harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f ( \omega ) \, = \, \frac{\omega^{2}\, - \, q^{2}\, - \, 2 \, q \, \omega \, \operatorname{c o t}\omega}{\omega^{2}\, - \, 2 \, q \, \omega \, \operatorname{c o t}\omega}\, \operatorname{s i n}\omega \,{.}
\sum_{A = 1}^{r}a_{I}^{A}q_{A}= \sum_{A = 1}^{r - 1}\tilde{a}_{I}^{A}\tilde{q}_{A}+ \tilde{a}_{I}^{0}\frac{\sum_{A = 1}^{r}m_{A}q_{A}}{\sum_{A = 1}^{r}m_{A}}
\vert F(z)\vert <\vert z\vert \varepsilon e^{(c-2t)\vert y\vert}\quad \textrm{or}\quad \vert f(z)\vert <\frac{Me^{2(1-t)\vert y\vert}}{\vert z\vert^{\alpha -1}},\quad z=x+iy,\quad \vert z\vert \to \infty
{\Upsilon}(s) =\sum_{\lambda}\int{d^{-\epsilon}k_{\omega}\over (2\pi)^{-\epsilon}}\left({k_{\omega}^2+\lambda\over \mu^2}\right)^{-s}= (1/4\pi)^{-\epsilon/2}{\Gamma(s+\epsilon/2)\over \Gamma(s)}\zeta_1(s+\epsilon/2),
- \int_{\Omega_{FEM}}\frac{\partial{\lambda_h}}{\partial t}(0) \bar{u}(x,0) ~dx - \int_{\Omega_{FEM_T}}\frac{\partial \lambda_h}{\partial t}\frac{\partial \bar{u}}{\partial t}~ dxdt &- \int_{\partial \Omega_{FEM}}\partial_n \lambda_h \bar{u}~ dxdt + \int_{\Omega_{FEM_T}}( a_h \nabla \lambda_h) (\nabla \bar{u}) ~ dxdt = 0.
E_1(F_3^2+E_1E_2F_3+E_1^2F_4)+E_2^2(E_0F_3+E_2F_1+E_1F_2)=0, &E_1(F_1^2+E_0E_1F_1+E_1^2F_0)+E_0^2(E_0F_3+E_2F_1+E_1F_2)=0.
\frac{\ddot{a}}{a}+ 2 \frac{\dot{a}^{2}}{a^{2}}+ D \frac{\dot{a}}{a}\frac{\dot{b}}{b}+ \frac{2}{a^{2}}= \frac{8 \pi \overline{{G}}}{D + 2}\left[ \rho + ( D - 1 ) p_{a}- D p_{b}\right]
\tilde \varphi^{in}_{\omega_0, u_0}(v,p) =\int\! d \omega \e^{i\omega u_0}{e^{-(\omega - \omega_0)^2/ 2 \sigma^2}\over ({2 \pi})^{1/4}\sigma^{1/2}}e^{-i \omega v}{\tilde{\psi}}_\omega^{in}(p)
|\nabla(\Gamma(t)f)|_q\leq \mathcal{B}e^{\sum_{i=1}^N\left[\beta_i(t)\theta_i-\frac{t}{2}\theta_i^2\right]}\left|\nabla f\right|_q, |\nabla(\Gamma^{-1}(t)f)|_q\leq e^{\sum_{i=1}^N\left[-\beta_i(t)\theta_i+\frac{t}{2}\theta_i^2\right]}\left|\nabla f\right|_q
S_{(f)\sigma}(z,z')=G_{(\sigma)}(z,z')-\sum_{r=1}^n\sum_{N_r=\mu_r,\nu_r}\int\limits_{C_b^{(r)}}G_{(\sigma)}(z,z'')\frac{d\theta'' dz''}{2\pi i}\tilde Y_{\sigma,N_r}^{(1)}(t'')\Psi_{\sigma,N_r}^{(0)}(z')
\tilde{\alpha}_j = \begin{cases}\tilde{c}_{\lambda_j + n - k - j}((D_j/D_{j-1})^{\vee}\otimes (D_{\gamma_j}^{\perp}/F^{\chi_j})) (-n\leq \chi_j<0) \displaystyle\frac{1}{(2+\beta \tilde{c}_1((D_j/D_{j-1})^{\vee}))}\tilde{c}_{\lambda_j + n - k - j}((D_j/D_{j-1})^{\vee}\otimes (D_{\gamma_j}^{\perp}/F^{\chi_j})) (0\leq \chi_j< n).\end{cases}
R(r) = \int_r^\infty \big\langle \psi_0 \big(\cdot,( 1 + \epsilon_{R(u)}) u \phi \big), \phi^* \big\rangle_m^{-1}du = \int_r^\infty u^{-\gamma_0}l(u) du = -\frac{1}{\gamma_0-1}\int_r^\infty l(u) du^{-(\gamma_0 - 1)}\stackrel[r\to 0]{}{\sim}C_X^{-1}(\gamma_0-1)^{-1}r^{-(\gamma_0 - 1)}.
