$\begin{align}\int_1^{\infty}f(x) dx &= \lim_{N\to \infty}\int_1^{N} x^{-r} dx\\
 &= \lim_{N \to \infty} \left[\frac{1}{1-r}x^{1-r}\right]_1^N \\
 &= \lim_{N\to \infty} \frac{1}{1-r}\left[\frac{1}{N^{r-1}} - 1\right] \\
 &= \frac{1}{r-1}.
 \end{align}
 $


$\sec\theta = \dfrac x3\quad$


$\forall x,y,z: \neg (x R y \wedge y R z)$


$\int_0^1 \frac{{f}(x)}{x^p} $


$\frac {(1-p)p^3}{1-(1-p^4)}$


$\|x-z\|<\frac{|\langle f,x\rangle|}{\|f\|}+\varepsilon,$


$\lim _{n\to \infty }\frac{\left[1x\right]+\left[2x\right]+\left[3x\right]+...+\left[nx\right]}{n^2} = \frac{x}{2}$


$<e^{\lambda x}(T \circledast S), \phi (x)> \\ = <T_{x}, <S_{y}, e^{\lambda (x+y)} \phi (x+y)>>  \\ =<e^{\lambda x} T_{x}, <e^{\lambda y}S_{y},  \phi (x+y)>> \\ =<e^{\lambda x} T \circledast e^{\lambda x} S, \phi (x)> \\$


$(av)^\gamma = a^\gamma v^\gamma$


$ a^{log_cb}=b^{log_ca} $


$y-\frac{b-c}{2} = -(ax+\frac{b+c}{2})$


$y(x) = a +(b-a)x$


$\mathbb R[x]/(x^2+n)$


$b=\{\{x\}\}$


$(ab, a+b)|(a, a+b)\cdot(b, a+b) = 1\cdot 1$


$\alpha = \omega + \alpha$


$(1-rs)(1+rs+\ldots+r^{n-1}s^{n-1})=1+rs-rs+\ldots-(rs)^n=1-(rs)^n=1-0=1$


$s_n = \frac{n(n+1)}{2}$


$ \sum_{k=n-m}^{n-1} \frac{k \sin \frac{k \pi}{n}}{1 + \cos^2 \frac{k \pi}{n}} = \sum_{k=1}^{m} \frac{(n-k) \sin \frac{k \pi}{n}}{1 + \cos^2 \frac{k \pi}{n}}. $


$|V| = (r+1)N^r - rN^{r-1}$


$ \alpha(\tau)=-\tfrac{1}{2}-i\tfrac{(2+2p-p^2) (1-2 p -2p^2) (1+4p+p^2)}{6 \sqrt{3} ~p(p+2) (2 p+1)(1-p^2)} $


$|KH|=qr$


$\int_0^\infty {R(x)dx}$


$\displaystyle \int_0^\infty x f(x) \; dx$


$\boxed{\begin{array}{ll} f_1=n-1\\ f_2=1 \\ f_3=n- 4 \\ f_4=0 \end{array}} $


$\frac{q}{\sin\gamma} = \frac{2 a c\cos\gamma}{a+c+f} \qquad \frac{r}{\cos\beta} = \frac{2 a b \sin\beta}{a-b+e} \qquad \frac{s}{\cos\gamma} = \frac{2 a c \sin\gamma}{a-c+f}$


$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) )$


$I_k=\{(x,y)\in\mathbb{R}^2:a_1< x< b_1, a_2< y< b_2\}$


$\mathcal{B}=\{\{a\},\{b\}\}$


$\lim_ {n\to \infty} \mu^*(E_n) \leqslant \mu^*(E) $


$k | ar+bs=1$


$\overline{\mathbb{Q}}\subset\overline{\mathbb{R}}=\mathbb{R}.$


$f(a)≤k≤f(b)$


$x = 1 + \sqrt{2} \to (x-1)^2 - 2 = 0 \to x^2 - 2x - 1 = 0$


$\displaystyle \sum_{k=1}^{\infty} \frac{\zeta(2k)}{k16^{k}}=4\int_0^{+\infty} \dfrac{\left(\sinh\left(\tfrac{x}{8}\right)\right)^2}{x(e^x-1)}dx$


$x= \frac{a+x}{1-ax}$


$\sum_{n=1}^\infty(a_n)=\sum_{n=1}^\infty(b_n)=\sum_{n=1}^\infty(c_n)$


$\{\alpha^\gamma \mid \gamma \in A \} \subseteq \{ \alpha^\gamma \mid \gamma < \beta \}$


$\frac43=\frac1x+\frac1y=\frac sp$


$f(x) = \frac{1}{(-1)^x+1}$


$\int   \frac{1}{2(x^2+1)^2}dx$


$\mu(A \cup B)= \lim_{n\to \infty}\mu_n(A \cup B)= \lim_{n\to \infty}(\mu_n(A) + \mu_n(B))= \lim_{n\to \infty}\mu_n(A) + \lim_{n\to \infty} \mu_n(B)= \mu(A) +\mu(B)$


$\zeta(s) = \frac{1}{\Gamma(s)} \int_0^{\infty} \frac{t^{s-1}}{e^{t}-1} \,dt$


$ |(D^\gamma\psi)(x)| <=R_\gamma|x|^{N+1-|\gamma|}. $


$E(x^2) = E[E(X^2|Y)]$


$P[m, n] \implies P[m, n+1]$


$ \begin{bmatrix} A & B \\ B & A \end{bmatrix} $


$ \int_0^\infty{|f(x+t)|^2dx} = \int_0^\infty{|f(x)|^2dx}$


$1 - \sin^2\varphi = \cos^2\varphi$


$f(f(x)) = xf(x)$


$\begin{align*} \mathbb{P}(X_T =a \mid X_0 = i) &= \mathbb{P} \left( \bigcup_{n \in \mathbb{N}_0} B_n \mid X_0 = i \right) \\&= \sum_{n \in \mathbb{N}_0} \mathbb{P}(B_n \mid X_0 = i) \\ &= \sum_{n \in \mathbb{N}_0} \mathbb{P}(\{X_n=a\} \cap \{T =n\} \mid X_0 = i). \tag{1} \end{align*}$


$\mathcal{G}=\left\{ \left\{ 0\right\} \right\} $


$f_{\theta}(x)=\frac{2x}{\theta^2}, x>0$


$\lim_{n\to\infty} \left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+...+\frac{1}{\sqrt{n^2+n}}\right)$


$\frac{\binom{n+2}{2}-3\left\lfloor \frac{n}{2} \right\rfloor - 1}{6} + \left\lfloor \frac{n}{2} \right\rfloor + 1$


$|a \cap b| = r \leq k$


$ \sum_{k=1}^\infty \|e_k-e'_k\|^2 < 1. \tag{*} $


$f, f_1, f_2, ...:S\to\overline{\mathbb{R}}$


$ ds = \sqrt{dx^2+dy^2} = \sqrt{1 + \left(\frac{dy}{dx}\right)^2}dx = \frac{rdx}{y}\tag{2} $


$f(n) = \binom{n}{d}$


$ \lim_{x\to x_0^-} f(x) = \lim_{x\to x_0^+} f(x)= \lim_{x\to x_0} f(x). $


$(a+b)^{r} \leq 2^{r} (a^{r} +b^{r})$


$-(x/L)^\gamma+(K-L)(-\gamma)x^\gamma L^{-\gamma-1}$


$\begin{cases}x^3-3x=y \\ y^3-3y=z \\ z^3-3z=x \end{cases}$


$(\gamma(b)-\gamma(a),\gamma(d)-\gamma(c))=(\gamma(b)-\gamma(a))\overline{(\gamma(d)-\gamma(c))}=0$


$\in C^n\setminus C^{n+1}$


$E[(X-E[X])^2] = E[X^2-2 X E[X] + E[X]^2] = E[X^2] -2 E[X] E[X] + E[X]^2 = E[X^2] - E[X]^2.$


$\omega=2\pi\left(1-\frac{\sqrt{d^2-R^2}}{d}\right)$


$d(x,u) \leq d(x,y) + d(y,u)$


$ \tilde{\fermi}\pars{s,x} = {\expo{x} \over s - 2} \quad\mbox{and}\quad \fermi\pars{t,x} = \expo{x}\int_{\gamma - \ic\infty}^{\gamma + \ic\infty} {\expo{st} \over s - 2}\,{\dd s \over 2\pi\ic}\quad\mbox{with}\quad\gamma > 2 $


$\begin{cases} x=t \\ y=-2t \\ z=t \end{cases} \implies \pmatrix{x \\ y \\ z} = t\pmatrix{1 \\ -2 \\ 1}$


$X = F \oplus Y$


$m\in\langle m\rangle$


$\sum_{n = 1}^\infty \frac{1}{n^2 + a^2} = \frac{1}{2}\sum_{n = -\infty}^\infty \frac{1}{n^2 + a^2} - \frac{1}{2a^2}$


$ \left(\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x    }^{x_x^x}}\right)_{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x    ^x\right)_{x_x^x}^{x_x^x}}}^{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^    {\left(x_x^x\right)_{x_x^x}^{x_x^x}}} $


$F(x)= \int_a^{x^2}f(t)dt$


$X^+$


$\Delta y=\Delta y(a,\Delta x)=f(a+\Delta x)-f(a)$


$\boxed{\zeta(-1) = 1 + 2 +3 + 4 + \dots =  - \frac{1}{2}}$


$\|A\| = \sqrt{\rho(A^*A)} = \sqrt{\rho(AA^*)} \leq 1$


$p\circ T$


$x_n \in V_n \subseteq \overline{V_n} \subseteq U_n$


$ \{x_1, x_2,x_3,....,x_n\} $


$ \|x\|\le \sum_{i=1}^n |x_i| \|e_i\| \le \|x\|_\infty \sum_{i=1}^n \|e_i\|. $


$ \liminf_{n\to\infty}\mu_n(X)\geq \mu(B(x,r)) $


$z = \mathbb{R}[x]/(x^2+1)$


$p_1p_2\mid n$


$\int_0^{2\pi}=\int_0^{\pi}+\int_{\pi}^{2\pi}$


$\frac{X_1+X_2+...+X_n}{n} $


$\forall a,b \in \mathbb{Z}\times\mathbb{Z}, f(a+b) = f(a)+f(b)$


$\lim_{n\to\infty}\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}}$


$\text{P}(B) = 1/6$


$\frac{|\langle Ax, y\rangle |}{\|x\|\cdot \|y\|}\le \frac{\|Ax\|}{\|x\|}.$


$e^{x} \geq \left (1 + \frac{x}{n} \right) ^{n}$


$\int_0^\infty g(x)\,\mathrm dx$


$a,\quad a+d,\quad a+2d,\quad\ldots$


$\omega \subset \alpha$


$ S=2\pi\int y\ ds \\ V=\pi\int y^2\ dx  $


$E[X\mid\mathcal F_{k_0}]=0$


$\int_0^1 f'(t) \phi(t)dt = 0$


$\left[\begin{array}{ccc|c}       2&   k&   2&  0\\       1&   -1&  1&  1\\      0&  1& -1&  k     \end{array}\right]$


$PV\int_{-\infty}^{\infty} \frac{1}{(x^2+1)(x^2+2x+2)}dx$


$\log_a(b)+\log_a(c) = \log_a(bc)$


$a_{k+p}=\frac{1}{N}\sum\limits_{n=-\infty}^{\infty}e^{-j\frac{2 \pi}{N}(z-p)n}=\frac{1}{N}\sum\limits_{n=-\infty}^{\infty}e^{-j\frac{2 \pi}{N}zn}e^{j\frac{2 \pi}{N}pn}$


$\frac{4^x}{6^x + 9^x} + \frac{9^x}{6^x + 4^x} = \frac{7}{6}$


$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3  \end{array} \right| $


$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx = \int_{0}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx + \int_{-\pi}^{0}\frac {\sin nx}{(1+2^x)\sin x}dx$


$B = \{v_1, v_2, \ldots, v_n\}$


$Q=\{(x,y) \in \mathbb{R}^2 : 0\leq x\leq 1, 0 \leq y \leq 1\}$


$R^*\equiv S^*$


$F_3 = F_2 + F_1 = 1 + 1 = \mathbf{2}\\F_4 = F_3 + F_2 = 2 + 1 = \mathbf{3}\\F_5 = F_4 + F_3 = 3 + 2 = \mathbf{5}\\F_6 = F_5 + F_4 = 5 + 3 = \mathbf{8}\\F_7 = F_6 + F_5 = 8 + 5 = \mathbf{13}\\\vdots$


$\left(\frac12\right)^n\times2=\frac1{2^{n-1}}$


$a_n = 1 + \frac 1{2^{n-2}}$


$xRy \rightarrow yRx.$


$B_1\times\cdots\times B_n$


$F(x) = \displaystyle \int_{a}^x f(t)dt=x^2-2x-3$


$F(n)=F(n-2)+F(n-3)+F(n-2)=2*F(n-2)+F(n-3)=2*(F(n-3)+F(n-4))+F(n-3)=3*F(n-3)+2*F(n-4)$


$\lim_{n\to\infty}\left(\sum_{k=1}^{2n}\frac{1}{2n}\left\lfloor\frac{2n}{k}\right\rfloor-\sum_{k=1}^{n}\frac{1}{n}\left\lfloor\frac{n}{k}\right\rfloor\right)$


$g(x) =\dfrac{c^{x}}{c^{x}+c^{1/2}} $


$\left\lVert e_n-e_m\right\rVert_p<\epsilon$


$\lnot F \lor \lnot\lnot F$


$\left\lfloor{a}\right\rfloor - \left\lfloor{b}\right\rfloor \ge \left\lfloor{a-b}\right\rfloor$


$ \lim_{N\to\infty}\sum_{k=1}^N\frac{9}{10^k}. $


$s(x)=\sum_{i=1}^n c_i\chi_{E_i}$


$E_3(n) = (\frac{n}{2})^{(n-1-\lfloor\frac{n}{2}\rfloor)}$


$F(a,b)=K(b)=F(a)(b)$


$f(z) = z \ ^ 2 + \lambda$


$\int \frac{1}{{(1+x^2)^3}} \ dx $


$C =\begin{pmatrix} A & -B \\ B & A \end{pmatrix} $


$A_0 \subseteq A_1 \subseteq \ldots \subseteq A_{\alpha} \subseteq \ldots (\alpha \in \omega_1)$


$g(n)=\sum_{j=1}^nA_j $


$\{\{A\}\} \subseteq B$


$\int_{-\pi}^{\pi}f(t)dt=2\int_{0}^{\pi}f(t)dt \neq 0$


$(x + a) (x - a) = 0$


$\frac{1}{16}e^\frac{-t}{4}(t-8) $


$x(t) = \frac{1}{\sqrt{1-\frac{\epsilon^2}{4}}} e^{\frac{-\epsilon t}{2}}\sin(t\sqrt{1-\frac{\epsilon^2}{4}})$


$\sum_{n=-\infty}^{\infty}\frac{1}{n^2+1}=\pi\sum_{n=-\infty}^{\infty}e^{-2\pi|n|}=\pi coth\pi$


$ \det \begin{pmatrix}A & 0 \\ 0 & B\end{pmatrix}=\det(A)\det(B) $


$ \Leftrightarrow (\sin\gamma)(\tan (a+\gamma))+\cos \gamma=\frac{b}{c}$


$\lim_{x\to a}f(x)g(x)=\lim_{x\to a}f(x)\cdot\lim_{x\to a}g(x)$


$|\mathbb{E}[XY]|\le\mathbb{E}[|XY|]\le(\mathbb{E}|X|^p)^{\frac{1}{p}}(\mathbb{E}|Y|^q)^{\frac{1}{q}}$


$f(ka+b)=kf(a)+f(b)$


$ \Delta a_{k+1} = r \Delta a_{k} + \gamma = r^2 \Delta a_{k-1} + r \gamma + \gamma = r^3 \Delta a_{k-2} + r^2 \gamma +  r \gamma + \gamma\\ =\ldots = r^{k-1} \Delta a_2 + \Gamma $


$(1,1,-1,1,-1,-1,-1,1,-1,1,-1,-1,-1,-1)$


$\gamma_1 \geq \dots \geq \gamma_n$


$ \zeta(s)\Gamma(s)= \int_{0}^{\infty}\frac{x^{s-1}}{e^x-1}\,dx \implies \frac{d}{ds}(\zeta(s)\Gamma(s))= \int_{0}^{\infty}\frac{x^{s-1} \ln(x)}{e^x-1}\,dx $


$A_n = \frac {n(n+1)}2 - 1$


$V= P \oplus K $


$a\in V\subseteq\cl V\subseteq U$


$\|x_m-x_n\| \geq 1$


$E|XY|^2 \leq (E|X|^p)^{1/p} E|Y|^2 \ $


$n! =\sqrt{2n\pi}\left(\frac{n}{e}\right)^n e^r$


$\int\int dx^2=\int \int 2xdx$


$\lim_{t\rightarrow 0} \frac{||\gamma(a+t)-\gamma(a)-\gamma'(a)t||}{|t|}=\lim_{t\rightarrow 0} ||\frac{\gamma(a+t)-\gamma(a)}{t}-\gamma'(a)||=||\lim_{t\rightarrow 0} \frac{\gamma(a+t)-\gamma(a)}{t} -\gamma'(a)||=0$


$\begin{bmatrix} \alpha & \beta \\ -\beta &\alpha\end{bmatrix}$


$ y^2-p(x) = (A(x)y+a(x))(B(x)y+b(x))=A(x)B(x)y^2+(A(x)b(x)+B(x)a(x))y+a(x)b(x) $


$A\subseteq B\subseteq C$


$A^{\bot\bot}\otimes B^{\bot\bot}\vdash(A\otimes B)^{\bot\bot}$


$(x- a)^{n-1}v,\ldots (x-a)v, v$


$B=\{e_1,e_2, ...., e_n\}$


$ x!\sim (x/e)^x\sqrt{2\pi x} $


$ 1+2+3+\cdots \rightarrow -\frac{1}{12} $


$ I(A) = \int_0^2 \sqrt{1 + \frac{dy}{dx}^2}dx  = \int_0^2 \sqrt{1 + (2x)^2} dx = \sqrt{17} + \frac{1}{4} \sinh^{-1}(4) \approx 4.64678. $


$\begin{align}\int_0^\infty f(x)\,\mathrm dx&=\int_0^1 f(x)\,\mathrm dx+\lim_{y\to a^-}\int_1^y f(x)\,\mathrm dx+\lim_{z\to a^+}\int_z^c f(x)\,\mathrm dx+\lim_{w\to\infty}\int_c^wf(x)\,\mathrm dx\\&=\int_0^1 f(x)\,\mathrm dx+\infty+\lim_{z\to a^+}\int_z^c f(x)\,\mathrm dx+\lim_{w\to\infty}\int_c^wf(x)\,\mathrm dx\\&\equiv\text{ undefined or diverges to infinity}\end{align}$


$ \left[ \begin{array}{ccc|c} 1 & a & 0 & 1\\ 0 & 1 & b & 1\\ 0 & -ac & 1 & 1-c \end{array} \right]. $


$p\in \overline{p_1 p_2}$


$\displaystyle \int_0^\infty \dfrac{\sin^3x}xdx=\dfrac\pi4$


$F=A\oplus R(T)$


$\Phi(s,n)=\phi(s,n)=s\varphi(n)$


$h(P) = (bg)(P)\implies b(P)g(P)=0$


$f(n) = \left(\frac{\sqrt 2}{2}(1+i)\right)^n$


$ \int_0^1 \dfrac{\sin(x)}{x} ~dx \approx 0.9460830704$


$P\{\sup_{t\geq 0} M_t > x \,| \,F_0\} \geq 1\wedge \frac{M_0}{x}.$


$\forall \epsilon, \exists \delta \quad 0<|x-a|<\delta \quad \implies \quad |f(x)-f(a)| < \epsilon.$


$s\equiv r\equiv 2s\pmod {4}$


$ \frac{d}{d\gamma} \int_{0}^{\gamma} (\gamma-s) \, dF(s) = F(\gamma) - F(0) = \int_{0}^{\gamma} dF(s) $


$\frac {(1 + \frac 1n)^{n + 1} - (1 + \frac {1}{n + 1})^{n + 1}}{b - a} < (n + 1)(1 + \frac 1n)^n$


$P(k)\Rightarrow P(k+1)$


$1+\frac{n}{1!}+\frac{n*(n-1)}{2!}+\frac{n*(n-1)*(n-2)}{3!}+...+\frac{n*(n-1)...3*2}{(n-1)!}+\frac{n!}{n!}=2^n$


$\sum_{i=1}^{n-1} \cos \frac{2ik\pi}{n}\sin \frac{2il\pi}{n}=0$


$xRy\wedge yRx \Longrightarrow x=y$


$W=P\oplus Q$


$P(E)=P(E\cap E)=P(E)P(E)$


$x(t)=|t|^{-n}$


$G(x) = \int_a^xf(t)\,dt$


$A=\{\langle x,y\rangle\in\Bbb R^2:y=0\}$


$p \nmid r_1r_2$


$\mathcal{C} = (e_1,e_2,e_3,e_4,e_5)$


$\lfloor \alpha \rfloor = \alpha-\{\alpha\},\lfloor \beta \rfloor = \beta-\{\beta\},\lfloor \gamma \rfloor = \gamma-\{\gamma\},0\leq \{\alpha\},\{\beta\},\{\gamma\}<1$


