ \ p_k^{N,M}>0, \ \ k=1,...,K^{N,M}\leq N+1;\ \ \ \ \ \ \sum^{K^{N,M}}_{k=1}p_k^{N,M}=1 .\end{align*}
\begin{align*}h\left(\sum a_{I}x^Idx_{k_1}\dots dx_{k_q}\right) := \sum a_{I} h(x^I dx_{k_1}\dots dx_{k_q}).\end{align*}
\begin{align*} \langle \lambda,\, n | \hat{L}^2 | \lambda,\, n \rangle = l(l+1) = \lambda^2 - \frac 14 = (n + |m|)(n + |m| + 1).\end{align*}
\begin{align*}P = \left( \begin{array}{cc}\begin{array}{ccc}P_{(+\,+\,+)} & & \\ &P_{(+\,-\,+)} & \\ & &P_{(+\,-\,-)}\end{array} & \mbox{\Huge 0} \\\hspace*{-19mm} \mbox{\Huge 0} &\begin{array}{ccc}\vspace*{-3.5mm}\hspace*{-0.5mm}\ddots & & \\ &\hspace*{-3.1mm} \ddots & \\ & &P_{(-\,-\,-)}\end{array}\end{array} \right)\end{align*}
\begin{align*}N_f = k_{\alpha-1}+ k_{\alpha+1},\end{align*}
\begin{align*}g^{-1}{\bf v}_LI={\bf v}"_LIr"(\tau")_A({\bf a}")_{\bf T}\end{align*}
\begin{align*}V=-{ {\cal L}+\Lambda\over 2}=w{d{\cal L}\over d w}-{\cal L}\, ,\end{align*}
\begin{align*}\left\{ \pi _{{\rm ab}},H_{{\rm C}}\right\} \approx 0\end{align*}
\begin{align*}W_{EH}=-{1\over 16\pi G}\int_{\Sigma_r}2K~~.\end{align*}
\begin{align*}{\cal L} _{p} \stackrel{\rm def}{=}Tr[F(2p)^2 +H(2p)^2] \geq \pm 2 Tr^{\star} F(2p)H(2p)\end{align*}
\begin{align*}j^{a}=\epsilon^{a}_{\;bc}q^{b}p^{c}+s^{a}\end{align*}
\begin{align*}g^{\mu \nu} = \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) \ , \qquad\epsilon^{\mu \nu} = \left( \begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right)\end{align*}
\begin{align*}H(m/n)=\max(|m|,|n|)\hskip 40pt\gcd(m,n)=1.\end{align*}
\begin{align*}\int_{-1}^{+1} dz f(z) \quad \rightarrow \quad \sum_{j=1}^{N_{1}} w_{z}(j) f(z_{j}).\end{align*}
\begin{align*}\Psi_{n,m} = e^{-\eta L_z}\Upsilon_{n,m}(\rho)e^{i\theta(m-n)}T_{n,m}(\mu),\end{align*}
\begin{align*}{\mathcal G}_{l+\nu} (r'',r';E) =\sum_{n=0}^{\infty}{\mathcal G}^{(n)}_{l+\nu} (r'',r';E) \; ,\end{align*}
\begin{align*}n(g,\mbox{\boldmath$A$})=1= -\frac{1}{2\pi}\int_{S^{2}} {\rm Tr}\left[ \sigma_{3} d(\mbox{\boldmath$A$}^{h^{-1}})\right].\end{align*}
\begin{align*}-A+2B+\frac{1}{2}M_{ij}^2+\frac{d(d-2)}{4}=-\lambda^{+-}(\lambda^{+-}+1-d)-\lambda(\lambda+d-3)+\frac{1}{2}m_{ij}^2\,.\end{align*}
\begin{align*}s=dm^i{\partial\over \partial m^i}=d \rho{\partial\over \partial \rho}+dx_0^\mu{\partial\over \partial x_0^\mu}.\end{align*}
\begin{align*}\tilde g_b\left( a\right) \equiv \left[ \lambda \left( b\right) \right]^{-1}g_b\left( a\right) ,\end{align*}
\begin{align*}\vec{r}_i=(t_i\cos \theta _i,t_i\sin \theta _i),\qquad i=1,2,3. \end{align*}
\begin{align*}L = \frac 12 \dot q^2 + \lambda(q_i q_i - R^2 ).\end{align*}
\begin{align*}\psi_1=f_1(x_1+\frac{2i\gamma_1\hbar}{\theta}x_2)e^{-\frac{3x_1^2}{8\gamma_1\hbar}-\frac{\gamma_1\hbar x_2^2}{2\theta^2}+\frac{ix_1x_2}{2\theta}},\end{align*}
\begin{align*}\Vert z\Vert=\max(|x|,|y|)\hskip 30ptH(z)=\max(H(x),H(y))z=(x,y).\end{align*}
\begin{align*}K(\theta,\beta,\lambda,\mu,\xi)=\left( \begin{array}{cc}\beta(\xi+\theta) & \mu \theta \\\lambda \theta & \beta(\xi - \theta)\end{array} \right)\;\;\;,\end{align*}
\begin{align*}Z[J,K]=e^{\frac{1}{\hbar}W[J,K]}=\int\!D\Phi\,e^{\frac{1}{\hbar}\left[-S[\Phi] +\frac{1}{2}\int\!\!J(x)\Phi^{2}(x)\,d^{n}x+\frac{1}{24}\int\!\!K(x)\Phi^{4}(x)\,d^{n}x\right]}.\end{align*}