\phi_{( a )}= \sqrt{{\frac{N V}{g_{0}^{2}}}\mathbf{F^{( 0 )}}{}_{c d \mu \nu}\mathbf{F}^{( 0 )}{}^{c d \mu \nu}}\equiv \sqrt{{\frac{N V}{g_{0}^{2}}}\mathrm{T r}\mathbf{F^{( 0 )}}{}_{\mu \nu}\mathbf{F}^{( 0 )}{}^{\mu \nu}}
18 = 2(a_{2,3}+ a_{2,4}+ a_{3,4}) - ((a_{1,2,3}+ a_{1,2,4}+ a_{2,3,4}+ a_{1,2,3}+ a_{1,3,4}+ a_{2,3,4}+ a_{1,2,4}+ a_{1,3,4}+ a_{2,3,4}) - (a_{1,2,3}+ a_{1,2,4}+ a_{1,3,4})) = 2(a_{2,3}+ a_{2,4}+ a_{3,4}) - (a_{1,2,3}+ a_{1,2,4}+ a_{1,3,4}) - 3a_{2,3,4}.
I_j^{3,1}\lesssim 2^{j}\sum_{j'-j=-1,-2}\|\dot{\Delta}_{j'}a\|_{L^{\frac{2p}{2-p}}}\|\dot{\Delta}_{j}\nabla\Pi\|_{L^2}\|\dot{\Delta}_j\Pi\|_{L^p}^{p-1}\lesssim d_j2^{j(\frac52-\frac{3}{p})}\|a\|_{\dot{B}_{q,1}^{\frac{3}{q}}}\|\nabla\Pi\|_{L^2}\|\dot{\Delta}_j\Pi\|_{L^p}^{p-1}.
\partial_r u\, \partial_r v = \left(\log\frac{r}{\rho}+ \beta\right)^{2\alpha}\, \partial_r f\, \partial_r g + \frac{\alpha^2}{r^2}\, \left(\log\frac{r}{\rho}+ \beta\right)^{2\alpha-2}\, f g &+ \frac \alpha r\, (f\, \partial_r g+g\, \partial_r f) \left(\log\frac{r}{\rho}+ \beta\right)^{2\alpha-1}.
\frac{1}{L}\sum_{i=1}^L H(Y_i |{\underline S}, Y^{i-1}) & = \frac{1}{L}\Big( \sum_{i=1}^{t-1}H(Y_i |{\underline S}, Y^{i-1}) & + \sum_{y^{t-1}}P_{Y^{t-1},{\underline S}}(y^{t-1},{\underline s}) \sum_{i=t}^L H(Y_i |{\underline s}, y^{t-1}, Y_t^{i-1})\Big).