$A =  \begin{bmatrix} 1&x&x^{2} \\ 1&y&y^{2} \\ 1&z&z^{2} \end{bmatrix}  $


$\int_{\Omega}|u_n|^{q_1\gamma} \le C\left(\int_{\Omega}|u_n|^{q_1} + |u_n|^{q_1-1}|\nabla u_n|\right)^{\gamma},$


$[p+e,p-e]$


$\lfloor\frac{a}{b}\rfloor - \lfloor\frac{a}{b}\rfloor x + \lfloor\frac{c}{d}\rfloor x + x - x^2$


$f'(c) = \lim_\limits{x\to c^-} p'(x) = \lim_\limits{x\to c^+} q'(x)$


$ | \int_{\gamma} f | \leq \int_{\gamma} |f| |dz| $


$\sigma = \gamma^{23} = (\gamma^2)^{11} \gamma = e^{11} \gamma = \gamma .$


$\rho(A^TA)$


$(x,y,z) \cdot (1,1,1)=0$


$ \sum_{n \in \mathbb{Z}} e^{-\pi n^2 t}= \frac{1}{\sqrt{t}}\sum_{n \in \mathbb{Z}} e^{-\pi n^2 / t}$


$F(x)=\displaystyle\int\limits_a^x f(t)\,\text{d}t.$


$z=\sqrt re^{it/2}$


$\lim_{ n \to \infty }f(\infty)=?$


$f_n(x,t) = \sum_{j=1}^n f_{j,n}(x)\mathbf{1}_{T_i}(t)$


$a^{log_b(c)}= p$


$f_X(x)=\frac{1}{100}e^{-\frac{x}{100}}$


$D_\dot\gamma \dot\gamma = \nabla_\dot\gamma \dot\gamma - A(\dot\gamma,\dot\gamma)\nu.$


$f(a)=\lim_{x\rightarrow a^+}f(x)=\lim_{x\rightarrow a^-}f(x)=\lim_{x\rightarrow a}f(x)$


$f'(x)=-\frac1{x\log^2x}+\frac1{(x-1)^2}=\frac{x\log^2x-(x-1)^2}{x(x-1)^2\log^2x}$


$\lfloor \alpha \rfloor+\lfloor \beta \rfloor +\lfloor \gamma \rfloor =$


$\tan(x+y)=a(\tan(x)+\tan(y))$


$R_N(x_N) = \frac{3^N}{3^N-5} > 1$


$d(x,z)\lt d(x,y)+d(y,z)$


$\lim_{x\to a^+}\frac{f(x)-f(x)}{x-a}=\lim_{x\to a^+}f'(x)=f'(a)=\lim_{c\to a^+}f'(c)$


$xRy, yRx$


$(a+b, ab)=1$


$s \equiv u \equiv t \Rightarrow s \equiv t \Rightarrow [s] = [t]$


$[x,y]=\langle x,y\rangle$


$\bigl\lfloor \frac{a}{bc} \bigr\rfloor = \Bigl\lfloor \frac{\bigl\lfloor \frac{a}{b} \bigr\rfloor}{c} \Bigr\rfloor$


$\sum|a_n-b_n|<\delta $


$\forall x\in A\forall y\in A(xRy\lor yRx)\tag{2}$


$ P(m) = \frac{(1-p)^{m-1}p}{1-(1-p)^{N}} $


$\frac{13}5=2+\frac35=2+\frac1{\frac53}=2+\frac1{1+\frac23}=2+\frac1{1+\frac1{\frac32}}=2+\frac1{1+\frac1{1+\frac12}}$


$T = ([n], E)$


$p_1p_2\cdots p_m\mid a$


$f(n) = \sin\left(\frac{n\pi}{2}\right)$


$f \sin(t) - \int_{0}^{\pi} f^{\prime} \sin(t) dt = 0$


$\|\delta x\|=\frac{\|\Delta x\|}{\|x\|},\|\delta A\|=\frac{\|\Delta A\|}{\|A\|}$


$ \frac{d}{d\gamma} \int_{0}^{\gamma} (\gamma-s) \, dF(s) \quad``\,=\text{''}\quad \underbrace{(\gamma - \gamma)F'(\gamma)}_{=0} + \int_{0}^{\gamma} dF(s).  $


$f_v(x)=\frac{\langle x, v \rangle}{\|v\|^2}$


$  (y-z)^2 = (y-x + x- z)^2 = (y-x)^2  + (x- z)^2 + 2 (y-x)(x-z)  $


$(A_1\times ... \times A_n)$


$\;\begin{cases} x=-3+t,\\y=4,\\z=1. \end{cases}$


$\frac{1}{m}\sum_{n = 1}^\infty \mu(A_n) \ge \mu(B_m)$


$\implies \left|\frac{ds}{dx}\right| = \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$


$aq_1q_2 |a$


$|d(x,y) -d(u,v)|\leq d(x,u) +d(y,v)$


$\forall \epsilon>0\quad\exists\delta>0\quad\forall x,y\quad|x-y|<\delta\Rightarrow|f(x)-f(y)|<\epsilon$


$y=\frac{ab-bx}{a}$


$[x,y]=\{x\}$


$ |g(y)-g(x)| \le  \int_x^y \gamma t^{\gamma-1}dt \le \gamma (x-y)^\gamma \cdot \left(\int_x^y |t|^{(\gamma-1)\cdot \frac{\gamma}{1-\gamma}} dt\right)^{\frac{1-\gamma}\gamma} =  \gamma (x-y)^\gamma ( y^{1-\gamma}-x^{1-\gamma})^{\frac{1-\gamma}\gamma} \le \gamma (x-y)^\gamma . $


$\displaystyle \int_0^1 \frac{\sin(x)}{x}~ dx$


$\frac{2^n}{1+2^n}=\frac{2^n}{2^n}\frac{1}{\frac{1}{2^n}+1}=1$


$z^{\frac{1}{n}} = r^{\frac{1}{n}}e^{\frac{\mathbb{i} \theta}{n} } $


$ \frac{1-a^{(1-s)/2}}{(1-a)a^{1/2}} = \frac{1-a^{1/2}}{(1-a)a^{1/2}} + \frac{1-a^{-s/2}}{1-a} = \frac{1}{\sqrt{a}(1+\sqrt{a})} + \frac{1-a^{-s/2}}{1-a}. $


$\displaystyle \int\frac{1}{\sqrt{x^8+1}}dx$


$\forall \epsilon>0, \exists \delta>0,\, |x-a|\lt \delta\implies |f(x)-f(a)|\lt \frac{\epsilon}{|\lambda|}$


$f_i:\mathbb{R}^{n+1}\to \mathbb{R}^n$


$\begin{align}\lim_{n \to \infty}\dfrac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\dots+\sqrt{n-1}}{n\sqrt{n}} &= \lim_{n \to \infty} \frac{\sqrt{n}}{(n+1)\sqrt{n+1}-n\sqrt{n}} \\&=\lim_{n \to \infty} \frac{1}{(n+1)\sqrt{1+\frac{1}{n}}-n}\\ &=\frac{1}{1+\frac{1}{2}}=\frac{2}{3}\end{align}$


$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{3^{mn}}$


$A=A_1\times...\times A_n$


$\left\lfloor \frac{k+1}{p} \right\rfloor=\left\lfloor \frac{k}{p} \right\rfloor$


$D= \{(x,y)\in \mathbb{R}^2: 0\le x\le 1,\text{ } 0\le y \le 1 \} $


$ \eqalign{   & {{1 - y^{\,3/4} } \over {\left( {1 - y} \right)y^{\,7/8} }} = {{y^{\, - 7/8}  - y^{\, - 1/8} } \over {\left( {1 - y} \right)}} =   \cr    &  = {{1 - y^{\, - 1/8}  - \left( {1 - y^{\, - 7/8} } \right)} \over {\left( {1 - y} \right)}} \cr}  $


$x_{1}, x_{2} \in \mathbb{N}$


$ \sum_{n\geq 1}\frac{1}{n^2}=\sum_{n\geq 1}\frac{(n+1)-n}{n^2}\color{red}{\neq}\sum_{n\geq 1}\frac{n+1}{n^2}-\sum_{n\geq 1}\frac{n}{n^2}. $


$\operatorname{\mathcal{Jac}}\left( \mathbb{Q}[x] / (x^8-1) \right)$


$\displaystyle S_N = \frac{N(N-1)}{2}$


$\nu^+$


$\int_{-\pi}^{\pi} f(t) \sin (nt) dt \rightarrow 0 , \int_{-\pi}^{\pi} f(t) \cos (nt) dt \rightarrow 0$


$\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{1}{(x^2+1)(x^2+9)}dx$


$|f'(x)| \leq c |f(x)|$


$P(E) = \frac{1}{4}$


$\rho(A) - \rho(A) = \infty - \infty).$


$u(x) = \sum_{j=1}^k \frac{1}{|I_j|} \chi_{I_j}(x)$


$ f(e^{-t}) = \sum_{n=-\infty}^{\infty } e^{-tn^2} = \sqrt{\frac{\pi}{t}}\sum_{n=-\infty}^{\infty } e^{-\frac{n^2}{t}} =\sqrt{\frac{\pi}{t}}f(e^{-\frac{1}{t}}) $


$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n \frac{1}{2+k/n}$


$\sum\limits_{n=0}^{\infty}\frac{1}{n^2+a^2}$


$\int \frac{1}{(x^2-1)^{3/2}}dx$


$S\to T_{0,1}|T_2\\T_{0,1}\to\epsilon|002|112|012\\T_2\to 00T_{0,1}2|011122|111122$


$P(\{(x,y) \in \mathbb{R^2}:x \in A,0<y \leq 1 \})=\lambda(A)$


$ \langle\dot\gamma(t),\dot\gamma(t)\rangle \ddot\gamma(t) = \langle\dot \gamma(t),\ddot \gamma(t)\rangle\dot \gamma(t) $


$P[X_1<\cdots < X_n] = 1/n!$


$\lim_{n\rightarrow\infty}\int^{1}_{k=0}x^{k}\sum^{n}_{k=0}\frac{x^{k+1}-(1-x)^{k+1}}{(2x-1)(1-x)^k}dx$


$2\beta(s)\beta'(s)=2\langle\gamma'(s),\gamma(s)\rangle-2\langle\gamma'(s),\gamma(0)\rangle=2\langle\gamma'(s),\gamma(s)-\gamma(0)\rangle$


$I=(x_1,x_2,x_3, ..., x_n)$


$S= \{s_{1}, s_{2}, s_{3},\dots,s_{n}\}$


$|AB|=R\theta$


$\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{a+b+d}{c}\right\rfloor+\left\lfloor\frac{a+d+c}{b}\right\rfloor+\left\lfloor\frac{d+b+c}{a}\right\rfloor$


$e_n - f(x_n)/f'(x_n)$


$T^{1,0}M\oplus T^{0,1}M$


$f(a)+f(b)\ge f(a+b)$


$\Bbb F_2[x]/(x^4+x^2)$


$g(x)=(1-x^2)\frac{\sin ^{-1}(x)}{x}$


$[rn, r(n+1))$


$xRy, yRz $


$\left(1 + \frac{x}{N}\right)^N \approx e^x,$


$(-)\times (\pm) = (\mp)\\  (\pm)\times(\pm) = (+)\\  (\mp)\times(\mp) = (+)\\  (\pm)\times(\mp) = (-) $


$q = \bigg\lfloor{\frac{\lfloor{x}\rfloor}{d}}\bigg\rfloor = \Big\lfloor{\frac{x}{d}}\Big\rfloor$


$ x'+y=y+x'=(y+x)'=(x+y)'=x+y' $


$R=P\oplus Q$


$S_N=\frac{X_1+X_2+ \cdots +X_N}{N}$


$ dl = \sqrt{ \left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2 } $


$[ ....0, \lambda, -(\lambda+\gamma), \gamma, 0, ...]$


$[T^{a,x}, (T^{b,y})^*]= [T^{a,x}, T_{b,y}]$


$||f|| = \int_a^b|f(t)| \, dt + \int_a^b|f'(t)| \, dt$


$D(x,B)\leqslant d(x,y)+D(y,B)$


$S=\big\{\{a\}:a\in A\big\}$


$(x_1,x_2,x_3,x_4)=x*(1,0,-1,1)+y*(0,1,2,-1)$


$\int_0^\infty \int_0^\infty |f(t,x)|dtdx < \infty$


$ y=\frac{b+dx}{1-b-dx} $


$ \overline{D}=D^*=(U^*AU)^*= U^*A^*U=U^*AU=D $


$U_t \subseteq V\subseteq \overline{V}$


$f_X(x)=\frac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}},x \in \mathbb{R}$


$|AB|=x$


$ar-bs = 1-nz.$


$I=\int_0^\infty\,\frac{1}{\left(x^2+1\right)^3}\,\text{d}x$


$a^{1+sk}\equiv b^{1+sk}\equiv b\cdot b^{sk}\equiv b\cdot (b^s)^k\equiv b \cdot 1\equiv b \bmod 9.$


$T_p \mathbb{R}^{n+1} = \mathbb{R}^{n+1}$


$\mathbf{type}(\gamma\times\gamma,\lhd)\leq \gamma$


$f(n) = 2f(\frac n2)$


$N! + 2, N! + 3, N! + 4, ..., N!+N$


$\alpha^{++}$


$ y=\frac{b-x}{1-bx} $


$\frac{-27}{-3} = 9$


$ A_1, A_2, ... , A_k, A_{k+1} $


$ S_n = \frac{n(n-1)}{2}$


$\sum_{k=1}^{M-1}\cos\frac{2\pi kj}{M}=-1.$


$f_{X,Y}(x,y) = \frac{1}{100}e^{\frac{-(x+y)}{10}}$


$\int_0^{\infty} dx \frac{x^{u-1}}{e^x-1} = \sum_{k=0}^{\infty} \int_0^{\infty} dx\, x^{u-1} \, e^{-(k+1) x}  = \Gamma(u) \sum_{k=0}^{\infty} \frac{1}{(k+1)^u} = \Gamma(u) \zeta(u)$


$V \oplus T$


$(b, a+b) = (b, a) = 1$


$ L'=\lim_{x \rightarrow \infty}f'(x)=\lim_{x \rightarrow \infty}f'(\xi_x)=\lim_{x \rightarrow \infty}[f(x+1)-f(x)]=0.$


$ \begin{vmatrix}     1  & 1  & 1 \\     x  & y  & z \\     yz & xz & xy  \end{vmatrix}  = 0 $


$\lim_{x\to 0} f'(x)=0=\lim_{x\to\delta} f'(x)$


$p_1 p_2 \ldots p_k + 1$


$\int_{0}^{\pi}\frac{\sin^n{x}}{(1+r^2-2r\cos{x})^{(n+2)/2}}\,\mathrm dx=\frac{1}{1-r^2}\int_{0}^{\pi}\sin^n{x}\,\mathrm dx$


$(x+y)^2(y+z)(x-z)+(y+z)^2(x+z)(y-x)+(x+z)^2(x+y)(z-y) = 2 x y^3+2 x^3 z + 2 y z^3-2 x^2 y z-2 x y^2 z-2 x y z^2.$


$\mathbb{P}(X_n\leq x)\rightarrow \mathbb{P}(X\leq x)$


$\sin a + \sin^2 a + \tan a - \tan a \sin a  = \sin a + \sin^2 a + \frac{\sin a} {\cos a} - \frac{sin^2 a}  {cos a}$


$a = \frac{187}{200} = \frac{1}{\frac{200}{187}} = \frac{1}{1+\frac{13}{187}} = \frac{1}{1+\frac{1}{\frac{187}{13}}} = \frac{1}{1+\frac{1}{14+\frac{5}{13}}} = \frac{1}{1+\frac{1}{14+\frac{1}{2+\frac{3}{5}}}}=\ldots$


$ T_n = n\cdot (n+1)/2 $


$\sum \limits_{j = 1}^n i_j \ge k$


$0^++m^+=(0^++m)^+=(0+m^+)^+=0+(m^{+})^{+}$


$\sum_{k \le x}\frac1k= \frac{[x]}{x}+ \int_1^x \frac1{u}du-\int_1^x \frac{\{u\}}{u^2}du = O(1) + \ln (x) - O(1/x).$


$f(x) = \frac{2x}{x^{2}+1}$


$\{\gamma,-\gamma\}$


$\beta^+$


$ \mathbb{E} \left[ 1_{\sup_{t \geq 0 } M_t > x}  \mid \mathcal{F}_0 \right] = 1 \wedge \frac{M_0}{x}.$


$|AB| = 2$


$T^+$


$a(n) =\dfrac{n(n+1)}{2} $


$f_{n}\left( x\right) =\dfrac {x^{n}} {1+x^{n}}$


$ \begin{cases} x=t\\ y=t\\ z=t\\ \end{cases} $


$=a+\sum_{i=1}^n[(b-a)^{-(n-1)} \binom{n-1}{i-1}(b-x)^{n-1-(i-1)}(x-a)^{i-1}](x-a)=$


$\frac{\pi^2}{6} = \sum_{n=1}^\infty \frac{1}{n^2} = \sum_{n=0}^\infty\frac{1}{(2n+1)^2} + \sum_{n=1}^\infty\frac{1}{(2n)^2}$


$g : B \rightarrow A_1 \times \dotsc \times A_n$


$\begin{cases} x_1=1+s+2t \\ x_2=s \\ x_3=t \end{cases}$


$\sum_{n=1}^\infty a_k(n \bmod 2^k) e^{- \pi n^2 x} = (2^k x)^{-1/2}  \sum_{m=1}^\infty \frac{h_k(e^{2i \pi m/2^k})}{2^k} e^{- \pi m^2 2^k/ x}$


$a_1 = g(f(a_1))$


$\frac{x!}{\left( \frac{x}{e} \right)^{x}}\approx \sqrt{2\pi\,x} $


$\int_0^\infty f(x) dx = \int_0^\infty g(x) dx$


$ u_1 = a - \frac{1}{u_0} = a - \frac 1a = \frac{a^2-1}{a},\\ u_2 = a - \frac{1}{u_1}= a - \frac{a}{a^2-1}=\frac{a^3-2a}{a^2-1},\\ u_3 = a - \frac{1}{u_2} = a - \frac{a^2-1}{a^3-2a}=\frac{a^4-3a^2+1}{a^3-2a},\\ u_4 = a - \frac{1}{u_3} = a - \frac{a^3-2a}{a^4-3a^2+1}=\frac{a^5-4a^3+2a-1}{a^4-3a^2+1}. $


$log_{b^{n}} x = \frac {log_{c}x}{log_{c} (b^ {n})}$


$\lim_{x\to p}f(x)g(x)=\lim_{x\to p}f(x)\cdot\lim_{x\to p}g(x)$


$\int_0^\infty f(t)g(t) dt \neq 0$


$z = \sqrt[n]{r} e^{2 \pi k i/n} \ \text{ for k= 0, 1,..., n-1 }$


$\lim_{n\rightarrow \infty}E(x_{n})\geq E(x)$


$\kappa_\gamma = |h(\dot \gamma, \dot \gamma)| \le C,$


$D_{\pm} \times \Bbb R^{n+1}$


$\frac{1}{2^n+1}+\frac{1}{2^n+2}+...+\frac{1}{2^{n+1}}\geq\frac{1}{2^{n+1}}+\frac{1}{2^{n+1}}+...+\frac{1}{2^{n+1}}=\frac{2^n}{2^{n+1}}=\frac{1}{2}.$


$\alpha+\beta+\gamma=-\frac ba, \alpha\beta+\beta\gamma+\gamma\alpha=\frac ca,\alpha\beta\gamma=-\frac da$


$(a^2 - b^2 - 2ab, a^2 - b^2 + 2ab) = 1$


$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{4}{5}$


$\begin{align*} \int \frac{1}{(x^2 - 1)^2}\,\mathrm{d}x &\equiv \int \frac{1}{\left((x + 1)\cdot(x - 1)\right)^2}\,\mathrm{d}x \\ &= \int \frac{1}{(x + 1)^2\cdot(x - 1)^2}\,\mathrm{d}x\tag{1} \end{align*}$


$\Big | \frac{S_v}{\sqrt{v}} \Big | = \frac{1}{\sqrt{v}}\Big |X_1 +...+ X_v\Big | \le \frac{1}{\sqrt{v}}\Big (\Big |X_1 \Big | +...+ \Big | X_v\Big | \Big) \le \frac{1}{\sqrt{v}}v = \sqrt{v}$


$\left(\frac{x}{60}e^{\frac{-47x}{60}}\right)^{2}$


$  R = R_1 R_2 R_3 = \left( \begin{array}{rr} -2 & 1 \\ 1 & -1 \end{array} \right) $


$z\dfrac{f^{'}(z)}{f(z)} = z^{2}\dfrac{f^{'}(z^{2})}{f(z^{2})}$


$\hat f_n(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\sin x\sin (nx)\cos(xt)}{x^2}dx-\frac{i}{\sqrt{2\pi}}\underbrace{\int_{-\infty}^{\infty}\frac{\sin x\sin (nx)\sin(xt)}{x^2}dx}_{=0\text{ by symmetry}}$


$\forall a \in D, \exists \epsilon \gt 0, \exists \delta \gt 0, \forall x \in D: |x-a| \lt \delta \Longrightarrow |f(x) - f(a)|\lt \epsilon  $


$ f (x ) = x^2 - x - 2$


$\frac{L(b)-L(a)}{b-a} = \frac{\sqrt{\frac{\gamma Rb}{2\mu}}-\sqrt{\frac{\gamma Ra}{2\mu}}}{b-a} =\sqrt{\frac{\gamma R}{2\mu}}\frac{\sqrt{b}-\sqrt{a}}{b-a}$