\begin{align*} \phi^{'} = g^{-1} \phi g \ \ .\end{align*}
\begin{align*}{\cal T} (u) = L (u) K_{-} (u - i x/2) \sigma_2 L^T (-u)\sigma_2\end{align*}
\begin{align*}X_f = \omega^{ij} {\partial f({\bf x},{\bf t})\over\partial x^i}{\partial\phantom{x^j}\over\partial x^j}\,,\end{align*}
\begin{align*}t_0-t= { 2 \over {n-2}} { M^{(n-4)/2} \over \sqrt{2\lambda}}\left(\phi(t)^{(2-n)/2}- \phi_0^{(2-n)/2} \right).\end{align*}
\begin{align*}{(t^{\beta})_B}^C \, {(t_{\beta})_E}^F = - \, \frac{2}{3} \, {\delta_B}^C \, {\delta_E}^F + 2 \, {\delta_B}^F \, {\delta_E}^C.\end{align*}
\begin{align*}\Delta =P^2\,,\qquad P=\gamma^a\nabla_a \,. \end{align*}
\begin{align*}O(4,4;{\bf Z}) \backslash O(4,4)/O(4) \times O(4),\end{align*}
\begin{align*}[L^{(1)}_{n},\eta(w)]=w^{n+1}\partial_{w}\eta(w)+\frac{1}{2}(n+1)w^{n}\eta(w),\end{align*}
\begin{align*}h(z)=\lim_{t\to\infty} \frac{1}{t}\,\log H(F^t(z))\end{align*}
\begin{align*} N_{l}= <0,in|a_{l}^{\dagger}(out)a_{l}(out)|0,in>=\left|g\left({}_{-}|{}^{+}\right)\right|^2.\end{align*}
\begin{align*}\beta _{\alpha }^{2}=\frac{\gamma T_{n}(\gamma a)}{\omega a(2\pi )^{N+1}},\end{align*}
\begin{align*}S_L= \int d^4x [-e{\overline\Psi}_L\gamma^\mu i\partial_\mu\Psi_L+ C_\mu{\overline\Psi}_L e\gamma^\mu \Psi_L]. \end{align*}
\begin{align*}\kappa = {4\tau\over \rho (1 + \tau^2) + 2 \tilde{\rho} \tau} ={4t\over \rho_0 (1 + t^2) + 2 \tilde{\rho}_0 t},\end{align*}
\begin{align*}n^3P_{\rm \Phi }=n^3\biggl \vert \frac{\mu }{a}\biggr \vert ^2=n^3\vert C\vert ^2.\end{align*}
\begin{align*}\psi_{m^{ij}}=\delta(y-m^{12})\,\delta(x_0-m^{13})\,\delta(x_1-m^{14})\,\delta(x_2-m^{23})\,e^{\frac{im^{24}\,x_3}{m^{12}}}\ ,\end{align*}
\begin{align*}S_{\rm HCD}={\Lambda^{-(n+1)}}\int d^3x FD^{n}F\end{align*}
\begin{align*}\partial ^{\mu } \partial _{\mu}\psi + {m ^{2 }c ^{2}\over {\hbar^2}}\psi=0,\end{align*}
\begin{align*}T_a(q) \;=\; \Theta(-q^0)\: \delta(q^2-a) \;\;\; , \end{align*}
\begin{align*}\langle S(x)\,S(\tilde{x})\rangle_w=\langle1\rangle_w\Omega^{-1/2}(x)\,\Omega^{-1/2}(\tilde{x})\bigtriangleup_F(x,\tilde{x})\end{align*}
\begin{align*}\Vert z\Vert_p=\max(|x|_p,|y|_p)\hskip 30pt \nu_p(z)=\min(\nu_p(x),\nu_p(y)).\end{align*}
\begin{align*}C^{-1}E_i=e_i, \; {\rm for}\; i=1,2,\ldots,N-1.\end{align*}
\begin{align*}\oint_0 \frac{d\zeta}{\zeta^{n-2}}\chi-\oint_{\infty} \frac{d\zeta}{\zeta^{n-2}} \chi=\frac{1}{2 k_1} \oint_{\Gamma_1}\frac{du}{\zeta^{n-2}(u)} \log\frac{\rho(u)}{\xi(u)} ,\end{align*}
\begin{align*}\qquad \partial_0 x^{\bar\mu}=-p^{\bar\mu}, \qquad \partial_0 p^{\bar\mu}=-\partial_1\partial_1 x^{\bar\mu}, \qquad (p^{\bar\mu}\pm\partial_1 x^{\bar\mu})^2=0;\end{align*}
\begin{align*}\nu^2=-2\kappa^2_5\Lambda_5/3.\end{align*}
\begin{align*}\exp\left(-\frac{16}{3\chi}\right),\quad \hbox{if} \quad \chi \ll 1.\end{align*}
\begin{align*}\frac{\partial x_{\alpha}^T(\xi)}{\partial \xi}\ =\ -f(r(\xi))x_{\alpha}^T(\xi)\ ,\ \ \ \ \ \ \ \\frac{\partial r(\xi)}{\partial \xi}\ =\ -f(r(\xi))r(\xi)\ .\end{align*}
\begin{align*}[K_{3}^{q}, K_{\pm}^{q}] = \pm K_{\pm}^{q} \quad , \quad[K_{+}^{q}, K_{-}^{q}] = - [2 K_{3}^{q}]\end{align*}