Df_2 = \left(v_1\cdot D\alpha_1 \dots v_r\cdot D\alpha_r\right) = \left( \left( \begin{array}{c}1 0 \end{array}\right)\cdot D\alpha_1 \cdots \left(\begin{array}{c}1 0 \end{array}\right)\cdot D\alpha_m ~ v_{m+1}\cdot D\alpha_{m+1}~\cdots ~ v_r \cdot D \alpha_r\right).
f\cdot \Big(\sum_{g\in \Gamma}\gamma_{g^{-1}}(c)F\gamma_{g^{-1}}(c)-F\Big) = f\cdot \Big(\sum_{g\in S}\gamma_{g^{-1}}(c)F\gamma_{g^{-1}}(c)-\sum_{g\in S}\gamma_{g^{-1}}(c^2)F\Big) & =f\cdot \Big(\sum_{g\in S}\gamma_{g^{-1}}(c)[F,\gamma_{g^{-1}}(c)]\Big),
\begin{array}{l}{{{\cal D}_{-}{\cal X}^{--}{\cal X}^{-}{\cal D}_{-}= \mu_{-}^{- 1}~ , \mu_{-}{\cal X}^{-}= q^{2}{\cal X}^{-}\mu_{-}~ ,}}\\{{\mu_{-}{\cal D}_{-}= q^{- 2}{\cal D}_{-}\mu_{-}~ , \mu_{-}^{- 1}\equiv 1 + ( q^{- 2}- 1 ){\cal X}^{-}{\cal D}_{-}~ ,}}\\ \end{array}
\begin{array}{l}\mathbf{F}=(q\cos\theta +g\sin\theta) [(\mathbf{E}\cos\theta+\mathbf{B}\sin\theta)+\mathbf{v}\times(-\mathbf{E}\sin\theta+\mathbf{B}\cos\theta)] =q\mathbf{E}+g\mathbf{B}+\mathbf{v}\times(q\mathbf{B}-g\mathbf{E}).\end{array}
\int_{- T / 2}^{T / 2}d s_{i}\simeq \int_{- T / 2}^{T / 2}d t_{i}\, \sqrt{1 + h_{1 1}[ x ( t_{i}) ]}\simeq T + \frac{1}{2}\int_{- T / 2}^{T / 2}d t_{i}\, h_{1 1}[ x ( t_{i}) ]
d_{\mu}( q ) = ( 1 - \delta_{q_{3}, 0}) \, \frac{1}{2 q^{2}\, q_{3}^{2}}\, q_{\mu}\left( 1 - \delta_{\mu , 3}\right) + \delta_{q_{3}, 0}\, \frac{q_{\mu}}{( q^{2}+ i \epsilon )^{2}}\, ( 1 - \xi ) \ .
\widetilde{\mathbb{Q}'}:=\{\widetilde{Q'}=\widetilde{Q'_{1}}\times\widetilde{Q'_{2}}\subseteq\mathbb{R}^{2}: \widetilde{Q'_{1}}:=2^{k'}[-\frac{1}{2},\frac{1}{2}], \, \widetilde{Q'_{2}}:=2^{k'}[\frac{1}{24},\frac{25}{24}], \, \forall k'\in\mathbb{Z}\},
\begin{array}{r l}{\cal H}=\frac{1}{2}\pi^2_1 +\frac{\pi}{2e^2}\left(\partial_1\pi_1\right)^2+\frac{e^2}{2\pi}\left(A^1-\frac{1}{2e\lambda}\partial_1(\phi_+-\phi_-)\right)^2 &+\frac{\pi}{2e^2}\phi^2-\frac{\pi}{e^2}\phi(\partial_1\pi_1)\ .\end{array}