$S= \{v_1,v_2,.......,v_k\}$


$\|A\|_1=\operatorname{trace}(\sqrt{A^* A})$


${\bf x} = (x,y)$


$x^2+y^2+z^2= (x-y)(y-z)(z-x)$


$x \to \alpha^{+}$


$\begin{pmatrix} a & b \\ b & -a\end{pmatrix} \tag 2$


$=\left[\begin{array}{ccc|c}1&-2&2&0\\2&k&-1&3\\0&-1&-1&5\end{array}\right]$


$\frac{n(n+1)}{2}+(n+1)=(n+1)\left[\frac{n}{2}+1\right]=(n+1)\left[\frac{n}{2}+\frac{2}{2}\right]=(n+1)\frac{n+2}{2}=\\=\frac{(n+1)(n+2)}{2}$


$ ay - xb = d$


${a^{Lo{g_a}x}} = x$


$I=\int_0^1\int_0^1 \frac{\sin^{-1}(xy)}{xy}$


$\gamma \subseteq \gamma_0\forall \gamma \in A$


$|\mathbb{R}^n|=|\mathbb{R}^{n-1}\times\mathbb{R}|$


$ \frac{3}{\beta} = \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{\alpha\beta + \beta\gamma + \gamma\alpha}{\alpha\beta\gamma} $


$q = (a, a+k, a+2k, ..)$


$x_3, x_4\in\mathbb{N}$


$d(\gamma(y_{i+1}),\gamma(a)) < d(\gamma(y_i), \gamma(a))$


$f'(t)=2(2r \cos(t)-1)(-2r \sin(t))+2r \sin(t)\cos(t)=$


$d(x,A)\leqslant|x-a|\leqslant|x-y|+|y-a|.$


$\frac{f}{||f||} = \frac{g}{||g||}.$


$f_0(x) = \frac{2}{x^2}$


$\left\lfloor \dfrac{\left\lfloor\tfrac{x}{m}\right\rfloor}{n} \right\rfloor = k = \left\lfloor\dfrac{x}{mn}\right\rfloor$


$\sqrt{2\pi x} \left(\frac{x}{e}\right)^x$


$ M(z) = \frac{z-\alpha}{1-\bar{\alpha} z} $


$\int_0^\infty f(x)dx=1$


$|f(x)| = |f(x) - c_0 + c_0| \geq \left||c_0| - |f(x) - c_0| \right|\geq |c_0| - |f(x) - c_0| > |c_0| - |c_0| = 0$


$\tilde\gamma = \arg\min_\gamma \int_0^T L(\gamma, \dot\gamma)\,dt\quad  \textrm{s.t.}\quad \gamma(0) = \gamma_0, \gamma(T) = \gamma_T$


$|f(z_0^2)| \le |f(z_0)|$


$ a = \int_{0}^{\infty} f(x) dx $


$f'(x)=1-{e^{-1\over x}\over x}-e^{-1\over x}$


$1 + 2 + 3 + 4 + 5 + ... = \frac{-1}{12}.$


$\lim_{x \to a} f(x)g(x) = \lim_{x \to a} f(x) * \lim_{x \to a} g(x) = 0 * G = 0, \forall G \in \mathbb R$


$mRm\Leftrightarrow P(m)=P(m)$


$ax+by=p$


$\left[ \begin{array}{ccc|c} 1 & 4 & -1 & 0 \\ 0 & -5 & 1 & 1 \\ 0 & 0 & 3 & 2 \\ 0 & 0 & 2 & -1 \end{array} \right]$


$\begin{align*}\int_0^1\frac{\log(x)\tan^{-1}(x)\tanh^{-1}(x)}{x}dx &= \frac{\pi^2}{16}G-\frac{7\pi\zeta(3)}{32} \tag{2}\\ \int_0^1\frac{\log^2(x)\tan^{-1}(x)}{1-x^2}dx &= -\beta(4)-\frac{\pi^2}{24}G+\frac{7\pi}{16}\zeta(3)\tag{3} \end{align*}$


$x^3 - 1, x^2 - 1 \in (x - 1)$


$\mathbb{R}^{n}\times\mathbb{R}^{n}\simeq \mathbb{R}^{2n}$


$\left[\matrix{\lambda & 0\cr 0& 1\cr}\right],\ \left[\matrix{1 & 0\cr 0& \mu\cr}\right],\ \left[\matrix{1 & 1\cr 0& 1\cr}\right],\ \left[\matrix{1 & 0\cr 1& 1\cr}\right].$


$\lfloor c \rfloor \geq \lfloor a \rfloor + \lfloor b \rfloor$


$d(x,A) = \inf\{d(x,y) : y \in A\}.$


$C=\{B\} = \{\{A\}\}$


$1 < \frac {x_1 + x_2 +... + x_n}{n} \leq 2$


$\dfrac{2r+1}{(r^2+r)^2+1}=\dfrac{(r+1)^2-r^2}{1+r^2(r+1)^2}$


$=\frac{(\sin 2^o\cos 1^o+\cos 2^o\sin 1^o)-(\cos 2^o\cos 1^o-\sin 2^o\sin 1^o)}{-(\sin 2^o\cos 1^o+\cos 2^o\sin 1^o)-(\cos 2^o\cos 1^o-\sin 2^o\sin 1^o)}$


$(1-\frac{x^2}{n})e^{x} \leqslant (1+\frac{x}{n})^n \leqslant e^{x}. $


$ax+by=r$


$ Q =  \begin{bmatrix} 0 & 1 & 2 & 1 & 2  \\ 0 & 1 & -1 & -1 & 1 \\ 0 & 2 & -5 & -4 & 1 \\ 0 & 4 & 2 & 0 & 6  \end{bmatrix} $


$\mathcal{L}(\gamma) = \int_\gamma \sqrt{g(\gamma', \gamma')}$


$\kappa^+$


$|f(z^2)| ≤ |f(z)|$


$ \lim_{n\rightarrow \infty} \mu(A_n)=A, $


$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}  > 4 \sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{(2m)^2+(2n)^2} $


$A_1\subseteq A_2\subseteq A_3\subseteq \cdots \subseteq A_n\subseteq A_{n+1}\subseteq \cdots$


$\left(1+\frac{x}{n} \right)^n \leq e^x \leq \left( 1 - \frac{x}{n} \right)^{-n} = \left( 1 + \frac{x}{n - x} \right)^n$


$\int_0^1f(t)P_n(t)dt\to \int_0^1f(t)g(t)dt$


$f(T, p) = p$


$ x11=\left(\begin{array}{rrr}% d&e\\% 0&f\\% \end{array}\right)% =df, x12=-\left(\begin{array}{rrr}% 0&e\\% 0&f\\% \end{array}\right)% =0, x13=\left(\begin{array}{rrr}% 0&d\\% 0&0\\% \end{array}\right)% =0 $


$ \left(  \left(  R^{=}\right)  ^{\sigma}\right)  ^{+}=\left(  \left( R^{\sigma}\right)  ^{=}\right)  ^{+}=\left(  \left(  R^{\sigma}\right) ^{+}\right)  ^{=}\supseteq\bigcup_{n=-\infty}^{\infty}R^{n}. $


$ \zeta = \int_0^\infty f(x)\,dx $


$F = K \oplus P$


$=\frac{1}{2}-\frac{n}{n+1}+\frac{n}{n+2}-\frac{1}{n+2}=\frac{1}{2}+\frac{-n}{(n+1)(n+2)}-\frac{1}{n+2}=\frac{1}{2}-\frac{2n+1}{(n+1)(n+2)}$


$\frac ad=\frac{1-\cos\gamma\cos(2\delta+\gamma)}{\sin\gamma\cos\gamma}\tag{1}$


$\displaystyle \int_1^\infty f(x) \quad dx$


$∪!\mathcal F = \{x | ∃!A(A ∈ \mathcal F ∧ x ∈ A)\}$


$ {}^s({}^ra) = {}^{sr}a $


$f_n(x)=\sum_{j=-\infty}^\infty c_j \, \mathbf 1_{A_j}(x)$


$(z-x^2, z+x^2)=d>1$


$ \lim_{n\to\infty} \frac {1+\sqrt[n]2 + \sqrt[n]3 + ... \sqrt[n]n} {n}  $


$\frac{a+b}{b+c}$


$g':B\rightarrow A_1 \times \dotsc \times A_n$


$ \begin{align} \int_0^1\left(\frac{\sin^{-1}(x)}{x}\right)^3\,\mathrm{d}x &=\int_0^{\pi/2}\left(\frac{t}{\sin(t)}\right)^3\,\mathrm{d}\sin(t)\tag{6}\\ &=-\frac12\int_0^{\pi/2}t^3\,\mathrm{d}\frac1{\sin^2(t)}\tag{7}\\ &=-\frac12\frac{\pi^3}8+\frac32\int_0^{\pi/2}\left(\frac{t}{\sin(t)}\right)^2\,\mathrm{d}t\tag{8}\\ &=-\frac12\frac{\pi^3}8-\frac32\int_0^{\pi/2}t^2\,\mathrm{d}\cot(t)\tag{9}\\ &=-\frac12\frac{\pi^3}8+3\int_0^{\pi/2}t\cot(t)\,\mathrm{d}t\tag{10}\\ &=-\frac12\frac{\pi^3}8+3\int_0^{\pi/2}t\,\mathrm{d}\log(\sin(t))\tag{11}\\ &=-\frac12\frac{\pi^3}8-3\int_0^{\pi/2}\log(\sin(t))\,\mathrm{d}t\tag{12}\\ &=\bbox[5px,border:2px solid #C0A000]{-\frac12\frac{\pi^3}8+\frac{3\pi}2\log(2)}\tag{13} \end{align} $


$P[X\geq b] =1$


$\lim_{x \rightarrow c}{g(x)}=L$


$\vartheta \mapsto \vartheta + \pi$


$\int_{-\infty}^{\infty}|f(t)|^2\,dt<\infty$


$f(a+b)\ge f(a)+f(b)$


$ar + bs= d$


$ \det A=\chi_B[X:=a] = (a-b)^{n-1}(a+(n-1)b). $


$A_1\subseteq A_2\subseteq\ldots\subseteq A_n$


$\displaystyle n!\approx\sqrt{2\pi n}\bigg(\frac{n}{e}\bigg)^n$


$ak + bs = d$


$(A,B;X,Y) = (A'',B'';X'',Y'') = (A',B';X',Y')$


$n^{\frac{n}{9}}\geq (1+\frac 1n)^n$


$I=\int\frac{(1+x^2)dx}{1-2x^2(1-2\sin^2(\frac{\alpha}{2}))+x^4}$


$\lvert AB \rvert = s$


$r\equiv\begin{cases} x-y=2 \\2x-z+1=0\end{cases}$


$\begin{pmatrix} A-B & B-A\\ B & A \end{pmatrix}$


$\gamma=\gamma_{R} \wedge \gamma_{2} \wedge \gamma_{\delta} \wedge \gamma_{3},$


$\det A=(a+(n-1)b)(a-b)^{n-1}.$


$ \begin{align} \sum_{m = 1}^\infty\sum_{n = 1}^\infty \frac{1}{nm(n+m)^2} & = \sum_{r = 2}^\infty \frac{1}{r^2}\sum_{s = 1}^{r-1} \frac{1}{s(r-s)} = 2 \sum_{r=2}^\infty \frac{1}{r^3} \sum_{s=1}^{r-1}\frac{1}{s}, \tag{3} \end{align} $


$(q,u) \in \mathbb{R}^{n+1} \times \mathbb{B}^n$


$|f(x_1)| \leq \frac{1}{2}|f(x_0)|$


$-31=\frac{-39.2}{x}(1-e^{\frac{x}{4}})$


$0 < i < \alpha^{+}$


$a^{\log_a(b)} = b$


$\quad a^{\large \log_a(b)} =  b.$


$d(x,y)= \sup _{f \in K} \lvert f(x) - f(y)\rvert$


$\sigma(a)\,\rho\,\sigma(b)$


$\mathbf Z/(3)$


$xRy\,\,\text{ and }\,\, yRx \qquad \Rightarrow \qquad x = y$


$p(m) \implies p(m + 1)$


$\left\lfloor\frac{\lfloor\frac ab\rfloor}c\right\rfloor$


$ y = \frac{1+8x}{3-3x}$


$\int_{-\infty}^{\infty} f(x)10^{x}dx$


$\begin{bmatrix}1 & 1 & -1 & 1 \\ 0 & 1 & a+2 & 1 \\ 0 & 0& (3+a) & 1 \end{bmatrix}$


$R_1-(k-1)R_2\to R_1\\ R_1-R_2\to R_2\\ R_1+R_3\to R_3\\ R_3-R_2\to R_3\\ -\dfrac{1}{k}R_2\to R_2\\ \dfrac{1}{k+1}R_3\to R_3\\ R_3-R_2\to R_2\\ (k-1)R_3+R_2\to R_3\\ R_1+R_3\to R_1\\ \dfrac{1}{k-1}R_3\to R_3\\ R_1-kR_2\to R_1$


$C_n=\dfrac {(2n)!}{n!(n+1)!}$


$(p^{e_i}), 1 \le e_i \le e$


$\sum_{k=1}^n\sin(\frac{2\pi k}{n})$


$F(x)=\int_a^xf(t)~dt \Rightarrow F'(x)=f(x).$


$\lvert\lvert A \rvert\rvert \leq \sqrt{\rho(A^TA)} $


$F=X+Y$


$Y = \{0,1,2,3,4\}$


$ \begin{cases} x=at\\[4px] y=bt\\[4px] z=ct \end{cases} $


$\exists! g':C_1' \to C_0'$


$f(x;\theta)=\frac{2x}{\theta^2}\mathbb{I}(\theta\geqslant x),$


$A_{1}\subset ... \subset A_{i}\subset A_{i+1}\subset ...\subset \mathbb{R}$


$\displaystyle \bullet\; a^{\log_{a}(x)} = x$


$I\;=\; \frac{x\sqrt{1-x^{2}\,}}{1-2x^{2}} + C$


$A=\text{im}(\gamma)$


$\left(\begin{array}{ccc|c} 1& -2& 1& 0 \\ 3& 0& -1& 4 \\ 1& 1& -1& k \end{array}\right).$


$\begin{bmatrix} 0 & 2 & 3 & 1 \\ -2 & 0 & 1 & 4 \\ -3 & -1 & 0 & 1 \\ -1 & -4 & -1 & 0   \end{bmatrix}$


$\frac{1}{\pi} \int_{0}^{\pi} \sin(x) dx + \frac{1}{\pi} \int_{\pi}^{2\pi} \sin(x) dx$


$s_n(x) = \sum_{i = 1}^n a_i \chi_i(x)$


$\int_0^1 \frac{\tanh ^{-1}(y)}y \,dy=\sum_{n=0}^\infty \frac {1}{(2n+1)^2}=\frac{\pi ^2}{8}$


$\sqrt{1+ \frac{dy}{dx}^2} \approx 1 + \frac{1}{2}\frac{dy}{dx}^2$


$d(x,y)+d(x,s)\leqslant d(y,s)$


$b, a+b, 2a+b, …$


$\Bbb C\cong \Bbb R[x]/(x^2+1)$


$X_1 \cdot ... \cdot X_n$


$\lim\limits_{n\to\infty} b_n = \sqrt2.$


$\displaystyle 9^x+3\leq a(3^x+1)\Rightarrow a\geq \frac{9^x+3}{3^x+1} = \frac{9^x-1+4}{3^x+1} = 3^x-1+\frac{4}{3^x+1}$


$\frac{a+b+\cdots}{\gamma}=\frac{a}{\gamma} + \frac{b}{\gamma}+\cdots.$


$cov(X,Y)=-0.2994\neq 0$


$T_x(X) + T_x(Z) = T_x(Y)$


$f(f(y) + xf(x)) = y + f^2(x)$


$[ \gamma (a), \gamma (b)] \subseteq \gamma [a,b]$


$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $


$\{x \in \mathbb{R}:a\leq x\leq b\}=\{y\in \mathbb{R}:\exists s,t\in [0,1]\; with\; s+t=1\; and\; y=sa+tb\}$


$n! = \sqrt{2\pi n} \left(\frac{n}{e}\right)^{n} e^{r_{n}}$


$f(x)=\frac{4x^2}{x^2+3} $


$\lim\limits_{n\to\infty}\sum\limits_{r=0}^n a_r$


$\frac{1}{\gamma}\int \frac{dt}{(t+1)(t+a\gamma)}$


$a_n = \frac{n(n+1)}{2} * (\frac{(2n+1 +3)}{6})$


$\{(x+y)(x+z), (x+y)(y+z), (x+z, y) \}$


$y'' + y' + y = y''$


$P(x)=\left(x^{\frac{p-1}{2}}+1\right)^2$


$[f(f^{-1}(x))]' = f'(f^{-1}(x))*{f^{-1}}'(x) = 1$


$\lim_{x\to a}f(x)=\lim_{x\to a^+}f(x)=\lim_{x\to a^-}f(x)$


$\|A\|_2 = \sqrt{\lambda_{\max}(A^*A)}.$


$\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx = \sum_{n=1}^\infty \int_1^\infty \frac{\chi_n(x)}{x^{s+1}}\,dx = \sum_{n=1}^\infty \int_n^\infty \frac{dx}{x^{s+1}} = \sum_{n=1}^\infty \left[-\frac{1}{sx^s}\right]_n^\infty = \frac{1}{s}\sum_{n=1}^\infty \frac{1}{n^s} = \frac{\zeta(s)}{s}$


$\sum_{n=1}^{\infty}p^n\cos(nx)=\frac{1}{2}\left(\frac{1-p^2}{1-2p\cos(x)+p^2}-1\right)$


$f_Y(x)=\frac{1}{\sqrt{2\pi}}(e^{\frac{-(\frac{x-\mu}{\sigma})^2}{2}}-e^{\frac{-(\frac{x+\mu}{\sigma})^2}{2}})$


$x+z=x+[y+(-x)]=x+[-x+y]=[x+(-x)]+y=0+y=y,$


$\ln\left(2\right) = \lim_{N \to \infty}{\sum_{n = 1}^{N}{\frac{1}{N + n}}}$


$\left(\frac{1+i}{2}\right)^{2^{n}} = \frac{1}{2^{2^{n-1}}},$


$ \left[     \begin{array}{ccc|c}       2&-2&1&10\\       3&1&-3&18\\       1&-5&5&k\\     \end{array} \right] $


$\gamma(ab)\neq\gamma(a)\gamma(b)$


$A = \sum_{k=0}^{n-1}\cos\frac{2k^2\pi}{n}$


$\lim_{n \to \infty} \frac{2-\sqrt{2+\sqrt{2 +\sqrt{2+ \cdots n \text{ times}}}}}{4^{-n}}$


$\mathbb{R}[x]/(x^2+1) \simeq \mathbb{C}$


$\mathcal{A}x = \mathcal{b}$


$x(x-a)^{n-1} = (x-a)^n + a(x-a)^{n-1}$


$T^{0,1} M \to T^{1,0} M$


$(\mathbb{Z}[x]/(x^2+1))/(p)=\mathbb{Z}[i]/(p))\cong \mathbb{F}_{p^2}$


$\displaystyle \frac{2x}{\theta^2}$


$ D: \begin{cases} x=-1+6t   \\[2ex] y=6-5t\\[2ex] z=1-2t \end{cases} $


$P(0) \implies P(1),$


$\mathbb E[s\mid\{\tau=0\}\cup\{s\geq z\}]=\frac{\mu_s-\gamma\int_{-\infty}^{z}xf_s(x)dx}{1-\gamma F_{s}\left(z\right)}=\frac{(1-\gamma)\mu_s+\gamma\int_{z}^{\infty}xf_s(x)dx}{1-\gamma F_{s}\left(z\right)}$


$\ds{\sum_{k = 1}^{m}\omega_{k} = \sum_{k = 1}^{m}\exp\pars{2\pi k\ic \over m} = \color{red}{0}}$


$ [I_U]^{f}_{f'} [L]^{e}_{f} [I_V]^{e'}_{e}[v]_{e'} = [I_U]^{f}_{f'} [L]^{e}_{f} [I_Vv]_{e} = [I_U]^{f}_{f'} [LI_Vv]_{f} = [I_ULI_Vv]_{f'} = [Lv]_{f'} = [L]^{e'}_{f'}[v]_{e'}. $


$\gamma(A)=\sup\{\gamma(K)\vert K\subseteq A , K \space\text{compact}   \}$


$ \begin{pmatrix} a & -b \newline b & a \end{pmatrix}.$


$U(a,b,c)=\{\langle x,y\rangle\in\Bbb R^2:a<x<b\text{ and }c<y\}\;.$


$\mu^{*}(A) =       \begin{cases}        0 &\quad\text{if } A = \emptyset \\        1 &\quad\text{if } A \text{ is bounded (and } A \neq \emptyset ) \\        \infty &\quad\text{if } A \text{ is unbounded and (} A \neq \emptyset )          \       \end{cases} $


$\begin{align} \int_1^L \frac{\sin^2(x)}{x}\,dx&=\frac12\int_1^L \frac{1-\cos(2x)}{x}\,dx\\\\ &=\frac12\log(L)-\frac12\int_2^{2L}\frac{\cos(x)}{x}\,dx\tag 1 \end{align}$


$p_1p_2 = a_3$


$\gamma_{a,b} + \gamma_{b,z} = \gamma_{a,z}$


$(S^{n+1} -S^n )$


$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{\sin(n-m)}{n^2+m^2}$


$\lim_{n \to \infty} \inf \mu_n(U) \ge \mu(U)$


$S=\{r\in\Bbb Q:r<2\}$


$x+h = (x_n)+(-x_n) = (x_n+[-x_n]) = (0) = 0$


$\begin{vmatrix} x_0 & y_0 &z_0 \\  x_1 & y_1 & z_1\\  x_2 & y_2 & z_2 \end{vmatrix}=0.$


$x + 9 -9(x+9)^{\frac{2}{3}} + 27(x+9)^{\frac{1}{3}} -27 = x -9$


$s(x)= \sum_{j=1}^p \alpha_j 1_{A_j} (x)$


$\gamma= \gamma_{1,+} \cup \ldots \cup \gamma_{n,+}\cup \gamma_{\sigma(1),-} \cup \ldots \gamma_{\sigma(n),-}$


$\frac 1 {(1-x^2)(1-2x)}=\frac {a+bx} {1-x^2} +  \frac c {1-2x}$


$ 0 \le x \le 1 \implies 0 \le x^{n-1} \le 1 \implies 0 \le x^n \le x.$


$|G(p)|^2 \ge |G(p^2)|$


$\|Af(x)\|_1 \leq \|f(x^2)\|_1$


$(x,y,z) = (1,1,1)$


$ [A]^\mathcal{B}_{\mathcal{B}'} \cdot [x]_\mathcal{B} = [A(x)]_{\mathcal{B}'}$


$y = \frac{ax+b}{cx+d}$


$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{n(m^2+n^2)}=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{n^2(m^2+n^2)}=\frac{\pi^4}{72}$


$f_X(x) = \frac{1}{2\pi}e^\frac{-(x)^2}{2}$


$P(2) = 1/6$


$=\sum_{k=1}^n\sqrt{\frac{\color{Red}{k^2(k^2+k+1)}+\color{Green}{k(k^2+k+1)}+\color{Blue}{(k^2+k+1)}}{k^2(k+1)^2}} $


$\frac{C}{1+r}+\frac{C}{(1+r)^2}+...+\frac{C+FaceValue}{(1+r)^2}$


$\alpha < \beta = \gamma^+ = \gamma \cup \{ \gamma \}$


$\frac{a + b}{b}$


$ f(9) = 3(9) = 27 $


$(a,b)=1\implies (ab, a+b)=1$


$ d_{t}=\frac{(1+\sqrt{2})^{t+1}-(1-\sqrt{2})^{t+1}}{2\sqrt{2}} $


$b^{log_b(z)} = log_b(b^z)$


$w=xyz\,[y,x]\,[z,y]\,[[z,y],[y,x]]\,[z,x]\,[[z,x],[y,x]]\,[[z,x],y]\,[[[z,x],y],[y,x]],$


$\ \overbrace{3\mid\color{#b0f}{n\!-\!1} = k(k\!+\!1)/2-1}^{\large n\text{ has remainder } 1}$


$=\frac{1-r^2}{2(1-2r\cos(\theta-\phi)+r^2)}$


$w = \sqrt[3]{2}\ e^{\frac{i\pi}{12}}$


$m+m>m$


$ x=t^{\frac{1}{n}}$


$1+2+3+4+...\stackrel{?}{=}-\frac{1}{12}$


$\mathcal C = (\{a\}, \Omega)$


$\sum_{n=1}^{\infty}\frac{n^3}{e^{n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{16}$


$P(X_n\leq x)=P(X_n<x)+P(X_n=x)$


$f_n(x)=(1+\frac{x}{n})^n, x \in \mathbb{R}$


$\tan \theta =\frac{x \pm \sqrt{x^2-4ky-4k^2}}{2k}$


$\int_0^\infty \frac{r^{p-1}}{(1+r)^n}$


$ \begin{cases} x=3\\ y=1+4t\\ z=-2+4t \end{cases} $


$\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}{\left(n^2+m^2\right)^{-{p}}}$


$\forall \varepsilon>0, \exists \delta>0: |z-y|<\delta\implies |f(z)-f(y)|<\varepsilon\ \ ?$


$\underline{x} = (x,y) $


$ax+by=c+d$


$\mathbf \Sigma^+$


$d(x,A) \leq d(x,y) + d(y,a)$


$\,f(n) = n^3-n\,$


$T=\{t_1, t_2,..,t_{m-n}\}$


$f(x) = \frac{(1-p) x}{\sqrt{1-4 p (1-p) x^2}}$


$ \begin{cases} x=-4s\\ y=s\\ z=-5s\\ \end{cases}$


$ \left[ \begin{array}{ccc|c}   1&1&-1&-2\\   2&-1&3&14\\ -1&-2&1&3 \end{array} \right] $


$\mathrm{erf}(x)=\frac{e^{-x^2}}{\sqrt{\pi}}\sum^{\infty}_{n=0}\frac{(2x)^{2n+1}}{(2n+1)!!}$


$det\begin{pmatrix}b^*&a\\ {-a^*}&b\end{pmatrix}=1$


$\tan \theta = \frac{x}{\sqrt{81-x^2}}$


$A= B\oplus C$


$a^r \equiv a^{s + k\varphi(n)} \equiv a^s \cdot (a^{\varphi(n)})^k \equiv a^s \cdot 1^k \equiv a^s \pmod n$


$\det \left( \begin{smallmatrix} A & -B \\ B & A \end{smallmatrix} \right) =|\det(A+iB)|^2$


$\cos{\theta} = \frac{x}{5}$


$ \forall \epsilon>0 \qquad \exists r_{\epsilon}>0 \qquad |y-x|\leq r_{\epsilon} \Rightarrow |h(x)-h(y)|\leq \epsilon   $


$S=\{\langle x,y\rangle:0\le x,y<1\}$


$\sum\limits_{\gamma<\alpha} a_\gamma<\prod\limits_{\gamma<\alpha} a_\gamma$


$\left\lfloor\frac{\left\lfloor\frac ac\right\rfloor}b\right\rfloor=\left\lfloor\frac{\left\lfloor\frac ab\right\rfloor}c\right\rfloor =\left\lfloor\frac a{bc}\right\rfloor.$


$ \sum\limits_{n} e^{-n^2 t} = \frac{1}{{2 \pi t}} \sum\limits_{n} e^{-n^2/4 t}.$


$\gamma,\tilde\gamma \in C^1([a,b], \Omega)$


$|f(x)|\leq\sup|f(x)|$


$\alpha\in \Omega\subset\Delta$


$I = \int^{1}_{0}\frac{\tan^{-1}(x)}{x}dx = \tan^{-1}(x)\ln(x)\bigg|^{1}_{0}-\int^{1}_{0}\frac{\ln(x)}{1+x^2}dx$


$ \begin{matrix}    & 1 &  x& x^2 &   & w & x w &   & w^2 &  & \\ 1  & 1 &  .& -3   &   & 2 & .   &   & 1   &  & \\ y  & . &  .&     &   & . &     & \\ y^2& . &   &     &   &   &     & \\ \\ z  & . & 1 &     &   & -2&     & \\ y z& . &   &\\ \\ z^2& 1 &   &\\ \end{matrix} $


$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} = 1$


$\|x_n - y_n\| < 1$


$ax + by = e$


$\left[\begin{array}{cccc|c}       1 & 1 & 1 & 0 & 75         \\[0.55ex]       0 & 0 & -1 & 1 & 5        \\[0.55ex]       0 & -1 & 0 & 1 & 0\\[0.55ex]       0 & 0 & 1 & 2 & 70 \end{array}\right]$


$A \subset V \subset \bar V \subset U$


$\frac{d}{dx} (-x)^{3/7} = \frac{-3}{7}(-x)^{-4/7} = \frac{-3}{7}(-x)^{3/7-1}=\frac{-3}{7}\frac{(-x)^{3/7}}{-x}=\frac{3}{7}\frac{(-x)^{3/7}}{x}(x<0)$


$\lim \limits_{n\to\infty}\frac{\sum\limits_{r=1}^{n}r^0}{\sum \limits_{r=1}^{n}r} =6$


$\varphi(2p) = \varphi(2) \varphi(p) = \varphi(p)$


$|CK| = r-s$


$|ab|=m\mid 6$


$C_0 < x < C_1$


$F(x) = \int_{-\infty}^{x}f(t)\text{ d}t$


$\frac 6a + \frac 6b =1$


$z=s^\frac 1n e^{(i\frac \phi n+\frac{2kpi}n)}$


$ (x-y)^2 (x+y) + (y-z)^2 (y+z) + (z-x)^2 (z+x) \geq  {(y-z)^2} (x+y+z) $


$C(a)=\frac12\int_0^1 u^{-1/2}(1-u)^{\frac{a-1}2}du$


$y = 31 - 10(e^{\large \frac{x}{20}} + e^{\large \frac{-x}{20}} ).$


$ax + by = d - cz_0$


$1 - \cos \varphi = 2\sin^2 \frac{\varphi}{2},$


$\displaystyle \lim_{x \rightarrow c}f(x)=l$


$=\left\lfloor \lfloor a\rfloor\lfloor b\rfloor+\{a\}\lfloor b\rfloor+\{b\}\lfloor a\rfloor+\{a\}\{b\}\right\rfloor-\lfloor a\rfloor\lfloor b\rfloor$


$ \left[     \begin{array}{ccc|c}       1&2&1&1\\       -1&4&3&2\\        2&-2&a&3     \end{array} \right] $


$\vartheta(a+\!\frac{z+b}\tau,-i\tau)$


$f_s(s |\{\tau=0\} \cup \{s \geq z\})=\frac{(1-\gamma)+1_{s\geq z}\gamma}{(1-\gamma)+(1-F_s(z))\gamma}f_s(s)$


$\left(\begin{array}{ccc|c} 1&-2&1&7\\ 1&-2&-k&k\\ k&-2&k&7 \end{array}\right)$


$F(x) = \int_a^x\!f(t)\, dt.$


$\mathbb{N}_{k,n}=\{k,k+n,k+2n,\ldots\},\ 1\le k\le n,\ \bigcup_{k=1}^n\mathbb{N}_{k,n}=\mathbb{N}.$


$n! \sim \sqrt{2\pi n}\cdot \left(\frac{n}{e}\right)^n,$


$f(x)=3^x / (3^x+1)$


$ax+by=M.$


$m!+2, \cdots, m!+m$


$|f(z_r)| \geq |f(z)|$


$G=\{\langle x,y\rangle\in\Bbb R^2:y<x\}$


$ I = \int\int_D \frac{1}{(x^2 + y^2)^{n/2}}  dxdy .$


$(a+b,a-b)=(x,y)$


$\mathbb Q(a)=\mathbb Q(a+c)$


$d(a,x)\le d(a,y)+d(y,x)$


$g(x) := \sum_{j=1}^n c_j 1_{A_j}(x)$


$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x} \quad\text{a.s.,} $


$\mathbb{R}[x]/(x^2+1) \approxeq \mathbb{C}$


$ \lim_{n\rightarrow\infty} \sum_{i=1}^{n} i^{-p}a_i < \infty$


$2^e p_1p_2\cdots p_k,$


$ \mathrm{dist}(x,M)=\frac{|F(x)|}{\Vert F\Vert}=|F(x)| $


$1 + 2 + 3 + 4 + \dots = -\frac{1}{12}$


$T f(p)=F(p-t)$


$\gamma=(\gamma_{1},\gamma_{2},...,\gamma_{n})$


$Cov(X,Y) = 0.$


$\lim_{n\to \infty} \frac{1}{n} (\sqrt{\frac{1}{n}} + \sqrt{\frac{2}{n}} + \sqrt{\frac{3}{n}} + ... + \sqrt{\frac{n}{n}})$


$ \exists \varepsilon > 0, \forall \delta > 0, \exists q_\delta\in(x-\delta,x+\delta)\text{ s.t. } |f(x)-f(q_\delta)|=f(q_\delta)=q_\delta^2 > \varepsilon$


$S = 1 + 2 + 3 + 4 + \ldots $


$13\mid n^{13}-n$


$F(\gamma) = \begin{cases} \frac{1}{\ln{\gamma}} & \gamma>e\\ \frac{\gamma}{e} & |\gamma|< e\\ -\frac{1}{\ln{-\gamma}} & \gamma<-e\end{cases}$


$F(x) = \int_{a}^{b} f(t)dt.$


$f (x)\le d (x,a)\le d (x,y)+d (y,a) $


$\{a, a+b, a+2b,..., a+ib\}$


$R \equiv R \land S \equiv R \lor S \equiv S$


$ \lim_{n \to \infty} \mu(A_i \cap E_n) = \mu(A_i),$


$(a+b,a-b)=g$


$d = ax +by$


$\min\left(\lfloor \frac{R}{\gamma}\rfloor,N\right)$


$e^x > \bigg(1+ \frac xn\bigg)^n$


$\int_0^\infty f(x)\ dx=0.$


$\exists c_0\in (a,x): g'(c_0)=0$


$\lfloor{a^b}\rfloor = c$


$R=\{x+iy:a\leq x\leq b, c\leq y\leq d\}$


$\frac{x_1 + x_2 +... + x_n}{n} = a$


$E_n, x_n$


$\textbf{x}= \gamma \textbf{p}_1$


$\Delta (1/|x|)= \delta(x)$


$ log_a(x) = b $


$\sum_{k=1}^n\cos(\frac{2\pi k}{n})$


$\int_{0}^{\pi} f(t) \sin (nt)\mathrm dt =0$


$\int_{1}^{\infty} f(x)dx$


$A = \{a\}$


$(x*y)*z =(x+y-[x+y])*z = x+y-[x+y]+z - [x+y-[x+y]+z]$


$p_1\nmid a, p_2\mid a$


$ I(a)=-\log(2\sin(\pi/a))-\gamma+\log(2 \pi/a)+\gamma=\\ \log\left( \frac{\pi/a}{\sin(\pi/a)}\right) $


$f(X) = X^2-X+41$


$\Phi(s \otimes n) = \Phi((s, n) + S\otimes_RN) = s\varphi(n) = s'\varphi(n') = \Phi(s' \otimes n')$


$\zeta(s)=\sum_{n=1}^m\frac 1{n^s}+\gamma(1-s)\sum_{n=1}^m\frac 1{n^{1-s}} + R(s)$


$\phi(n):=n^{-s}$


$\{e_i, 1\le i \le n\}$


$0 = L(P_n,f)\le U(P_n,f) = \sum_{k=1}^{n^2}M_k\cdot \frac{1}{n^2} = \sum_{k=1}^{n}M_k\cdot \frac{1}{n^2} + \sum_{k=n+1}^{n^2}M_k\cdot \frac{1}{n^2}.$


$f_{X,Y}(x,y) = \frac{2}{\theta^2}$


$\begin{align}     (m_1 - m_2)^2 &= 4 \frac{\tan^2 \theta}{\sin^4 \theta} - 4 \frac{(\tan^2 \theta + \cos^2 \theta)}{\sin^2 \theta} \\         &= 4 \frac{\tan^2 \theta - \sin^2 \theta (\tan^2 \theta + \cos^2 \theta)}{\sin^4 \theta} \\         &= 4 \frac{\tan^2 \theta (1 - \sin^2 \theta) - \sin^2 \theta \cos^2 \theta}{\sin^4 \theta} \\         &= 4 \frac{(\tan^2 \theta - \sin^2 \theta) \cos^2 \theta}{\sin^4 \theta} \\         &= 4 \frac{(\sec^2 \theta - 1) \cos^2 \theta}{\sin^2 \theta} \\         &= 4 \frac{\tan^2 \theta \cos^2 \theta}{\sin^2 \theta} \\         &= 4 , \end{align}$


${n+1\choose 2}\frac 1{2^{n-1}}$


$\frac{1}{2}\int_{-\infty}^{\infty} \frac{\cos(ax)}{(1+x^2)^2}\mathrm dx$


$(A, -B)$


$ ||A||^{2}_2 = \rho(A^{T}A) $


$f_Z(z) = \frac{4 \log z}{z^3}, \quad z > 1.$


$\overline{X}=\overline{Y}=\mathbb{R}$


$P \equiv R \equiv S$


$ f_x(x) = \frac {2x + 1}{10^{2}} $


$(a,a+k,a+2k,a+3k, ...)$


$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{10^{mn}}$


$ \begin{align}    \int_{-\infty}^{\infty}|(\lambda I-L)^{-1}f|^{2}\,dx & \le     \int_{-\infty}^{\infty}\left(\int_{-\infty}^{x}e^{-\Im\lambda(x-t)}|f(t)|\,dt\right)^{2}\,dx \\     & \le \int_{-\infty}^{\infty}\int_{-\infty}^{x}e^{-\Im\lambda(x-t)}\,dt         \int_{-\infty}^{x}e^{-\Im\lambda(x-t)}|f(t)|^{2}\,dt\,dx \\     & \le \frac{1}{\Im\lambda}\int_{-\infty}^{\infty}\int_{-\infty}^{x}e^{-\Im\lambda(x-t)}|f(t)|^{2}\,dt\,dx \\     & = \frac{1}{\Im\lambda}\int_{-\infty}^{\infty}\int_{t}^{\infty} e^{-\Im\lambda(x-t)}\,dx\,|f(t)|^{2}dt \\     & = \frac{1}{(\Im\lambda)^{2}}\int_{-\infty}^{\infty}|f(t)|^{2}\,dx            =\frac{\|f\|^{2}}{(\Im\lambda)^{2}}. \end{align} $


$w = e^{\frac{2\pi i} {n}}$


$  \left[\begin{array}{rr|r}7.5 & -5 & 0 \\k & 5 & 0 \\\end{array}\right]$


$||A||_{2}=\sqrt{\rho(AA^{t})}=\sqrt{\rho(A^{2})}=\rho$


$\frac{ax+b}{c-x}=y,$


$\Bbb R[x]/(x)$


$ \frac{\tan(\gamma)+\tan(a)}{1-\tan(a)\tan(\gamma)}=\frac{k - \cos(\gamma)}{\sin(\gamma) }$


$\frac{1}{\pi} \left( \arccos(1 - 2p) - \frac{1}{2} \sin 2 \arccos (1 - 2p)\right), 0 < p < 0.5$


$F = L \oplus M \oplus N$


$S=\{S_1,S_2,S_3,... S_n\} $


$\sum_{n=1}^\infty n =1+2+3+4+\dots = -\frac{1}{12}.$


$\left(\frac{n(n+1)}2\right)^2-\left(\frac{(n-1)n}2\right)^2=n^2\left(\frac{n+1}2+\frac{n-1}2\right)\left(\frac{n+1}2-\frac{n-1}2\right).$


$(a-b,a+b) =2$


$A_1 \supseteq A_2 \supseteq \cdots \supseteq A_n \supseteq A_{n+1} \supseteq \cdots$


$f(x)= \frac{1}{x^2}$


$\sum_{n = 1}^{\infty} 1 = \infty$


$\cos^2\varphi = 1 - \sin^2\varphi$


$(\forall x \in A)(\forall y \in A) \rightarrow xRy \lor yRx \lor x=y$


$\sum_{n\in \mathbb{N}} \frac{1}{n^2} = \sum_{n\in \mathbb{N}}^{} \frac{1}{(2n)^2} + \sum_{n\in \mathbb{N}} \frac{1}{(2n - 1)^2}$


$\forall x\forall y (xRy\lor yRx)\to \forall x(xRx)$


$f(a+b) = fa + fb$


$(P, *, *, *, E, E, E) \\ (E, P, *, *, *, E, E) \\ (E, *, P, *, *, E, E) \\ (E, *, E, P, *, E, *) \\ (E, E, *, *, P, *, E) \\ (E, *, E, *, *, P, E) \\ (E, *, *, E, E, *, P) \\ $


$\eta(s) = \frac{1}{\Gamma(s)}\int_{0}^{\infty} \frac{x^{s-1}}{e^{x}+1} \, dx \, , \quad \text{Re}(s) >0. $


$\sum_{k\in\mathbb Z}\binom nkk^2x^k=nx\bigl[(1+x)^{n-1}+(n-1)x(1+x)^{n-2}\bigr]=F(x)\,.$


$ P[Z\geqslant sa]=P[s^Z\geqslant s^{sa}]\leqslant s^{-sa}E[s^Z]=\mathrm e^{-aI(s)}, $


$f(a+(f(x)-x))=f(a)+f(x)$


$x = \frac{n(n + 1)}2$


$\frac{1}{x}+\frac{4}{y} = \frac{1}{12}$


$\{a,a+b,a+c,a+b+c\}$


$\inf_{k \in \mathbb N} \mu(A_k)=\lim_{k \to \infty} \mu(A_k)=0$


$p_1p_2\mid \eta(1)$


$\phi(x)=\frac{1}{2\pi}e^{\frac{-x^2}{2}}$


$\begin{cases}   x=u\\   y=\cos v\\   z=\sin v \end{cases}$


$|ab|=\infty,$


$\frac{\Bbb{R[x]}}{\langle x^2+1\rangle}$


$n_1=n_2=\ldots =n_l=n_{l+1}+1=n_{l+2}+1=\ldots =n_k+1$


$1 + 2 + 3 + 4.... = -\frac 1 {12}$


$(a-\Delta x,f(a-\Delta x))$


$ \cos^2\varphi = \frac{\sqrt{1 - \cos^2 l \cos^2\theta} - \sin l}{1 + \sqrt{1-\cos^2 l\cos^2\theta}},$


$\frac{k(k+1)(4k-1)}{3}+(2k+1)2(k+1)=\frac{(k+1)(4k^2+11k+6)}{3}=\frac{(k+1)(k+2)(4k+3)}{3}$