\!\!\!\!I\!=\!\int_{\Sigma}\! d^3 \! x \!\sqrt{\gamma}\left(\!R(\gamma)\!-\!{1 \over 2}\!\left( \left(\!DU \right)^2 \!-\! e^{-2U}\!\!\left(\!DV \right)^2 \right) \!-\!{1 \over 2}\!\left(\left(\!D \Phi \!\right)^2 \!-\! e^{- 2 \Phi}\!\left(\!D \Psi \!\right)^2 \right)\!-\! \left(\!D T \right)\!^2 \!+\! e^{-(U\!+\Phi)}V(T) \!\right)
A_{n_j}= \frac{2i}{N}\sum_{k=0}^{N-1}e^{-2\pi i kj/N}2\sin(\pi kv_i)\{[\!\! \prod_{l, m \not= i}2 \sin ( \pi k v_l ) ] Tr\gamma_{k,9}+9 Tr \gamma_{k, 5_i}\}
{\cal Z}= \left[ \frac{2}{\beta}I_{1}( \beta ) \right]^{\# \mathrm{\; o f \; p l a q u e t t e s}}\sum_{J_{P}, \; j_{l}}\quad \prod_{\mathrm{p l a q u e t t e s}}( 2 J_{P}+ 1 ) \; T_{J_{P}}( \beta ) \; ( - 1 )^{2 J_{P}}\prod_{\mathrm{l i n k s}}( 2 j_{l}+ 1 )
\int_{\triangle_{\lambda_{0}. . . \lambda_{k}}^{\prime b}}F = \int_{\triangle_{\lambda_{0}. . . \lambda_{k}}^{b}}F + \sum_{i = 0}^{k}( - 1 )^{i}\int_{W_{\lambda_{0}. . . \hat{\lambda}_{i}. . . \lambda_{k}}^{b}}F ,
\frac{d^2}{dt^2}\Big|_{t=0}\lambda=\int_M \langle \mathcal{N}(h,K,v_{h,K}),(h,K,v_{h,K}) \rangle e^{-f}dV_g=\Big(\mathcal{N}(h,K,v_{h,K}),(h,K,v_{h,K})\Big)_{f}
\varepsilon_{ext, e}(\tau) = \mathcal{K}_\beta(\tau, \tau')\varepsilon_{ext,e}(\tau') + \int_{\tau'}^\tau \mathcal{K}_\beta(\tau,\tilde \tau) \left[ \mathcal{C}(\varepsilon_{ext}) +\mathcal{N}(\varepsilon_{ext}) \right](\tilde \tau) d \tilde \tau.
w = \int_{\vec{x}}z_{\vec{x}}\,d\mu(\vec{x}) &= \int_{\vec{x}}(1+\lambda)w_{1,\vec{x}}- \lambda w_{2,\vec{x}}\,d\mu(\vec{x}) &= (1+\lambda)\int_{\vec{x}}w_{1,\vec{x}}\,d\mu(\vec{x}) - \lambda\int_{\vec{x}}w_{2,\vec{x}}\,d\mu(\vec{x}) &= (1+\lambda)w_1 - \lambda w_2
p_{\nu_{\beta}}\left(t\right)=\left(tr\right)^{\frac{1}{\beta-1}}\frac{\sin\left(\frac{\beta}{\beta-1}f(r)\right)}{\sin(f(r))}\frac{\left|1-re^{if(r)}\right|^{2}}{\left|1-\eta_{\nu}\left(re^{if(r)}\right)\right|^{2}}\,p_{\mu}\left(t\right).
\int_{\gamma_\lambda}\frac 1{1-|z|^2}\frac{\partial}{\partial n}|z|^2\,\frac{ds}{4\pi}= \int_{\gamma_\lambda}\frac{|z|^2}{1-|z|^2}\frac{\partial}{\partial n}\log |z|^2\, \frac{ds}{4\pi}&= \int_{\gamma_\lambda}\left(\frac{|f(z)|^2}{\lambda}-1\right) \frac{\partial}{\partial n}\log |z|^2\, \frac{ds}{4\pi}.