$n=1 \ldots 128$


$\frac7x + \frac2y = \frac{11}{12}$


$\Delta c_0 = c_1 - c_0$


$Log_{a} (x^{n})=bn$


$\frac{\sqrt{1-\rho^2}}{2\pi(1-\rho \sin(2\theta))}$


$\mathcal{A}\mathcal{X}=\mathcal{B}$


$\,f(x) = x^2 -x + 1.\,$


$\left[\begin{array}{ccccc|c} 1 & 2 & -2 & 2 & -1 & 0\\ 1 & 2 & -1 & 3 & -2 & 0\\ 2 & 4 & -7 & 1 & 1 & 0 \end{array}\right]$


$K=\mathbb{Q}(\sqrt{d})$


$\lim_{n\to \infty} n\sum_{k=1}^n \frac 1{n^2+k^2}$


$G:=\left[\matrix{1&0&-1\cr 2&1&1\cr 3&2&1\cr}\right]\ .$


$ f(x)=9^x - 3^x+1$


$x \, \mapsto \, x^{3}-x-1$


$\lim_{N\to\infty}\sum_{i=1}^N\frac1i=\infty.$


$f(x) = \frac{1}{N}e^x - \frac{1}{N} - e^{x/N}+1$


$\tilde X=\cases{z_1^2+z_2^3+z_3^4+z_4^5+z_5^6=1\\ z_1z_2z_3z_4z_5=1\\ (z_1z_2+z_2z_3+z_3z_4+z_4z_5) \cdot z_6 =1 }$


$\int_{0}^5 \frac{1}{10}e^{-\frac{1}{10}x} = 1-e^{\frac{-1}{2}}=.3934$


$\int_1^\infty \frac{\sin(x)}{x} dx$


$\delta(x) = \frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{inx}$


$\exists L >0,\|\gamma(x)-\gamma(y)\| \leq L |x-y|.$


$A =\frac{αa + βb + γc}{α + β + γ}$


$\mathcal G=\{\{1\}\}$


$ax+by=c$


$\sum_{n\ge1}\frac{1+(-1)^nni}{n^2}=\sum_{n\ge1}\frac1{n^2}+\sum_{n\ge1}\frac{(-1)^ni}n$


$\displaystyle P(B)=\frac{1}{6}$


$|f(a)-f(b)|>k$


$f(x) = \left\{ {\matrix{    {{x^\beta }\sin \left( {{1 \over {{x^\alpha }}}} \right)}  \cr     {0,x = 0}  \cr   } } \right.$


$\alpha^\omega > \alpha$


$\sum_{n=1}^{\infty} (2n - 1)  - \sum_{n=1}^\infty n = \sum_{n=1}^\infty n$


$d(x,a) \leq d(x,y) + d(y,a)$


$\le\sum_{n=1}^\infty\frac{\sqrt n}{n^2}=\sum_{n=1}^\infty\frac1{n^{3/2}}=\zeta(3/2)<\infty$


$ \| \gamma(a)-\gamma(b)\| =d(\gamma(a),\gamma(b)) \le L(\gamma)=\int_a^b \| \gamma '\|.$


$|f(a)−f(b)|≤K|a−b|$


$F=A\oplus X$


$\{a < X \leq b, c < Y \leq d\}$


$\left[\begin{array}{cccccc|c} 1  & 2  &  0  &  0  & a     & 1   & -2 \\ 0  & 0  & 1  & -2 & 2    & -2  & -4 \\ 0 & 0 & -1 & 2  & a^{2}+2a &   2  & 3  \\ 0 & 0 &  0  &  0  & -1  & 0  & 1  \\ \end{array}\right]$


$\sum\|f_n-e_n\|^2< 1$


$\|T\|=\sup_{\|x\|=1}\|Tx\|=\sup_{\|x\|=1}\|x+M\|=\sup_{\|x\|=1}\inf_{m\in M}\|x-m\|\leq\sup_{\|x\|=1}\|x\|=1$


$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{k=1}^n \frac{1}{1+\frac{k^2}{n^2}}$


$\phi(x) = \sum_{j=1}^N a_j \chi_{B_j}$


$D_1, D_2\in{\cal D}\quad\Longrightarrow\quad  D_1\cap D_2 \in \cal D$


$\sum_{n \in \mathbb{N}}\|e_n-f_n\|^2 <1.$


$K \subseteq U \subseteq \overline{U} \text{ compact} \subseteq O$


$A_1 \subseteq A_2 \subseteq A_3 \subseteq\ldots$


$\sum_{k=1}^{\frac{n-1}{2}}\cos\left(\frac{2\pi k}{n}\right)=-\frac{1}{2}$


$\det(B)=\det\begin{pmatrix}     r & a & a & \cdots &a \\     a & r& a & \cdots & a \\      \vdots  & \vdots& \vdots & \ddots & \vdots \\      a & a & a & \cdots & r          \end{pmatrix}=(r+(n-1)a)(r-a)^{n-1}$


$\frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^2} \left ( \sum_{m=-\infty}^{\infty} \frac{1}{m^2+n^2} - \frac{1}{n^2} \right )$


$\frac{a+c}{b+d}=\frac{a+xa}{b+xb}=\frac{a(1+x)}{b(1+x)}=\frac{a}{b}$


$f(n)=\bigl(\frac{n-1}{n}\bigr)^{\log n}$


$x = \frac{a^2 + b^2 + ab}{a + b}$


$c_1x + c_0$


$xSy\iff xRy\land yRx$


$ \left(1+\frac{x}{n}\right)^n\ge\left(1+\frac{x}{m}\right)^m\tag{3} $


$\det(A) = (a+(n-1)b)(a-b)^{n-1}$


$ a_m = \frac{2}{2 \pi} \int_{-\pi}^{\pi} \sum_{n \in \mathbb{N}} 10^{-n} \frac{\sin(nx)}{\sin(x)} \cos\bigg(m \frac{2 \pi x}{2\pi}\bigg) \mathrm{d}x = \frac{2}{\pi} \sum_{n \in \mathbb{N}} \int_{0}^{\pi}  10^{-n} \frac{\sin(nx)}{\sin(x)} \cos(m x) \mathrm{d}x $


$f(x)= \binom{x}{2} = \frac{x^2-x}{2}$


$\alpha^+ = \alpha\cup\{\alpha\}$


$2\times1\times2\times1\times2\times1 \cdots=2^{n/2}$


$n^2-1,\quad n\in\mathbb{Z}$


$P(m) \Rightarrow P(m+1)$


$\{\{a\}\}$


$\lim_{n \to \infty} \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\binom{n-k}{k}\frac{1}{2^{n-k}}}.$


$\lVert x_k - x_m\rVert < 1$


${\large\int}_{-1}^1\frac{dx}{\sqrt[3]{9+4\sqrt5\,x}\ \left(1-x^2\right)^{2/3}}=\frac{3^{3/2}}{2^{4/3}5^{5/6}\pi }\Gamma^3\left(\frac13\right)$


$\mathbf{C}[\gamma^2,\gamma^3]\subset \mathbf{C}[\gamma]$


$\mathrm{gcd}(a+b,a-b)$


$\lim_{x\to a}f(x)=L \iff \lim_{x\to a^+}f(x)=L=\lim_{x\to a^-}f(x)$


$|ab| = |cd|$


$p\mid pn_2$


$M=\pmatrix{A & B\cr B & C\cr}.$


$R= \lbrace (x,y) \in \mathbb{R}^2 \mid a \leq x \leq b, \ c \leq y \leq d \rbrace \subset \mathbb{R}^2$


$ \zeta(s+1)=\frac1{\Gamma(s+1)}\int_0^\infty\frac{x^s}{e^x-1}\:dx, \qquad s>0, \tag1 $


$\mathbb Z[i] = \mathbb Z[x]/(x^2+1)$


$n!\sim\sqrt{2\pi n} \left(\frac{n}{e}\right)^n$


$P[X_n\geq1\space\forall n\geq1\mid X_0=0]=\dfrac{6}{\pi^2}$


$\frac{1}{x} + \frac{1}{y}= \frac{1}{13}$


$φ_a (z)=\dfrac{z-a}{1-\overline{a} z}  ,a∈\mathbb{D}$


$\pi (a,n) > m e^{ \gamma } \prod_{5 \leqslant p \leqslant n} (1 - 1/p)$


$\cos 2 \varphi = 2 \cos^2 \varphi - 1 \\ \sin 2 \varphi = 2 \sin \varphi \cos \varphi,$


$\sum_{k>n}\Vert a_k-b_k\Vert^2<1$


$\lim_{x \to 0} f(x)g(x) = \lim_{x \to 0} f(x) \cdot \lim_{x \to 0} g(x).$


$\int_0^\infty f^2(x)~dx=1$


$||\hat{y}|| = \frac{1}{||f||}$


$f(ax+by)=f(c).$


$d(f,g) = \frac{1}{\sup_x |h(x)|} - 1.$


$ \begin{array}{c} A^\top \gamma\,P + \gamma\,P\,A \leq -\gamma\,I < 0, \\ \gamma\,P \geq \gamma\,I > 0. \end{array} $


$ P(m), m<n \implies P(n) $


$\frac{x^2 y^2 z^2+x^2 y^2 z-x^2 y z^2-x y z-x z^2+y-z+1}{(x y+y+1) (x z+x+1) (y z+y+1)}$


$L:\begin{cases} x=1+2t \\ y=1+2t \\ z=2-t \end{cases}$


$\tag 1 J = (a, \gamma) \sqcup  (\gamma, b)$


$[T_X^{0,1},T_X^{0,1}]\subset T_X^{0,1}$


$Z = \{\{x\}\}$


$ds = \sqrt{1+\left ( \frac{dy}{dx} \right )^2} dx = \frac{dx}{\sqrt{1-x^2}}$


$ p + e = S(p)$


$\mid\mid\dot\gamma\times\ddot\gamma\mid\mid = \mid\mid\dot\gamma\mid\mid\ddot\gamma\mid\mid = \mid\mid\ddot\gamma\mid\mid=\kappa$


$\{u_n\} \subset W_0^{1,1}((0,1))$


$\gamma(s) = (\gamma_x(s), \gamma_y(s), \gamma_z(s))$


$\lim_{n \to \infty} \dfrac{\sqrt 1 + \sqrt 2 + \dots + \sqrt{n}}{n\sqrt{n}}=\lim_{n \to \infty} \dfrac{1}{n}\sum_{r=1}^n\sqrt{\frac{r}{n}}=\int_0^1\sqrt{x}=\frac{2}{3}$


$\lim_{n \rightarrow \infty} f(x_n)\ge f(x)$


$ \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{1}{{\rm lcm}(m,n)}. $


$ \sqrt{\cos^2{u}} = |\cos(u)| $


$m!+2, m!+3,\dots,m!+m$


$\frac4x + \frac9y + \frac{16}z = D$


$\int_0^x f(t)g(t)\,dt = \int_0^x f(t+a)g(t+a) \,dt$


$G_1:=\{x\in H_1: y\in H_2, x=2y\}$


$\gamma(t) = (\gamma_x(t), \gamma_y(t)) = (\gamma_x, \gamma_y)$


$0\in V\subseteq\bar V\subseteq W\subseteq \bar W\subseteq U$


$x_{1}\cdot...\cdot x_{n}$


$(k+1)^3=k^3\cdot\frac{(k+1)^3}{k^3}=k^3\left(\frac{k+1}k\right)^3\;.$


$ \sum_{n\geq 1}\arctan\frac{1}{2n^2}\leq \sum_{n\geq 1}\frac{1}{2n^2}=\frac{\pi^2}{12}<1\tag{1}$


$f\text{ is continuous at }c\iff (\forall\epsilon>0,\exists\delta>0:|x-c|<\delta\implies|f(x)-f(c)|<\epsilon)\tag{1}$


$\begin{bmatrix} a & b \\ -b & a\end{bmatrix}$


$6^x + 4^x =9^x$


$\text{Li}_{s,N}(z) = \frac{z^{N+1}}{\Gamma(s)}\int_0^{+\infty}\frac{x^{s-1}e^{-Nx}}{e^x-z}dx$


$\binom{n}{2}-(n-1) = \binom{n-1}{2}$


$A := \{ \omega \in \Omega: \lim_{n \rightarrow \infty} X_n(\omega) = X(\omega)\}$


$3|14^{2(k+1)}-1 \implies 3|14^2*14^{k+1}-1$


$\int_0^\infty  {xf(x)\,{\rm d}x}$


$\frac{(1+x^2)\cos(x\sqrt{2})-1)}{x^4(1+x^2)}$


$\zeta(s)\int_{0}^\infty \frac{x^{s-1}}{e^x}dx=\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$


$cov(X) - cov(X,Y) cov(Y,Y)^{-1} cov(Y,X)$


$\left \| A \right \|_2 = \sqrt{\lambda_{max} \left ( A^H A \right )}$


$|G| = p\cdot q\cdot r$


$C_{n} = n^2 - n + 2$


$a^{\log_a b} + a^{-\log_a b} > a^{\log_a b} = b$


$ \int_a^b \sqrt{1 + (\frac{dy}{dx})^2} $


$\int \kappa ds = 2\pi$


$|ab| = 35,$


$a^n = (1 + \delta)^n = 1 + n\delta + \cdots + \delta^n > 1 + n\delta.$


$\sum_{k=1}^n \cos(\frac{2 \pi k}{n}) = 0$


$d(x,M)=d(x,P(x))$


$ \zeta(s)=\frac{1}{\Gamma(s)}\left(1-\frac{2}{2^s}\right)^{-1}\int_{0}^{+\infty}\frac{x^{s-1}}{e^x+1}\,dx \tag{2}$


$\frac{\partial l}{\partial \theta} = -(x-\theta) = 0 \implies \theta = x$


$f(x) = \frac{4}{x}+2x+10+\frac{3+x}{4x^2+1}$


$\forall\varepsilon\gt0,\exists \delta\gt 0, \text{ such that } |x| \lt \delta ,|f(x)| \lt \varepsilon \tag{1}$


$P(m)$


$\langle x^3-Q(x), 1 \rangle =0$


$\forall \epsilon>0, \exists \delta_{f,\epsilon} >0, |x| \leq \delta \implies |f(x)-f(0)| \leq \epsilon$


$v(x)=\int_a^x f(t)dt$


$\sum_{n} ||a_n - b_n ||^2 < 1$


$-\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{\cos(\frac{\pi k}{3})\cos(\frac{\pi n}{3})}{n^{2}(k+n)}$


$f^{-1}(x) = \frac{3+2x}{1-x}$


$\int_{0}^{1} \frac{(\sin(x) )}{ x} \,\mathrm dx$


$f(a_1) > a_1.$


$\overline{V} \subseteq U$


$\sum e_n x_n$


$=\int_{y=\frac{1}{2}}^{y=1} 2 \cdot \frac{1}{2} \pi (1-y^2) dy $


$[X,Y]=XY-YX.$


$ \sum_{n=1}^{\infty} \sum_{m\geq n^{\mu-1+\delta}} \frac{\sin(\sin(nm))}{n^2+m^2}. $


$p(e) = p'(e')$


$(\det{\alpha}) (\det{\gamma}) = (\det{\alpha})(\det{\beta}) = dD$


$M=J \oplus T$


$\int \frac{\cos(x)}{(1+\cos(x))^3} \, dx$


$AB=BA=A^*B^*=B^*A^*=B^*A=A^*B=0\quad \text{and}\quad AB^*x=0$


$\frac{w^n-w^{n+1}-w^nz^n+w^{n+1}z^n + z^n-z^{n+1}-w^nz^n+z^{n+1}w^n } {1-w-z+zw^n+wz^n - w^n z^n}.$


$F(a+b) \leq F(a) + F(b)$


$y=\frac{a+bx}{x+c}$


$d(x,b)\leqslant d(x,y)+d(y,b)$


$f(a+b)=f(a)+'f(b)$


$\mathbb A = \{ \{(a,-b),(-c,d) \} | \forall a,b,c,d \in \mathbb{R} \}$


$\log(2)=\lim_{N\to \infty}\sum_{n=1}^N\frac{1}{n+N}$


$ \frac{x}{2} = \tan u$


$U\subseteq V\subseteq X\times X$


$\lfloor\alpha\rfloor+\lfloor\beta\rfloor+\lfloor\gamma\rfloor$


$t(a, b) = (a + b, a - b)$


$ \begin{align} \left(\frac{n!}{n^{n/2}}\right)^2 &=\overbrace{\frac{1(n-0)}{n}\frac{2(n-1)}{n}}^{\ge1}\overbrace{\frac{3(n-2)}{n}\cdots\frac{(n-2)3}{n}}^{\ge\left(\frac32\right)^{n-4}}\overbrace{\frac{(n-1)2}{n}\frac{(n-0)1}{n}}^{\ge1}\\ &\ge\left(\frac32\right)^{n-4} \end{align} $


$1+2+4+8+\cdots = -1$


$F_{X_n}(x)=P(X_n\leq x)$


$ \int_0^\infty f(t)\ dt = \int_0^3 f(t)\ dt = 3.5 $


$\lim\limits_{n\to\infty} \sum\limits_{r=0}^{\lfloor\frac{n}{2}\rfloor}\frac{1}{n}f(\frac{r}{n})$


$ -x^2 y^2 z^2-2 x y z (x+y+z)^3+(x y+x z+y z)^2 (x+y+z)^2+4 x y z (x y+x z+y z) (x+y+z)-2 (x y+x z+y z)^3= $


$f(1)+2^2 f(2)+3^2 f(3)+..+(n-1)^2 f(n-1)+n^2 f(n)=n^3 f(n)$


$\gamma(st)=\gamma(s+...+s)=\gamma(s)+...+\gamma(s)$


$ M=\begin{pmatrix}A&B\\ -B &A\end{pmatrix}, $


$\ds{\int_0^\infty f(x)\, dx}.$


$I_n = \frac12 \int_{-\pi}^\pi \frac{\cos (nx)}{a+\cos x}\,dx,$


$\mathsf P(n)\implies\mathsf P(n+1)$


$\begin{pmatrix} 1 & x &x^2\\1&y&y^2\\1&z&z^2\end{pmatrix} \begin{pmatrix} a_0\\a_1 \\ a_2\end{pmatrix} = \begin{pmatrix} c_0\\c_1 \\ c_2\end{pmatrix}$


$x = a^{\log_a(x)}$


$\int\frac{\tan{x}}{(x^2 +1)^2}dx$


$\int{\frac{1}{(x^2 + a^2)^n}}dx, n,a>0$


$\sin(\theta)=\frac{\tan(\theta)}{\sqrt{1+\tan^2(\theta)}}$


$P(X_{n+1}=x'|X_t=x)\geq \epsilon>0$


$=2\pi\lim_{n\to\infty}\sum_{k=1}^n \left(\frac{f(x_k^*)-mx_k^*-b}{\sqrt{m^2+1}}\right)\sqrt{1+\left[f'(x_k*)\right]^2}\left(\frac{1+mf'(x_k^*)}{\sqrt{m^2+1}}\right)\Delta{x}$


$x_n \to x \implies \dfrac{x_1+x_2+\ldots +x_n}{n} \to x$


$\lambda^+$


$S= \int 2 \pi y ds$


$ \begin{align} \int_0^1\frac{\sin(x)}{x^a}\,\mathrm{d}x &\le\int_0^1x^{1-\alpha}\,\mathrm{d}x \end{align} $


$ \bigl(\gamma(a_1)\,\gamma(a_2)\,\dots\,\gamma(a_k)\bigr). $


$\star x=\{x,\top\}$


$f(x) = \frac{x^3}{1+x^3}$


$B = A_1 \times A_2 \times \cdots \times A_n \times \cdots$


$ \lnot (\exists c_0) (\forall x) (x = c_0) $


$ \int_0^{\infty} \int_0^{\infty} f(x,y) dydx $


$\forall \epsilon>0, \exists\delta>0,|x-7|<\delta\implies|f(x)-f(7)|<\epsilon.$


$\lim_{x\to a} f(x) = L \iff \forall \varepsilon \gt 0 \ \exists \delta  \gt 0 \ni \left| x-a \right| \le \delta \Rightarrow  \left| f(x)-L \right| \le \varepsilon$


$B(x, R-||x||_2)\cap F = \{x, e_2, e_3, \dots, e_n\}$


$d(x,y)\leq d(x,x_{n})+d(x_{n},y_{n})+d(y_{n},y).$


$0\le e^x-\left(1+\frac xn\right)^n\le\dfrac{x^2e^x}{2n}$


$\displaystyle P\bigg(\int_0^1e^{as}dB_s\le\int_0^2e^{as}dB_s\bigg)=P\bigg(\int_1^2e^{as}dB_s\ge 0\bigg)=1/2$


$(p^{2})^{2} = p\sin(\phi) \cos(\theta)$


$\lim_{n\to\infty} \sum_{r=1}^n \frac 1 {2^r}.$


$(78,14) =2$


$(x^n - y^n) = (x+y)^{n-1}(x-y)^{n-1}$


$p_1 p_2\mid a$


$f(x,y) = \frac{1}{2\pi\sqrt{\sigma^2(1-p^2)}} \exp\left[\frac{-(x^2-2py+y^2)}{2\sigma^2(1-p^2)}\right]$


$d(x',a)-d(x,a)\leqslant d(x',x)$


$I_n = \int_{-\pi/2}^{\pi/2} (\cos{x})^n$


$6^{x+1} - 6^x = 3^{x+4} - 3^x$


$ g^1 \gamma -\gamma = \partial(H\gamma) - H(\partial \gamma).$


$|f(z)| = 2|g(z)|$


$dl=\sqrt{r^2+\left(\frac{dr}{d\phi}\right)^2}$


$S=\{\{a\},\{b\}\}$


$\|\psi_i^n-e_i\|\le 1/n$


$2J_n^{(0, -1)}(x)= J_n^{(0, 0)}(x)+J_{n-1}^{(0, 0)}(x) $


$d(x,A) \leq d(x,y) + d(y,p)$


$\operatorname{dVol}(e_1,e_2,e_3,\ldots,e_n) = 1$


$\text{Spec } \mathbb{R}[x]/(x^2 + 1)$


$\{ x^2, xy, y^2, xz, yz, z^2 \}$


$xRy=yRx$


$\sum_{k=1}^\infty \dfrac{1}{n}\cos\left(\dfrac{k\pi}{2n}\right)$


$H(x) = \int_a^{x}f(t)g(t) dt$


$\|A\| = \sqrt{\mbox{tr}(A^H A)}$


$(a^{1/n} - \epsilon')^n < a - \delta < x < a + \delta < (a^{1/n} + \epsilon')^n$


$\displaystyle{d(a, H)=\frac{|f(a)|}{\|f\|}}$


$Pr(A) = 1/6$


$\lim_{N\to\infty}\sum_{n=1}^{N}\frac{a_n}{n^2}$


$ \begin{align} &A_x=\{(a,b) \in\mathbb{N}^2 : a+b=x,\,\,1<a\le b\} \\ &B_x=\{(a,b) \in\mathbb{N}^2  : ab=x,\,\,1<a\le b\} \\ &C_x=\{ab : (a,b)\in A_x\} \\ &D_x=\{a+b : (a,b)\in B_x\} \\ &R=\{p,\,pq,\,p^3 \,:\, p,q\,\text{  are prime}. \} \\ &T=\{x : C_x \cap R=\emptyset\} \end{align} $


$ \zeta(s)=\left(1-\frac{2}{2^s}\right)^{-1}\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^s}=\left(1-\frac{2}{2^s}\right)^{-1}\frac{1}{\Gamma(s)}\int_{0}^{+\infty}\frac{t^{s-1}}{e^t+1}\,dt $


$U(A)\subseteq V$


$\boldsymbol{x} = (x,y)$


$log(2^x + 3^x - 4^x + 6^x - 9^x) = log (1296^x) = x log(1296)$


$(6^k + 3) - (4^{k-1} + 2)$


$\begin{cases} x=1+16t\\ y=7t\\z=-1-19t \end{cases}$


$\delta^{(n)}(-x) = (-1)^n\delta^{(n)}(x)$


$x^2(x-y)(x-z)+y^2(y-z)(y-x)+z^2(z-x)(z-y)\ge0$


$ f(0) = 0\\ f(\pi) = 0\\ f(2\pi) = 0\\ $


$={9\over 9+3}$


$\sum_{n=1}^{\infty}\frac{n^7}{e^{n\pi}-1}-16^2\sum_{n=1}^{\infty}\frac{n^7}{e^{4n\pi}-1}=\frac{17}{32}$


$\pi^{-(1-s)/2}\Gamma(\frac{1-s}{2})\zeta(1-s)=\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)$


$\frac{(1-p)p}{1-(1-p)^2} = \frac{(1-p)p}{1-1+2p-p^2}=\frac{1-p}{2-p}$


$\|A\|=\sqrt{\lambda_\max(A^\ast A)}$


$\|x\| \geq \sup_{y \neq 0} \frac{|\langle x,y \rangle|}{\|y\|}$


$U^{\bot\bot} \cap V^{\bot\bot} \subseteq(U^{\bot\bot}\cap V)\subseteq (U\cap V)^{\bot\bot\bot\bot}$


$s(x)=\sum\limits_{n=0}^k \alpha_n\chi_{A_n}(x)$


$(A_1\times ... \times A_{n+1})$


$F(a+b)=F(a)+F(b)$


$\gcd(a, p_1 p_2)\ne 1$


$F_a(x) = \left\{ \begin{array}{lr}   f_a & 0 \le x \le \gamma \\ 0 &  x > \gamma \\ \end{array} \right.$


$\frac{\sum\limits_{r=1}^{\infty}r^0}{\sum\limits_{r=1}^{\infty}r}$


$\log_a B$


$\overrightarrow{RP}=\pmatrix{2\\-6\\-1}$


$\forall \delta \gt 0, \exists\epsilon \gt 0 : |x-a| \lt \delta \implies |f(x)-f(a)|\lt \epsilon$


$\left[\begin{array}{rrr|r}1&a&0 & 1\\0&1&b & 1\\c&0&1 & 1\end{array}\right]$


$ \begin{vmatrix}     1  & 1  & 1 \\     x  & y  & z \\     yz & xz & xy  \end{vmatrix}  $


$\sum ^\infty_1 \| e_n - f_n \| < 1$


$\lfloor\frac{\lfloor\frac{x}{n}\rfloor}{m}\rfloor = \lfloor\frac{x}{mn}\rfloor$


$ \begin{array}{ccc} \left(\alpha\beta\gamma\right)^{2}-a_{0} & = & 0\\ -2\alpha\beta\gamma\left(\alpha\beta+\gamma\beta+\alpha\gamma\right)-a_{1} & = & 0\\ (\alpha\beta+\gamma\beta+\alpha\gamma)^{2}+2\alpha\beta\gamma(\alpha+\beta+\gamma)-a_{2} & = & 0\\ -2\alpha\beta\gamma-2(\alpha+\beta+\gamma)(\alpha\beta+\gamma\beta+\alpha\gamma)-a_{3} & = & 0\\ (\alpha+\beta+\gamma)^{2}+2(\alpha\beta+\gamma\beta+\alpha\gamma)-a_{4} & = & 0\\ -2(\alpha+\beta+\gamma)-a_{5} & = & 0 \end{array} $


$|F(z)|\leq |f(z)||f(−z)|\leq 2\cdot 1.$


$\frac{1}{2^{n-1-1}} = \frac{1}{2^{n-2}}$


$3^x + 4^x + 5 ^x = 6^x$


$\lim_{x\to c^{+}} ~ f(x) =l$


$\frac{\sqrt[n]{x}}{(1+\frac{x}{n})^n}\sim \sqrt[n]{x}e^{-x}$


$\lfloor \bar{.9}\rfloor = \left\lfloor \sum_{i=1}^{\infty}\frac{9}{10^{i}}\right\rfloor=\lfloor 1 \rfloor = 1.$


$\mathbf P[X_1 < X_0 \, | \, X_0 = p] = p.$


$\leq \lim_{n\rightarrow \infty}\inf\int_{E_n} f d\mu = \lim_{n\rightarrow\infty}\inf\int f 1_{E_n} d\mu$


$f(y^2f(x)+x^2f(y))=xy(f(x)+f(y)),\;\forall x,y\in [0,+\infty)$


$\gamma^3+\gamma(\gamma+1)=\gamma^3\gamma^4+\gamma^5+\gamma^4=(\gamma^2+\gamma)(\gamma^3+\gamma^2)=\varphi(\alpha)\cdot\varphi(\alpha\beta)$


$f_Y(x) = \frac{1}{\sqrt{2\pi}} e^\frac{-x^2}{2}$


$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$


${C_R}: |z| = R$


$a_n = \frac{1}{2}n(n+1) + 1.$


$V \subseteq \overline{V} \subseteq U$


$\{(x,y):a<x\le b, c<y\le d\}.$


$\tan(\theta) = x/2$


$\sum_{n\in \mathbb Z} e^{-n^2\pi x} = \frac{1}{\sqrt{x}}\sum_{n\in \mathbb Z}e^{\frac{-n^2\pi}{x}} $


$\frac{\pi^2}{6}=\sum_{n=1}^\infty\frac{1}{n^2}=\sum_{n=1}^\infty \frac{1}{(2n-1)^2}+\sum_{n=1}^\infty\frac{1}{(2n)^2}$


$\mathbb{R}[x]/( x^2 + x + 1 )$


$\frac 1 3 < x < \frac 1 2 \implies 2\pi < \frac{\pi}{x} < 3 \pi \implies 1 < 1+ \sin \frac{\pi}x <2$


$ \det A=c^{n(p-1)}(c-p)^{n-1}(c+p(n-1))\;. $


$k^3-1 = (k-1)(k^2 + k +1) , (a-d)/(b-c) = (k^2 +k+1)/k >= 3$


$\Bbb{P}[X_t \in B | X_s] = \Bbb{P}[X_{t-s} \in B | X_0 = x]_{x= X_s}$


$K=\Bbb R[x]/(x^2+1)$


$ f(2n)=f(2)f(n)-f(n+1)=3(n+1)-(n+2)=2n+1 $


$\displaystyle \int_0^\infty f_X(x)\,dx=1$


$\phi\circ\gamma(a+b) = \phi(\gamma(a+b)) = \phi(\gamma(a)+\gamma(b)) = \phi(\gamma(a)) + \phi(\gamma(a)) = \phi\circ\gamma(a)+\phi\circ\gamma(b)$


$t = \frac{1+2x}{1-x}$


$\sum_{n\geq 1}\frac{4}{(2n+2)^2-4}=\sum_{n\geq 1}\frac{1}{n(n+2)}$


$\sigma\gamma = \gamma\tau\gamma^{-1}\gamma = \gamma\tau\circ e = \gamma\tau$


$[X,Y]=X*Y-Y*X$


$\gamma^{4}+\gamma+1=(\gamma +b)(c\gamma^{3}+d\gamma^{2}+e\gamma+\gamma)$


$f(a+bx)=f(a)+f(b)f(x)$


$\gamma \cap \{a\} = \emptyset$


$z=\sqrt[4]{3}e^{\frac{\pi ki}{2}}\Longleftrightarrow$


$\lfloor-p^\gamma\rfloor-\lfloor-(p+1)^\gamma\rfloor$


$f(x)=\sqrt{2\pi x}\left(\frac{3}{e}\right)^x$


$\left(1+\frac{x}{N}\right)^N \approx e^x$


$[A \mid b] = \begin{bmatrix}-3 & 6 & -1 & 1 & -7\\ 1 & -2 & 2 & 3 & -1\\ 2 & -4 & 5 & 8 & -4 \end{bmatrix}$


$ \mathbb E(|X+\mathbb E(Y)|)\leqslant\mathbb E(|X+Y|). $


$ \zeta(s)=\sum_{n\geq 1}\frac{1}{n^s}=\left(1-\frac{2}{2^s}\right)^{-1}\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^s}\stackrel{\mathcal{L}^{-1}}{=}\frac{1}{\Gamma(s)}\left(1-\frac{2}{2^s}\right)^{-1}\int_{0}^{+\infty}\frac{t^{s-1}}{e^t+1}\,dt $


$\prod\limits_{k = 1}^{n} \sin \dfrac{k\pi}{n} = 0$


$ \frac{d}{dx}[f'(x)^2 + f(x)^2] = 2f''(x)f'(x) + 2f'(x)f(x) = 2f'(x)[f(x) + f''(x)] $


$\displaystyle \sum_{k=-9}^{9} e^{-ikx} =  e^{9ix} \sum_{k=0}^{18} e^{-ikx} $


$=-2\int \frac{d(1-\tan x)}{\left(1-\tan x\right)^2}$


$ b = \min\{s \in [a,1] \mid \gamma(s) = -\gamma(a)\} $


$\det\begin{pmatrix}A&B\\0&D\end{pmatrix}=\det(A)\det(D)$


$\tilde{\nabla}_\dot{\gamma}\dot{\gamma}=(\nabla_\dot{\gamma}\dot{\gamma})^\top=(\partial_\dot{\gamma}\dot{\gamma})^\top=\ddot{\gamma}^\top,$


$\mathbb{R}^n \sqcup \mathbb{R}^n$


$k!+2, k!+3, \dots, k!+k$


$K \subseteq V \subseteq \operatorname{Cl}(V) \subseteq U$


$V = \lbrace v_1, v_2, v_3,....,v_n\rbrace$


$(3, 7) = 1$


$A = A_0 \subseteq A_1 \subseteq A_2 \subseteq ...$


$\left(1+\frac xn\right)^n<e^x<\left(1-\frac xn\right)^{-n}$


$x_0, x_1, x_1 \in \Bbb N$


$x! \approx \sqrt{2\pi x}\left(\frac{x}{e}\right)^x, \text{ for } x \to \infty$


$\begin{bmatrix} a & -\overline b \\ b & \overline a \end{bmatrix}$


$ !Z= \begin{bmatrix} A & -B\\B&A \end{bmatrix} \ , $


$\det M(A,B)=\det M(A,-B)$


$ \underline{Lim} A_n = \{ x \in X : \exists n_0 \in \omega \forall n \geq n_0, x \in A_n \} $


$\displaystyle i\sum_{n=-\infty}^\infty \sum_{m=1}^\infty \frac 1{n+n^3}e^{inx}$


$\deg(a) = 4, \ \deg(b) =3,\ \deg(c) = 3, \ \deg(d) = 3,\ \deg(e) = 3, \ \deg(f) = 3, \ \deg(g) = 3, \ \deg(h) = 2, \ \deg(i) = 2.$


$\forall \epsilon\gt0, \exists \delta\gt0$


$f_X(0;\theta) = \theta$


$B = \{ \{d\} ,\{s\}\}$


$\|e_n - e_m\| = 1$


$A\vec v = \left[\matrix{Ry_1+Qy_1'+Py_1''\cr Ry_2+Qy_2'+Py_2''\cr Ry_3+Qy_3'+Py_3''\cr }\right]=\left[\matrix{0\cr 0\cr 0\cr}\right],$


$t=1 \implies \frac x 2=\frac{\pi}4+k\pi \implies x=\frac{\pi}2+2k\pi $


$[x, y] = \{x, y\}$


$ [F_n, F_n ] \subseteq F_n $


$\bigg\lfloor\frac{\lfloor \frac{x}{m}\rfloor}{n}\bigg\rfloor=\bigg\lfloor \frac{x}{mn}\bigg\rfloor$


$\frac{x}{2}=\sin \theta.$


$R = A_1 \times \dotsc \times A_s$


$\displaystyle\sum_{k = -(n-1)}^{n-1}e^{-k^2\tfrac{\pi}{n}} = \sum_{k = -\infty}^{\infty}\sqrt{n}e^{-n\pi k^2} - \displaystyle\sum_{|k| \ge n}e^{-k^2\tfrac{\pi}{n}}$


$f(x) = \limsup_n \frac{x_1+x_2+\cdots+x_n}{n}$


$E(|X+Y|^r)\leq c_r[E(|X|^r)+E(|Y|^r)]$


$\lim_{k\to \infty}\mu(A_k)=\mu(A)$


$f_2(3n,\ 3k) = \binom{n}{k}$


$f(x) = \frac{a+bx}{1+cx}$


$ \|A^TA\|_2=\rho(A^TA)=\|A\|_2^2. $


$P(t|F) = p, P(f|T)=q, P(t|T)=r, \text{ and } P(T) = P(T|f)P(f) + P(T|t)P(t)$


$L\{f(t)*g(t)\}=\int_0^t f(t-\tau)g(\tau)d\tau=\int_0^t f(\tau)g(t-\tau)dt=\int_0^tf(t)dt$


$d(\gamma(a),\gamma(b)) \le L(\alpha)$


$\lim_{n \to \infty} \inf s_n \le \lim_{n \to \infty} \inf t_n.$


$ \lim_{n\rightarrow\infty}[(1+i)^n+(1-i)^n]. $


$W^*(N) = W^*(C^*(N,I))$


$\lim_{n\to\infty} \mu(A_n)=0$


$J(y)=\int_{0}^{1}{ \sqrt{1+\frac{dy}{dx}  ^{2}}  dx}$


$ f(f(x)^2y)=x^3f(xy)\space\forall x,y\in\mathbb{Q^+} $


$[x,y] = [-1,i]$


$\text{Spec}(\mathbb{C}[s]/(s^2 - 1)) = \mathbb{Z}/2$


$\   M=   \left[ {\begin{array}{ccccc}    1 & \alpha & -1 & 2 \\    2 & -1 & \alpha & 5 \\    1 & 10 & -6 & 1 \\   \end{array} } \right]$


$D=\{\langle x,y\rangle\in E^2:x^2+y^2\le 1\}$


$d(x,a)-d(x',a)\leqslant d(x,x')$


$|X+Y|^2 \le |X|^2 + |Y|^2$


$\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{\left(-1\right)^{k+1}}{k^{2}+x^{2}}$


$\lim \limits _{n \to \infty} \sum _{i=0} ^{n} \frac{1}{2^i}$


$F(x) = \int_a^x f(y) \textrm{d}\mu$


$x^3 - x^2 - x - 1$


$(\frac 12)^2$


$Ax + By = C$


$\displaystyle \int |u|^p dμ \le C^p(\int |u|^r dμ)^{p/r} $


$\operatorname{support}(\widehat{f}) \subseteq [-\gamma, \gamma]$


$\sum_{n=1}^\infty \|e_n - f_n\| \leqslant \frac{1}{8M\|P\|},$


$\int_a^b f(t) \sin(t) \mathrm{d}t=\int_a^b f(t) \cos(t) \mathrm{d} t=0. $


$P \subseteq V$


$\begin{bmatrix}a & -b\\b & a\end{bmatrix},$


$Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z)$


$J_n=\int_{-\pi}^{\pi}\frac{\cos(nt)}{1+r^2-2r\cos t }dt$


$P = (C+E)\dfrac{r(1+r)^N}{(1+r)^N-1}$


$f'(x)=2 (x-1)-\frac{2}{5} e^{-\frac{x^2}{10}} \left(2+e^{-\frac{x^2}{10}}\right) x$


$[x,y] = x^2 = (a\ b\ c)$


$n!+2, \dots n!+n$


$\mathrm{P}[X\geq x]=0$


$ 1 + 2 + 3 + 4 + \cdots = \sum_{k=1}^\infty k = -\frac{1}{12}. $


$[ 2 (k + 1)^3 + 3 (k + 1)^2 + (k + 1) ] / 6 $


$x! \approx. \sqrt{2\pi x} \left(\frac{x}{e}\right)^x$


$\lim \limits_{n\to +\infty } \left(1+\frac{x}{n}\right)^n=\text{e}^x$


$|f(x_n)| \leq 2^{-n} |f(x)|$


$\lim_{x\to 0^+} f'(x)=\lim_{x\to 0^-} f'(x)=0$


$\|x \| = \sup_{y \neq 0} \frac{|\langle x,y \rangle|}{\|y\|}$


$x > \frac{1}{\pi} \implies \frac{1}{x} < \pi \implies 0<\sin \frac{1}{x} < 1 \implies \left\lfloor \sin \frac{1}{x} \right\rfloor = 0$


$=\sum \limits_{n=1}^\infty (-1)^{n+1}\frac{1}{ \left\lfloor(n+1)/2\right\rfloor}$


$\frac{\alpha}{||F||}$


$\mathrm{dist}(x,\ker f)=\frac{|f(x)|}{\|f\|}$


$\Bbb R^{n+m}=\Bbb R^n$


$\Bbb E[|XY|] \leq \sqrt{\Bbb E[|X|^2] \Bbb E[|Y|^2]}$


$[x,1] = x$


$ (x, y) + [(x', y') + (x'', y'')] = (x, y) + (x' + x'', y' + y'') $


$\log_a b+\log_a c =\log_a (bc)$


$\begin{align*}
 \lim_{n\to\infty}\frac{\ln\left(\frac{1}{2} + \frac{1}{\pi}\arctan\frac{nx}{t}\right)}{\frac{1}{n}} &\stackrel{\mathrm{L'H}}{=} \lim_{n\to\infty}\left(\frac{1}{\frac{1}{2} + \frac{1}{\pi}\arctan\frac{nx}{t}}\right)\left(\frac{x}{t\pi}\right)\left(\frac{-t^2n^2}{t^2+n^2x^2}\right)\\
 &=\frac{1}{1}\times\frac{x}{t\pi}\times\frac{-t^2}{x^2}\\
 &= -\frac{t}{x\pi}.
 \end{align*}$


$x_i ==> \infty$


$\mbox{denote }a=\frac{x_1+x_2+\ldots+x_n}{n}$


$f(a \cdot b)=f(a)+f(b)$


$\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{n^2k^2(n+k)^2}$


$a, a+d, a+2d,...,$


$\{a+b,a-b\}$


$ f(x+\delta)=(x+\delta,A(x+\delta))=(x,Ax)+(\delta,Ax)+(x,A\delta)+(\delta,A\delta). $


$\displaystyle \sum_{x=1}^n \frac{1}{x^2} = \sum_{x=1}^\infty \frac{1}{x^2} + O\left(\frac{1}{n}\right) = \frac16 \pi^2 + O\left(\frac{1}{n}\right)$


$\sin(x)\int_x^{x+2\pi}f(t)\cos(t)dt-\cos(x)\int_x^{x+2\pi}f(t)\sin(t)dt=0$


$\bigoplus\limits_P E_n:=\{(x_n):x_n\in E_n,\sum\limits_n\lVert x_n\rVert^pw_n<\infty\}$


$f(b)<k<f(a)$


$ \frac { \left( 1- \left( \tan \left( \frac x 2 \right)  \right)^2 \right)^2}{ \left( 1+ \left( \tan \left( \frac x 2 \right)  \right)^2 \right)^2 (1+\tan(x)) } $