\left|\zeta(s_0, \mathbb{X}_\alpha)(\omega)-p(s_0)\right| = \left|\frac{1}{2\pi i}\oint_{|z-s_0|=r}\frac{\zeta(z, \mathbb{X}_\alpha)(\omega)-p(z)}{z-s_0}\,dz\right| &\leq \frac{1}{2\pi}\int_{0}^{2\pi}\left|\zeta(r e^{i \theta}, \mathbb{X}_\alpha)(\omega)-p(r e^{i \theta})\right| \,d \theta
{^C_0}D^\alpha_tf(x,t)=\frac{1}{\Gamma(m-\alpha)}\int^t_0\frac{\partial^m f(x,\xi)}{\partial \xi^m}\frac{d\xi}{(t-\xi)^{1+\alpha-m}}, &{^{RL}_0}D^\alpha_tf(x,t)=\frac{1}{\Gamma(m-\alpha)}\frac{\partial^m}{\partial t^m}\int^t_0\frac{f(x,\xi)d\xi}{(t-\xi)^{1+\alpha-m}},
K_{t}(w; z_{\ell(1)}, \ldots , z_{\ell(t)}) =\sum_{s=0}^{t-2}g(w, z_{\ell(1)})g(w, z_{\ell(t)})\prod_{i=1}^{s}g(z_{\ell(i)}, z_{\ell(i+1)})\prod_{i=s+2}^{t-1}g(z_{\ell(i+1)}, z_{\ell(i)}).
\exp\Bigl\{\frac{i}{\hbar}\int_{0}^{t}\theta_{n}(t^{\prime}) d t^{\prime}\bigr\}=\exp\Bigl(-i(n+\frac{1}{2})\varphi(t)\Bigr)=\Bigl(\frac{k_{1}+k_{2}-k_{3}\tilde{\varphi}(t)+2i\sqrt{\kappa}\tilde{\varphi}(t)}{k_{1}+k_{2}-k_{3}\tilde{\varphi}(t)-2i\sqrt{\kappa}\tilde{\varphi}(t)}\Bigr)^{- (n+\frac{1}{2})/2}.
\widehat{P}_{n}^{(\alpha,\beta)}(x)=-\frac{1}{2}(x-b)P_{n-1}^{(\alpha,\beta)}(x)+\frac{bP_{n-1}^{(\alpha,\beta)}(x)-P_{n-2}^{(\alpha,\beta)}(x)}{\alpha+\beta+2n-2}\qquad(n\in\mathbb{N};P_{-1}^{(\alpha,\beta)}(x)=0),
\mu\left(\bigcup_{j=0}^{\lfloor\varepsilon h_n(v)/\delta_0\rfloor}T^{-h_n(v)-j+1}B_n(v)\right) = \left(\left\lfloor\frac{\varepsilon h_n(v)}{\delta_0}\right\rfloor + 1\right)\mu_n(v) > \frac{\varepsilon}{\delta_0}\mu(\tau_n=v)\geq \varepsilon.
\int_{B_{R}(x_0)}|u(x)|^p\,\left(\int_{\mathbb{R}^N\setminus B_{2\,R}(x_0)}\frac{1}{|x-y|^{N+s\,p}}\,dy\right)\,dx \le \frac{N\,\omega_N\,2^N}{s\,p}\,R^{-s\,p}\,\int_{B_R(x_0)}|u|^p\,dx &\le \frac{N\,\omega_N\,2^N}{s\,p}\,R^{-s\,p}\,\int_{\Omega}\frac{|u|^p}{d_\Omega^{s\,p}}\,dx\,\|d_\Omega\|^{s\,p}_{L^\infty(B_R(x_0))}.
(m-2)(m-3)^{3}(2m-2) + m(2m-2) > (m-2)^{2}(m-3)^{3}+ m(m-2)(m-3) + m(m-2)(m-3)^{2}+ m^{2}& = ((m-2)(m-3)+m)((m-2)(m-3)^{2}+m).