$\int f(x) = \frac14 \left [-4e^\frac{-x}{2}(-\frac{x}{2} - 1) + C \right] $


$I_n=(-(2\pi n + \pi), 2 \pi n).$


$\tan\theta = \displaystyle\frac{x}{2}$


$\Gamma(s) \zeta(s) = \lim_{N \to \infty} \Gamma(s) \sum_{n=1}^N n^{-s} = \lim_{N \to \infty} \int_0^\infty x^{s-1} \frac{1-e^{-Nx}}{e^x-1}dx = \int_0^\infty \frac{x^{s-1}}{e^{x}-1} dx$


$\gcd(r,s) = 1, z = rs$


$P[X=x|X+Y=k]$


$xRy, yRz \Rightarrow xRz$


$\int_0^\infty f_n(x)\,dx = 1$


$f[x,y]=xy$


$(k+1)^3 - (k-1)^3 = 6 k^2 + 2$


$z=e^{i\vartheta}$


$\lim_{n \rightarrow \infty} \sum_{k=n}^\infty 1$


$a^n - a^{n-1} > \delta$


$P(X_n\leq x) = x$


$[x,y]=h$


$Cov(X,Y)=Cov(X,g(Y)),$


$A=\{(x,y)\in \mathbb{R}^2:a\leq x\leq b, f(x)\leq y\leq g(x)\}$


$\int_0^\infty f^2(x)dx$


$f^{-1}(f(x) + f(e)) = x$


$\vartheta(u+\frac 1\tau,-i\tau)=e^{\pi/q+2\pi\tau}\vartheta(u,-i\tau)$


$\begin{align} \int\frac1{(1+\sqrt x)^4}\,dx&=2\int \frac{u}{(1+u)^4}\,du\\\\ &=2\int \frac{t-1}{t^4}\,dt\\\\ &=\frac{2}{3t^3}-\frac{1}{t^2}+C\\\\ &=\frac{2}{3(1+u)^3}-\frac{1}{(1+u)^2}+C\\\\ &=\frac{2}{3(1+\sqrt x)^3}-\frac{1}{(1+\sqrt x)^2}+C \end{align}$


$p\mid n_1n_2$


$\left\lfloor\frac{a}{b}\right\rfloor \geq \left\lfloor\frac{a}{b+1}\right\rfloor$


$E[(X+Y)^p] \le 2^{p-1} (E[X^p] + E[Y^p])$


$\frac{n}{{2^{n - 1} }}$


$(f'(x+1)+f'(x-1))f(x)-(f(x+1)+f(x-1))f'(x)=0$


$ |E_A|\leq \gamma_2|A|, \quad |E_B|\leq \gamma_2|B|. $


$\lVert A \rVert_F=\sqrt{\operatorname{tr}{(A^TA)}}$


$d(x,y)\le d(x,a) + d(a,y)$


$\lim\limits_{n \to ∞} \frac{{\sqrt 1}+\cdots +{\sqrt n}}{n{\sqrt n}}$


$\zeta\left(s\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}dx,\, s>1.$


$\int_0^{\infty} |f(x)|$


$u=\left(ac^{\frac{1-\gamma}{\theta}}+b d^{\frac{1-\gamma}{\theta}}\right)^{\frac{\theta}{1-\gamma}}\Rightarrow u^{\frac{1-\gamma}{\theta}}=ac^{\frac{1-\gamma}{\theta}}+b d^{\frac{1-\gamma}{\theta}}$


$L^{AB}_1 \subseteq L^{AB}_2\subseteq \cdots \subseteq L^{AB}_{k-1}\subseteq L^{AB}_{k}\subseteq \cdots\subseteq L^{AB}_{n}$


$C_R\cap \mathbb{R}$


$M=N\oplus K$


$\lim_{n\to\infty} \mu(A(n)) = \mu(A(\infty))$


$ r^n e^{rx} = \pm k^2 e^{rx} \quad \Longrightarrow \quad r^n = \pm k^2 \quad \Longrightarrow \quad r = \sqrt[n]{\pm k^2} e^{\frac{2\pi i j}n}, \quad 0 \le j \le n-1. $


$\lim_{n\to \infty} \sum_{r=1}^n {\frac{r^4}{4r^2-1}}$


$\int_0^\infty x dF(x),$


$(e_i)_{1\le i\le n}$


$f(t)=\sum_{j=1}^{n}\alpha_{j}\chi_{S_{j}}(t)$


$\varphi(x) = \sum_{n = 1}^{N} c_{n} \chi_{E_{n}}(x)$


$ D = \{(x, y) | a \leq x \leq b, \ c \leq y \leq d\}.$


$\ker A:=\{\vec{x} \in \mathbb R^n \mid A\vec{x}=0\}$


$i < \alpha^{+}$


$P_{m - 1} \implies P_m$


$\int_0^\infty f(x) x \ dx$


$A_1\supset A_2\supset\cdots A_n\supset A_{n+1}\supset\cdots$


$m_1m_2\mid n$


$\int\limits_{0}^{1200} \int\limits_0^{1200-x}(\frac{1}{1200})^2 e^{\frac{-x}{1200}} e^{\frac{-y}{1200}} dydx = \int\limits_{0}^{1200}(\frac{1}{1200})e^{\frac{-x}{1200}} \int\limits_0^{1200-x}(\frac{1}{1200}) e^{\frac{-y}{1200}} dydx = \int\limits_{0}^{1200}(\frac{1}{1200})e^{\frac{-x}{1200}} [-e^{\frac{-y}{1200}}]^{1200-x}_0 dydx = \int\limits_{0}^{1200}(\frac{1}{1200})e^{\frac{-x}{1200}}[1-e^{-(1-\frac{x}{1200})}]dx = \int\limits_{0}^{1200}(\frac{1}{1200})e^{\frac{-x}{1200}}-\frac{1}{1200}e^{-1}dx = [1-e^{-1}]-e^{-1} =1-2e^{-1} = Poisson(N\geq2, \lambda = 1)$


$Cov(Y, X) = Cov(Y', X)$


$\lim_{k \rightarrow \alpha^+} e^{ik}$


$ \lim_{n\to\infty}   \frac{\dfrac{\sqrt{n}+1}{n\sqrt{n}+\sqrt{n}+1}}{\dfrac{1}{n}}= \lim_{n\to\infty}\frac{n\sqrt{n}+n}{n\sqrt{n}+\sqrt{n}+1}= \lim_{n\to\infty}   \frac{n\sqrt{n}}{n\sqrt{n}}   \frac{1+\dfrac{1}{\sqrt{n}}+\dfrac{1}{n\sqrt{n}}}        {1+\dfrac{1}{n}+\dfrac{1}{n\sqrt{n}}}=1 $


$ \begin{align}  &\coth(x \pi)=\frac{x}{\pi}\sum_{n=-\infty}^\infty\frac{1}{x^2+n^2}   & \cot(x \pi)= \frac{x}{\pi}\sum_{n=-\infty}^\infty\frac{1}{x^2-n^2}         \\  & \text{csch}(x \pi )=\frac{x}{\pi} \sum_{n=-\infty}^\infty{\frac{(-1)^n }{x^2+n^2}}   &\csc(x \pi)= \frac{x}{\pi} \sum_{n=-\infty}^\infty{\frac{(-1)^n }{x^2-n^2}} \\ & \tanh(x \pi)=\frac{4x}{\pi}\sum_{n=-\infty}^\infty{\frac{1}{(2n+1)^2+4x^2}} &\tan(x \pi) = \frac{4x}{\pi}\sum_{n=-\infty}^\infty{\frac{1}{(2n+1)^2-4x^2}} \end{align} $


$\hat{\beta}=\frac{X_1+X_2+\ldots+X_n}{n}$


$\ ax + by = j$


$df_\lambda(x,y,z)=\left[\matrix{2x &2y &-2z\cr 1&1&\lambda\cr}\right]\ .$


$\beta^2(s)=||\gamma(s)-\gamma(0)||^2=\langle\gamma(s)-\gamma(0),\gamma(s)-\gamma(0)\rangle\\=\langle\gamma(s),\gamma(s)\rangle-2\langle\gamma(s),\gamma(0)\rangle+\langle\gamma(0),\gamma(0)\rangle$


$f(x)= \frac{1}{1000}e^{\frac{-x}{1000}}$


$f(x) = x^2 \frac{1-\frac{1}{(1+x^2)^{n+1}}}{1-\frac{1}{1+x^2}}=x^2\frac{(1+x^2)^{n+1}-1}{x^2(1+x^2)^{n}} = \frac{(1+x^2)^{n+1}-1}{(1+x^2)^{n}}$


$\displaystyle \int_0^\infty f(t)g(t)dt\le 0$


$g : \overline{\mathbb R}\to \overline{\mathbb R}$


$\sum_{n=1}^\infty a_n\leq\sum_{n=1}^\infty\frac{1}{n^2}+\sum_{n=1}^\infty \frac{1}{n^{4/3}}=\frac{\pi^2}{6}+\zeta(4/3)<\infty.$


$\color{green}{\left\{ \begin{array}{rcl} s = x + 1 \\ t = y + 3 \end{array}\right.} \iff \left\{ \begin{array}{rcl} x = s - 1 \\ y = t - 3 \end{array}\right. $


$P(\bar{A} \cap B) = 1/6$


$ds=\sqrt{(dx)^2+(dy)^2}=\sqrt{\left(-\frac12\sin t\right)^2+\left(\frac12\cos t\right)^2}\,dt=\frac12dt$


$\sum_{n=1}^{\infty} n-\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty} (n-n)=\sum_{n=1}^{\infty} 0=0.$


$\sum_{n=1}^\infty \frac{1}{n^2(n+1)} = \sum_{n=1}^\infty \frac{1}{n^2} - \sum_{n=1}^\infty \frac{1}{n(n+1)} = \frac{\pi^2}{6} - 1$


$\mathbb{R}^{n+1} = \mathbb{R}^{m+1} \times \mathbb{R}^{n-m}$


$\int_{-\infty}^{\infty}\frac{\sin^2{x}}{x^2}dx$


$R^+$


$|f(z^2)| ≤ 2|f(z)|$


$\frac{n+1}{2}C_n, \frac{n+3}{2}C_n, \ldots, (n-2)C_n, (n-1)C_n$


$ \frac {4}{n}+\frac {10}{n}=$


$ |f(re^{i\gamma})|=r^ae^{-r\sin\gamma}\le \Bigl(\frac{a}{\sin\gamma}\Bigr)^ae^{-a},\quad r>0. $


$\frac{1}{n^2}(n+2)\cdot 2^{n-1}$


$x^2+3x-10+\frac{3}{x}+\frac{1}{x^2}=0.$


$\bar \partial \pi :T^{0,1}M\rightarrow T^{0,1}N$


$A\simeq A_1\times\cdots\times A_n$


$E[X] = \int P[X \geq x] dx$


$\forall \epsilon>0,\;\exists \delta>0,\;|x-a|<\delta\implies |f(x)-g(x)| <\epsilon|g(x)| $


$\int_{0}^{1} \frac{\sin(x)}{x^{1.5}} \ dx $


$(k+1)!+2,\ (k+1)!+3,\ ...\ ,\ (k+1)!+(k+1)$


$M^+$


$ \lim_{n\to\infty} \left(1 + \frac{x}{n}\right)^{n} = e^{x}, $


$n! = \sqrt{2 \cdot \pi \cdot n} \cdot (\frac{n}{e})^n  $


$\frac{3}{9^{3/2}} = \frac{3}{27} = \frac{1}{9}$


$T=\frac{r(nr^{n+1}-(n+1)r^n+1)}{(1-r)^2}$


$\int_0^1\int_0^1 \frac{\sin^{-1}(xy)}{xy} dx dy$


$A_1 \times A_2 \times ...$


$\begin{align*} \int_0^\infty \dfrac{\sin^3x}xdx &=\dfrac14 \int_0^\infty\dfrac{3\sin x}xdx-\int_0^\infty\dfrac{\sin3x}xdx\\&=\dfrac{3\pi}8-\int_0^\infty \dfrac{\sin y}ydy\\&=\dfrac\pi4\end{align*}$


$f(x) = \frac{2^{\frac{1}{2}-n} n e^{-\frac{x^2}{2}} \left(1+\text{erf}\left(\frac{x}{\sqrt{2}}\right)\right)^{n-1}}{\sqrt{\pi }}$


$R=\{(t,y)\in \mathbb{R}^{2}: \alpha<t<\beta,\, \delta <y<\gamma\},$


$p_1p_2\cdots p_k\mid x$


$ax+by=k$


$\lfloor a\rfloor+\lfloor b\rfloor=\lfloor c\rfloor$


$I = \int\limits_0^\pi \frac{(\pi-x)\sin x}{1+\cos^2 x}dx\tag 2$


$p(\lambda)=(\lambda+1)(\lambda-1)^{n-1}$


$f(x)=\sum_{m\in M}\langle x,m\rangle f(m)$


$ds=\sqrt{1^2+\left(\frac{dy}{dx}\right)^2}dx$


$\lim_{n \to \infty} \sum_{k=1}^n\frac{1}{k\cdot 2^k}$


$ \frac{1}{2^1} + \frac{2}{2^2} + ... +\frac{n-1}{2^{n-1}} + \frac{n}{2^n} <2 $


$|ab|=|a||b|,$


$\int\frac{1}{(x^3+1)^2}\mathrm dx \cdots$


$g = \sum_{j=1}^{N} c_j \chi_{Q_j}$


$\det \left(\begin{array}{cc} 0 & A\\ -B & I \end{array}\right) = \det(AB) $


$\int_0^T u(t) \phi'(t) dt = \int_0^T u'(t) \phi(t) dt$


$\bigwedge^{m}(T^{1,0})\otimes \bigwedge^{l}(T^{0,1})$


$\sum\limits_{n=1}^\infty\left|\frac{1}{n^{1/p}}\right|^r=\sum\limits_{n=1}^\infty\left|\frac{1}{n^{r/p}}\right|,$


$Cov(X,Y') = Cov(X,Y) + Cov(X,Z) = Cov(X,Y)$


$\sum_{j=1}^{\infty}\sum_{i=1}^{\infty} \cfrac{(-1)^{(i+j)}}{i^2+2nij+j^2} $


$\frac {12}{y} =1$


$R \, : \, \mathbb{R}^{n+1} \, \rightarrow \, \mathbb{R}^{n+1}$


$J_n=\int_0^{\pi}\left(\frac{\sin nx}{\sin x}\right)^2dx$


$\lim_{n\rightarrow\infty} \sum_{i=1}^{\infty} h_n(i) = \sum_{i=1}^{\infty} h(i) $


$\kappa = \langle dn\gamma', d\phi \gamma'\rangle = - \mathrm {I\!I}(\gamma',\gamma')$


$\sum\limits_{k=1}^{n}k \cos{\frac{2k\pi}{n}} =-\frac {n}2$


$a_n=\int_0^1 \frac{f(x)}{(1+x^n)(1+x^{n+k})}dx$


$f(x)=\sum_{j=1}^{n}\chi_{E_{j}}(x)y_{j}$


$\lim\limits_{n\to\infty} \frac{3+\sqrt{3}+\sqrt[3]{3}+\dots+\sqrt[n]{3}-n}{\ln n}$


$\implies \lfloor\frac{a}{b}\rfloor \lfloor\frac{c}{d}\rfloor + \lfloor\frac{a}{b}\rfloor y + \lfloor\frac{c}{d}\rfloor x + xy$


$C_1\subseteq V\subseteq\overline{V}\subseteq U$


$\sqrt{1!\sqrt{2!\sqrt{3!\sqrt{\cdots\sqrt{n!}}}}} <3$


$ \bbx{\zeta\pars{1 \over 2}  = \lim_{N \to \infty}\pars{\sum_{k = 1}^{N}{1 \over \root{k}} - 2\root{N}}} $


$C_R' \subseteq C_R$


$\sum_{n=1}^\infty \frac{1}{n(n!)} = \int_0^1\frac{e^x-1}{x}dx$


$ \left[     \begin{array}{ccc|c}       k&1&1&-2\\       k&k^2&k&k\\       k&k&k^2&k     \end{array} \right] $


$\mathbb{Q}^{+}$


$\displaystyle \omega^2 = e^{\frac{4\pi i}{3}}$


$x!\approx \sqrt{2\pi x}~x^xe^{-x}$


$ax+by=d\mid n$


$a_j, i_j$


$v \in T_p \mathbb{R}^{n+1} = \mathbb{R}^{n+1}$


$S_n = \frac{X_1 + X_2 + \dots + X_n }{n}$


$\mathbb{var}[\overline{x}]= \mathbb{E}[(\overline{x}-\mathbb{E}[\overline{x}])^2]= \mathbb{E}[\overline{x}^2]-\mathbb{E}[\overline{x}]^2= \mathbb{E}[\overline{x}^2]-m_1^2$


$G = \{ z \mid |z| >2, 0 < arg(z) \leq \pi\}$


$X\subseteq\overline{V}\subseteq X$


$\sum_{n=0}^{\infty}\frac{2^{n}}{(2n)!\cdot2^{n}} = \sum_{n=0}^{\infty}\frac{1}{(2n)!}$


$ \limsup \frac{x_1+x_2+...+x_n+x_{n+1}}{x_n} \geq 4.$


$ \begin{align} \sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right) &=\mathrm{Re}\left(\sum_{k=0}^{n-1}e^{\frac{2\pi ik}n+i\phi}\right)\\ &=\mathrm{Re}\left(e^{i\phi}\sum_{k=0}^{n-1}e^{\frac{2\pi ik}n}\right)\\ &=\mathrm{Re}\left(e^{i\phi}\frac{1-e^{\frac{2\pi in}n}}{1-e^{\frac{2\pi i}n}}\right)\\ &=\mathrm{Re}\left(e^{i\phi}\frac{1-e^{2\pi i}}{1-e^{\frac{2\pi i}n}}\right)\\[6pt] &=\mathrm{Re}(0)\\[12pt] &=0 \end{align} $


$P(m, n) \implies P(m, n + 1)$


$R_{n}\left( x\right) =\dfrac {f^{n+1}\left( a\right) \left( x-a\right) ^{n+1}} {\left( n+1\right) !} $


$\theta^a=-\frac{1}{2}(\gamma^{-1})^{ab}\rho(G_b)_c^{\phantom{a} d}(u^{-1})_d^{\phantom{a} e}\mathrm du_e^{\phantom{a} c}$


$2\pi\int_0^{\sqrt{1-m^2/3}} (1-r^2)\rho\mathrm d\rho= 2\pi\int_h^1(1-r^2)r\mathrm dr=2\pi\left(\frac{1-h^2}2-\frac{1-h^4}4\right).$


$\mathbb{E}\vert g(X_1,\ldots, X_m)\vert^r < \infty$


$-\sum |x_n|\le\sum x_n \le \sum |x_n|$


$\frac{\partial^2}{\partial x^2}f(x,t,a) = -t(1+t)(1+x)^{-2-t}-2a \frac{(1+x)^{2+t}-(2+t) x \big(1+\frac{1+t}2x\big)-1}{x^3(1+x)^{2+t}}. $


$\begin{pmatrix} x & y  \end{pmatrix} \begin{pmatrix}  1& 1 &\frac{-z}{x} \\   2& \frac{z}{y} &1  \end{pmatrix}=\begin{pmatrix}  x+2y&x+z &y-z  \end{pmatrix}$


$Ax + By = D - Cz$


$c=\frac {1- \gamma a}{\gamma} $


$(\Bbb R[X] / (X^2)) \setminus (X)$


$f(x)=\frac{1}{\sqrt{2\pi t}}e^{\frac{-x^2}{2t}}$


$ M(t)=\|p-f(t)\|^2=(p-f(t),p-f(t)) $


$9^{x}=6^x +4^x$


$1 + 2 + 3 + \cdots = -\frac{1}{12}.$


$(x,y) = (a+b, a-b)$


$\lim_{x \to c^+}f'(x) = f'_+(c) = f'(c)  = f'_-(c) = \lim_{x \to c^-}f'(x)$


$\{A, -B\}$


$ \left[     \begin{array}{ccc|c}       1 & 1 & 1&4\\         0 & 0 & 1&2 \\         0 & 0&a-4&a-2\\     \end{array} \right] $


$g(x) = (x^3+x^4+x^5+x^6+x^7+x^8+x^9)(1+x+...+x^9)^4$


$\sum_{k=0}^{n-1}\left\lceil \frac{\left\lceil y\right\rceil+k}{n}\right\rceil=\sum_{k=0}^{n-1}\left\lceil \frac {y+k}{n}\right\rceil$


$N = (n+1)^2 - n^2 = 2n+1$


$P(m− 1)\to P(m)$


$ \gamma\xi\gamma^{-1}(\gamma(a_1))= \gamma\xi(a_1)=\gamma(a_2). $


$X=\lbrace 0,e_0,e_1,\ldots,e_n \rbrace$


$ \left\{    \begin{array}{l l}      x=1 + \frac{4t}{\sqrt{32}}  \\     y=2 - \frac{4t}{\sqrt{32}}\\   \end{array} \right.   $


$\left(\begin{array}{r}-B\\A\end{array}\right)$


$ \int_0^\infty f(x) \, dx = C,$


$\mathcal{P}(k)\implies\mathcal{P}(k+1)$


$ (\star) \quad {[{\gamma^{*}}(\nabla)]_{\frac{\partial}{\partial x}}}({\gamma^{*}}(X)) = {\gamma^{*}} \left( {\nabla_{{d \gamma} \left( \frac{\partial}{\partial x} \right)}}(X) \right) = \gamma^{*} \left( {\nabla_{\gamma'}}(X) \right). $


$F_0 = P\oplus T$


$S=\{z \in \mathbb C \: | (z+i)^n  = (z-i)^n\}; \: n \in \mathbb N, n \ge 2$


$ \cases{ x=st \cr \displaystyle y={1\over2}s(t^2-1)\cr \displaystyle z={1\over2}s(t^2+1)\cr } $