R_{00}= -\frac{1}{4}\sum_{i=1}^3 \frac{f_i'^2}{f_i^2}\;\; , \;\; R_{ii}= -\frac{f_i''}{2V^2}+ \frac{V'}{2V^3}f_i' + \frac{f_i'^2}{4 V^2 f_i}\;\;\; ; \; \; i=1, 2, 3
V_{v i s}( \tilde{q}) = \frac{| \frac{\partial W_{v i s}}{\partial \tilde{q}}+ \frac{\partial f_{v i s}}{\partial \tilde{q}}\overline{{F_{\Phi}}}|^{2}}{\frac{\partial^{2}f_{v i s}}{\partial \tilde{q}\partial \tilde{q}^{\dagger}}}- 6 \mathrm{R e}W_{v i s}( \tilde{q}) F_{\Phi}- f_{v i s}| F_{\Phi}|^{2}.
\Delta s = \left( \frac{g_{\star \mathrm{R H}}}{g_{\star \mathrm{d o m}}}\right)^{1 / 4}\, \left( \frac{H_{\mathrm{d o m}}}{H_{\mathrm{R H}}}\right)^{[ 4 - 3 ( 1 + w ) ] / [ 2 ( 1 + w ) ]}\, \Omega_{r}^{- 3 / 4}.
\langle C_n^{is}\xi,\eta\rangle-\langle C^{is}\xi,\eta\rangle = \langle C_n^{is}(1-\phi(C_n))\xi,\eta\rangle+\langle C^{is}(\phi(C)-1)\xi,\eta\rangle +\langle (C_n^{is}\phi(C_n)-C^{is}\phi(C))\xi,\eta\rangle.
Q(x) V(x, \cdot)(i)= \sum_{j\in \mathcal M}q_{ij}(x)V(x, j)=\sum_{j\in \mathcal M,j\neq i}q_{ij}(x)(V(x, j)- V(x, i)), ~ x\in \mathbb R^d, ~i\in \mathcal M .
\sum_{\lambda\in\Lambda^{(m,n)}_{\texttt{c}}}| C_{\texttt{c}}(\boldsymbol{\xi}^{(m,n)}_{\texttt{c};\lambda}) |^{-2}\Delta^{(m,n)}_{\texttt{c};\lambda}= \prod_{1\leq j\leq n}\frac{1-q}{1-q^j}
e^{a_{+}a_{-}/ 2}e^{b_{+}b_{-}/ 2}\frac{e^{( a_{+}a_{-}Q^{2}+ a_{+}b_{-}Q + b_{+}a_{-}Q + b_{+}b_{-}Q^{2}) / ( 1 - Q^{2})}}{1 - Q^{2}},
\hat{m}(s,x,t,\phi'(t))e^{-\frac{1}{2}\int_{s}^{t}\phi'(u)^2 du}-\hat{m}(s,x,s+\epsilon,\phi'(s+\epsilon))e^{-\frac{1}{2}\int_{s}^{s+\epsilon}\phi'(u)^2 du}=\int_{s+\epsilon}^{t}e^{-\frac{1}{2}\int_{s}^{u}\phi'(\omega)^2 d\omega}\Upsilon(u)du
K = K_0 + W(X) = K_0^\mathrm{w}(X, D_X) + W(X) = K^\mathrm{w}(X, D_X), h = h_0 + \mu(x)^2 = h_0^\mathrm{w}(x,D_x) + \mu(x)^2 = h^\mathrm{w}(x,D_x).
M^{2}\left( \begin{array}{l}{\phi^{\prime}}{{\overline{{\phi}}}^{* \prime}}\end{array}\right) = \left( \begin{array}{l l}{m^{2}}{m^{2}x}{m^{2}x}{m^{2}}\end{array}\right) \left( \begin{array}{l}{\phi^{\prime}}{{\overline{{\phi}}}^{* \prime}}\end{array}\right) \ .
L(E_8) \downarrow E_7 = L( E_7)/V_{E_7}(\lambda_7)^{2}/0^{3}, L(E_7) \downarrow E_6 = L(E_6)/V_{E_6}(\lambda_1)/V_{E_6}(\lambda_6)/0, \quad V_{E_7}(\lambda_7) \downarrow E_6 = V_{E_6}(\lambda_1)/V_{E_6}(\lambda_6)/0^{2}, L(E_6) \downarrow F_4 = L(F_4)/W_{F_4}(\lambda_4), \quad V_{E_6}(\lambda_1) \downarrow F_4 = W_{F_4}(\lambda_4)/0.
h(x,t) = e^{2\pi i (t+ t_0)\nu_0}\hskip-.25in \sum_k\sum_{(q,m)\in\Gamma}\big[b_{(q,m),k}\,Hg(t - (q-k)T) \hskip-.25in e^{2\pi i m(x-t)/LT}\,\phi(x-(t+t_0)+(q-k)T)\, r(t-q T)\big]
( d G )_{1 1 \bar{I}\bar{J}\bar{K}\bar{L}}= - 4 \sqrt{2}\pi \left( \frac{\kappa}{4 \pi}\right)^{2 / 3}( J^{( 1 )}\delta ( x^{1 1}) + J^{( 2 )}\delta ( x^{1 1}- \pi \rho ) )_{\bar{I}\bar{J}\bar{K}\bar{L}},
P^{(\alpha,\beta)}_{2,k}(x)=\frac{(|\alpha|+1)_k}{k!}{}_4F_{3}\left(\begin{array}{c}-k,k+|\alpha|+\beta+1,\frac{|\alpha|}{2}+\frac{1}{2},\frac{|\alpha|}{2}+1 begin{align*}3pt] |\alpha|+1,\alpha_1+1,\alpha_2+1\end{array};\,2(1-x)\right).
\langle P(u_t(s)), \phi(s)\rangle_V \Bigr|^t_0 = \int^t_0 \Bigl[ \langle P(u_t(s)) , \phi_t(s) \rangle_V -\langle A(u(s)) , \phi(s) \rangle_X - \langle B(s , x, u_t(s)) , \phi(s) \rangle_W \Bigr] ds
J_f^{B,c}[\alpha,\beta] :=\mathbf{E}\bigg[ \int_0^1 e^{-\frac{1}{2}\int_0^t\|\beta_s\|^2ds}\langle B^*B\alpha_t,\beta_t\rangle\,dt &+e^{-\frac{1}{2}\int_0^1\|\beta_t\|^2dt}f\bigg(BW_1 + \int_0^1 B\alpha_t\,dt +\frac{\Phi^{-1}(c)}{2}\int_0^1 B\beta_t\,dt\bigg)\bigg]
\alpha_{0}^{2}e^{\psi_{2 0}}\Omega^{2}= \frac{2}{\mu l^{2}e^{- \psi_{2 0}}- 1}| x - 2 \psi_{0}/ 3 |^{\mu l^{2}\exp ( - \psi_{2 0}) - 1}+ c_{1},
N_m=\left[\frac{2P_1m!\Gamma(2P_1-m)}{2^{2P_1-2m-2}(m+\lambda+\frac{1}{2})(m+\delta+\frac{1}{2})\Gamma(m+\lambda-\frac{1}{2})\Gamma(m+\delta-\frac{1}{2})}\right]^{\frac{1}{2}}
\frac{1}{2\pi i}\int_{(\sigma)}L(s, S_f^\nu) V(s) X^s ds = \sum_{n \geq 1}S_f^\nu(n) v(\tfrac{n}{X}) = \sum_{n \leq X}S_f^\nu(n) + \sum_{X < n \leq X + X/Y}S_f^\nu(n) v(\tfrac{n}{X}) = \sum_{n \leq X}S_f^\nu(n) + O\bigg( \frac{X}{Y}X^{\kappa(f) + \frac{1}{3}+ \frac{1}{3}\alpha(f) + \epsilon - \nu}\bigg),
F[a,b;x] = \frac{\Gamma (b)}{\Gamma (a)}(I_k + J_k) &= x^{-a}\frac{\Gamma(b)}{\Gamma(b-a)}(\sum_{n=0}^{[L-\delta]}\frac{(a)_n(1+a-b)_n}{n!}x^{-n}+ \frac{\Gamma(b-a)}{\Gamma(a)}J_k(x)x^a).