$\gamma=nq\gamma_1+n\gamma_2 \implies \frac{\gamma}{n}=q\gamma_1+\gamma_2$


$\therefore f(a+b)=f(a)+f(b)$


$\overline {\mathbb{R}}$


$\,[x,y] = {\rm lcm}(x,y)$


$ A \subseteq V \subseteq \overline{V} \subseteq U $


$ \int_{0}^{1} \frac{\tan^{-1}(x)}{\sqrt{1-x^2}} dx$


$N= \{\omega: \lim_{n \rightarrow \infty} X_n(\omega) \ne 0\}$


$|ab|=n$


$\begin{cases} x_1=1+1s+2t \\ x_2=0+1s+0t \\ x_3=0+0s+1t \end{cases}$


$\{a\} \ne \{\{a\}\}$


$ \begin{cases} r\equiv 1\mod p-1\\ r\equiv 1\mod q-1 \end{cases} $


$\sum_{n=-\infty}^{\infty} e^{-\frac{1}{2} An^2+iBn} = \sqrt{\frac{2\pi}{A}} \sum_{l=-\infty}^{\infty} e^{-\frac{1}{2A}(B-2\pi l)^2}$


$R_\varphi(0,1) = (\cos(\varphi+\frac\pi 2),\sin(\varphi+\frac\pi 2)) = (-\sin\varphi,\cos\varphi)$


$\textrm{cov}(X,Y) = 0$


$\liminf \limits_{n \to \infty} \, \mu_n(G) \ge \mu(G)$


$x! \approx \sqrt{2 \pi x} \cdot \left(\dfrac{x}{e}\right)^x $


$ \begin{eqnarray*} P\left(\dfrac{Y^2}{X^2}<z\right){}={}P\left(Y^2<z\,X^2\right)&{}={}&P\left(Y<\sqrt{z}\,X\right)\,,\,\,\,\,\,\,\mbox{ since }Y\,,\,X\,,z>0\newline &{}={}&\int\limits_{0}^{\infty}P\left(Y<\sqrt{z}\,x\,\bigg|\,X=x\right)f_{X}(x)\,\mathrm{d}x\newline &{}={}&\int\limits_{0}^{\infty}P\bigg(Y<\sqrt{z}\,x\bigg)f_{X}(x)\,\mathrm{d}x\newline &{}={}&\int\limits_{0}^{\infty}\bigg(1-e^{-\sqrt{z}\,x}\bigg)e^{-x}\,\mathrm{d}x\newline &{}={}&\left(1-\dfrac{1}{1+\sqrt{z}}\right){\bf{1}}_{\left\{z>0\right\}}\,. \end{eqnarray*} $


$ \left|^\mathbb{N} \mathbb{R}\right| = |\mathbb R| < |P(\mathbb R)| \leq \left|^\mathbb R \mathbb N\right|. $


$\overline{ \mathbb{R}}$


$a_n = n^2 -n +2.$


$p_j \mid p_1 p_2 \cdots p_k$


$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x) \sin x} $


$H = [a,b]\times [c,d] = \{(x,y)| a \le x \le b; c \le y \le d\}$


$\mathcal B=\{v_1,v_2,....,v_n \}$


$x^6-6 x^4+9 x^2-x-2 = 0,$


$\int_{0}^{\pi }f(x+t)dt=\int_{0}^{\left [ n\pi \right ]}f(x+t)dt=0$


$\left | \left | A \right | \right |_{2}=\sqrt{tr(A^{t}A)}$


$|\theta-\sin \theta| < |2\tan\frac\theta 2 - \sin\theta| < 4|\sin^3\theta|<4|\theta|^3$


$P(16)\implies P(20)$


$X_1\subset X_2\subset \dots$


$\text{Max}[f(x)+f(y)]=\text{Max}f(x)=\text{Max}f(y).$


$\forall \epsilon>0,\exists\delta>0: |x-c|<\delta \implies |f(x)-f(c)|<\epsilon.$


$\langle a,b\rangle=\det(\langle a_i,b_i\rangle)$


$ -\frac1{12}=1+2+3+4+\dots\tag{5} $


$|\mathbb{E}|X_n|^p-\mathbb{E}|X|^p|\le \mathbb{E}|X_n-X|^p\to 0$


$P(X ≤ 10) = 1/6$


$ \int_{-\pi}^{\pi}{\cos\left(\, ax\,\right) \over 1-bx^{2}}\,{\rm d}x $


$\tan \theta=\frac{3}{2}$


$ \begin{pmatrix}   1 & 1 & 1 \\   0 & y-x & z-x \\   0 & y^2-x^2 &z^2-x^2\\ \end{pmatrix} = \begin{pmatrix}   1 & 1 & 1 \\   0 & y-x & z-x \\   0 & (y-x)(y+x) &z^2-x^2\\ \end{pmatrix}  $


$|xy| = pq$


$T_n \gamma =b.(\gamma -Sd \gamma -T_{n-1}\partial_n \gamma)$


$\{0\}\subset V_0\subset V_1\subset \cdots \subset V_r\subset k^n,$


$S=\{e_{1}, e_{2}, e_{3}, e_{4}\}$


$\mathbb R^n → \mathbb R^n$


$P(1) \implies P(2) \implies  P(3) \implies \dots \implies P(k) \implies P(k+1) \implies \dots $


$\begin{bmatrix}     1 & -2 \\     2 & 3 \\     \end{bmatrix}$


$(\overline{S} \cap \overline{T})^\perp \subset \overline{S \cap T}^\perp$


$2(n-1) = {n\choose2} = \frac{n(n-1)}{2}$


$\lim_{x\to 0} f(x)g(x) = \lim_{x\to0} f(x)\cdot \lim_{x\to0}g(x)$


$A\subseteq V_1\subseteq \overline{V_1}\subseteq U$


$p(n)=\binom{n}k$


$\begin{bmatrix}a&b\\b&-a\end{bmatrix}$


$F = x - y$


$\left(1+\frac{x}{n}\right)^n\leq e^{x}.$


$  k^2 = 2 \; \left( \begin{array}{c} k \\ 2 \end{array} \right) + \left( \begin{array}{c} k \\ 1 \end{array} \right),  $


$x^2 + y^2 + z^2 = x y + y z + x z$


$ax+by=m$


$\cdots \subseteq A_0 \subseteq A_1 \subseteq A_2 \subseteq \cdots \subseteq A_n \subseteq \cdots .$


$\frac{sr}{1-s} = \frac{(1-p)^N}{1-(1-p)^N}$


$\mathbb{R}^{n+1}\times \mathbb{R}^{n+1}$


$M=\begin{pmatrix} 1 & 1 & 1\\ x & y & z\\ x^2 & y^2 & z^2 \end{pmatrix}$


$f(x + 2x f(x)^2) = y f(x) + f(f(x) + 1)$


$\{e_i: 1 \leq i \leq n\}$


$\left|
 \begin{array}{cccc}1 & 1 & 1 & 1\\
 x & y & 1 & 0\\
 x^2 & y^2 & 1 &0\\
 x^3 & y^3 & 1&0
 \end{array}
 \right|$


$z=a e^{i \pi/4}$


$\lim_{n \to \infty} \inf s_n \le \lim_{n \to \infty} \inf t_n$


$ xf'(x)+f(x)=(x-y)f'(x+y)+f(x+y). $


$\mathbf{B}=\left[ \matrix{ \mathbf{A}_1&\mathbf{A}_2&\ldots&\mathbf{A}_{p-1}&\mathbf{A}_p \\  \mathbf{I}_M&\mathbf{I}_M&\ldots&\mathbf{I}_M&\mathbf{O}_M \\  \vdots &\vdots &\ddots& \vdots& \vdots\\  \mathbf{I}_M&\mathbf{I}_M&\ldots&\mathbf{I}_M&\mathbf{O}_M \\ } \right]:(pM \times pM)$


$C-\sum_{p\leq x}\frac{1}{p^{2}}=\sum_{p>x}\frac{1}{p^{2}}$


$ Cov(X,Y+Z)=Cov(X,Y)+Cov(X,Z)$


$f(a,b,c,\gamma)=\int\left(\gamma -a-b\cos \gamma -c\sin \gamma\right)^2\,d\gamma$


$(y-x)z+(y-x)(y+x)=(y-x)(z+y+x)$


$ \lfloor \frac{\lfloor\frac{a}{b}\rfloor}{c} \rfloor = \lfloor \frac{a}{b \times c} \rfloor $


$d(x,z)>d(x,y)+d(y,z)$


$\mathbb{E}|X|^r<\infty$


$Cov(Y,Z) =0$


$\operatorname{ord}_p(n) = k$


$\forall \epsilon>0 ,\exists \delta > 0 :|x-a|<\delta \implies |f(x) - L|<\epsilon$


$(-1)^n\bigl(x - (n-1)\bigr)(x+1)^{n-1}$


$∇^ 2u=X''(x)Y(y)Z(z)+X(x)Y''(y)Z(z)+X(x)Y(y)Z''(z)=0$


$\phi(n) = n + I$


$\det \begin{pmatrix}A & 0 \\ B & C\end{pmatrix} = \det (A) \det (C)$


$\|A\| = \sqrt{\lambda_{\max} (A^T A)}$


$\lfloor c^{a+b} \rfloor$


$|f'(z)| = \frac \pi 2 |f(z)|$


$ \lvert \gamma \times \gamma' \rvert^2 = (\gamma \cdot \gamma)(\gamma' \cdot \gamma') - (\gamma \cdot \gamma')^2 = (\gamma \cdot \gamma)(\gamma' \cdot \gamma')  \neq 0. $


$\int_0^{\infty} \frac{\tan^{-1}(x^2)}{x^2+a^2} \ dx$


$\frac{y+4}{5+4} = \frac{x+10}{8+10}$


$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{\cos(nx)\cos(my)}{n^2+m^2},$


$\in \mathbb{R^+}$


$I = \int_0^1 \frac{\sin^{-1}(t)}{t} A'(t) dt = - \int_0^1 \frac{\sin^{-1}(t)}{t}\log t dt = -\frac12 \int_0^1 \sin^{-1}(t) (\log^2(t))' dt$


$9^x + 12^x = 16^x$


$ \gamma(a) + \gamma(b) = \gamma(a+b). $


$M = F' \oplus T(M)$


$\lim\limits_{x \to a^+} f(x)=\lim\limits_{x\to a^-}f(x)=L$


$L_\alpha = [H_i,L_\alpha] \subseteq [L_i,L_\alpha] \subseteq L_i.$


$\frac{f(f(x)y)}{x^2} =  \frac{f(f(x) + f(y))}{x + y }\tag{1}$


$\lim_{x\to1^+}f'(x)\neq\lim_{x\to1^-}f'(x)$


$p|m \implies p |n$


$ \ \tan \theta = \frac{x}{a} \ $


$ \frac1{\binom{n+a}{a}}=\frac{a}{a-1}\left[\frac1{\binom{n+a-1}{a-1}}-\frac1{\binom{n+a}{a-1}}\right]\tag{1} $


$f''(x) = n(n-1)(x+1)^{n-1}$


$\,p_1 \mid q_{2}\cdots q_{n}\,$


$\sum_{n=-\infty}^{\infty}e^{-a|n|}=\sum_{m=-\infty}^{\infty}\dfrac{2a}{a^2+(2\pi m)^2}$


$z^{1/n} = r^{\frac 1n} e^{\frac{\theta}ni + \frac{2k}n \pi i} = [r^{\frac 1n} e^{\frac{\theta}ni}]*e^{\frac{2k}n \pi i}$


$\zeta(s)=\frac{1}{\Gamma(s)}\int_0^{\infty}\frac{x^{s-1}dx}{e^x-1},$


$\begin {array}{ll} x_1 = a\lambda /(\lambda+1)\\ y_1 = b\lambda /(\lambda+1)\\ (a-x_1)^2+(b-y_1)^2-r^2 =0 \\ \end{array}$


$\int_\gamma \! \frac{1}{z+1} \, \mathrm{d}z = \int_0^1 \! \frac{\gamma'(t)}{\gamma(t)+1} \, \mathrm{d}t$


$x-a,\:\:x\in \left[a,\:\infty \right]$


$ (a_1) \subseteq (a_2) \subseteq \dots \subseteq (a_i) \subseteq (a_{i+1}) \subseteq \dots\tag{seq} $


$[\mathbb{x}]_d^t:=[1,x,y,x^2,xy,y^2,x^3,x^2y,xy^2,y^3,\dots,y^d]$


$xRy \rightarrow yRx?$


$|f(z)|\le N|g(z)|$


$COV(X,Y) = COV(X, E(Y|X))$


$\Vert \cdot \Vert_2:=\sqrt{\lambda_{\max}(A^TA)}$


$ \sum_{k = -\infty}^{\infty} e^{-\pi k n^2} = \frac{1}{\sqrt{k}} \sum_{k=-\infty}^{\infty} e^{-\pi n^2/k} $


$t = f(x) = f(lim_{n \rightarrow \infty }x_n) = lim_{n \rightarrow \infty }f(x_n) = lim_{n \rightarrow \infty } t_n$


$\mathcal{X} = \mathcal{A}$


$P(k + 1): k + 1 = (k + 1 – 3) + 3$


$M^*M =MM^*$


$\lim_{n\to\infty}J(v_n)=x=\inf_{v\in K}J(v),$


$\int {1 \over (x^2-1)^2}dx$


$\lim_{n\to \infty} \sum_{r=1}^n \frac {1}{2^r}\tan \left(\frac {1}{2^r}\right)$


$             \|x\| \ge \frac{|\langle x,y\rangle|}{\|y\|} \\             |\langle x,y\rangle| \le \|x\|\|y\| $


$f(f(x)+f(y))=af(y)+f(a)+f(y)-f(x)y.$


$ \tan \theta = \dfrac{x}{2}.$


$\displaystyle\sum_{k=1}^{n-1}\cos\frac{2k\pi }n=$


$f(f(x)^2 + xf(x)) = xf(x) + x^2$


$\frac{P}{M} = \frac{\left(1 + J\right)^N - 1}{J(1+J)^N}$


$\forall x(xRb\lor bRx)$


$z^{\frac{1}{n}}=\sqrt[n]{r}e^{i \varphi /n}$


$F: \Bbb R^{n + 1} = \Bbb R \times \Bbb R^n \to \Bbb R^n \tag 2$


$|f(z^{2^n})| \le |f(z)| \to |f(0)| \le |f(z)|$


$\ddot x = x^3 -x$


$\zeta(s)=2^{s} \pi^{s-1} \sin \frac{\pi s}{2} \Gamma(1-s)\zeta(1-s).$


$A=\{(a,b,c,d)|a,b,c,d\in\mathbb{R}_{\ge0}; a+b+c+d=m\}$


$(1 + \tfrac{x}{n})^n < \mathrm{e}^x$


$\left(|a|^r+|b|^r\right)(1+1)^{r-1}\geq\left(\left(|a|^r\cdot1^{r-1}\right)^{\frac{1}{1+r-1}}+\left(|b|^r\cdot1^{r-1}\right)^{\frac{1}{1+r-1}}\right)^{1+r-1}=\left(|a|+|b|\right)^r$


$f(k)\leq k^2+k+2$


$ \Gamma(x)\approx\sqrt{\frac{2\pi}{x}}\left(\frac{x}{e}\right)^x $


$R=\{\langle x,y\rangle\in X\times X:\exists i\in I:\{x,y\}\subseteq U_i\}$


$\intop\left(1-x^{2/3}\right)^{3/2}dx=3\intop\cos^{4}\left(y\right)\sin^{2}\left(y\right)dy.$


$F_n\neq \cfrac{n(n+1)}{2}$


$P\oplus T$


$b^{log_b(n)} = log_b(b^n) = n$


$\|g\|_{L^1} = \sum_{n=-\infty}^\infty \int_{-\infty}^\infty e^{-n^4 (x-n)^2} dx =  \sum_{n=-\infty, n \ne 0}^\infty \frac{1}{n^2}\int_{-\infty}^\infty e^{-x^2} dx = 2\frac{\pi^2}{6}\sqrt{2\pi}$


$e^x > \frac{x^{n+1}}{(n+1)!}$


$p_1p_2...p_k - 1$


$u = 1-\displaystyle\sum_{k=1}^n\frac{1}{2^k}$


$|x+a||x-a|=(x+a)(x-a)$


$\begin{cases} 2x \equiv 1 (29) \\ 2x \equiv -1 (29) \end{cases} $


$n^2-\left(\frac{n-1}{2}\right)^2-\left(\frac{n+1}{2}\right)^2+\frac{n-1}{2}=\frac 1 2 (n^2+n-2),$


$df: T^{0,1}D \to f^*(T^{0,1}D)$


$d(x) = \inf_{m \in M} d(x,m)$


$ \begin{cases} y=5\\ x=3\\ z=7 \end{cases}\Longleftrightarrow $


$P_{\Theta|X}(\theta=0|x) = \frac{P(\theta=0)f_{X|\Theta}(x|\theta=0)}{f_X(x)} = \frac{1}{f_X(x)}((1-p)*1)$


$cov(x,y)=E(xy)-E(x)E(y)$


$W^{\bot}=\{\frac{1}{2}(1,-1,-1,1), \frac{1}{2}(1,-1,1,-1)\}$


$\big\{\langle x,y \rangle | \langle x,y \rangle \in \mathbb{R}^2,\quad\exists\;a,b,c,d \in \mathbb{R}^2 such \; that \; a<x<b \quad c<y<d\big\}$


$ \mathbb{P}\{X > (1+\gamma)P\} \leq \left(\frac{e^\gamma}{(1+\gamma)^{1+\gamma}}\right)^P \tag{1} $


$\mathbb{P}(X_n \geq K \;| \;n \geq N) = 1.] $


$4^x+6^x=9^x\implies 1+u=u^2$


$\lim_{n \to \infty}\dfrac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n-1}}{n\sqrt{n}}.$


$\alpha^+\times \alpha^+$


$X \text{ ind } Y \implies Cov(X,Y)=0$


$ \left[\begin{array}{ccc|c} 1 & 4 & 5 & 2\\ 2 & 2 & 4 & 10\\ 2 & 1 & 1 & 1\\ \end{array}\right]. $


$\frac{s(n)}{2}=\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{n}}-\frac{n}{2^{n+1}}\\ s(n)=\left[\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{n-1}}\right]-\frac{n}{2^{n}}$


$ \begin{split} &\hat{\nabla}_{(\gamma \circ \tau)'(s)} (\gamma \circ \tau)' \\&=\nabla_{(\gamma \circ \tau)'(s)} (\gamma \circ \tau)' + 2 d \sigma((\gamma \circ \tau)'(s)) \cdot (\gamma \circ \tau)'(s) - \| (\gamma \circ \tau)'(s) \| grad \sigma \\&=\nabla_{(\gamma \circ \tau)'(s)} (\gamma \circ \tau)' + 2 d \sigma((\gamma \circ \tau)'(s)) \cdot (\gamma \circ \tau)'(s) \\&=\frac{\tau ''(s)}{\tau'(s)} \cdot (\gamma \circ \tau)'(s) + 2 d \sigma((\gamma \circ \tau)'(s)) \cdot (\gamma \circ \tau)'(s). \end{split} $


$ \varphi_0(f)=\sum_{j=1}^hc_jf(n_j) $


$|y-y'| < |y| = -y$


$\phi_1(t,t_{0,1},x_{0,1})=e^{A_1(t-t_{0,1})}x_{0,1},$


$E_n = X_n$


$y',y'',y''',y''''$


$\cot(\sin^{-1}(z))=\frac{\sqrt{1-z^2}}{z}$


$\vert f(z) \vert \leq \vert g(z) \vert$


$S = \{\{a\}, \{a\}\}$


$x_\phi=e^{2\pi i/\phi}\sqrt[k]a$


$ \gamma_R=\gamma_{R,1}+\gamma_{R,2}-\gamma_{R,3} $


$L:ax+by=c$


$N=rs+r+s$


$ \frac{dy}{dx}=\frac{\sqrt{1-x^2}}{x} $


$g(\pi) = f(\pi) - f(0) = - (f(0) - f(\pi))$


$p_1p_2...p_k \mid m$


$2\int_0^1\frac{\sin(x)}{\arcsin(x)} \, dx$


$0\leq \sum_{n\geq 1}\frac{1}{\sqrt{n}}\,\sin\frac{1}{n}\leq \sum_{n\geq 1}\frac{1}{n\sqrt{n}}=\zeta(3/2).$


$e^x> \frac{x^{n+1}}{(n+1)!}.$


$ \zeta(s) = \pi^{s - \frac12} \frac{\Gamma(\frac{1-s}{2})}{\Gamma (\frac s2)} \zeta(1-s) $


$res_{z_0}f = \text{lim}_{z\rightarrow z_0}{1 \over (n-1)!}\bigg({d \over dz}\bigg)^{n-1}(z-z_0)^n[f(z)-G(z)].$


$y= c_1\left | x \right |^r+ c_2\left | x \right |^rln(\left | x \right |)$


$\sum_{k = 0}^{n - 1} \sin \frac{2\pi k}{n} = 0$


$f(x) = \frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}},$


$\int_{(i)}f(z)dz=\int_{(ii)}f(z)dz=\int_{-1}^{1}\frac{k}{cos(k)-1}dz$


$\Omega - \{ \omega \in \Omega: \lim_{n \rightarrow \infty} X_n(\omega) = X(\omega)\}$


$\begin{array}{11} x=u+1 \\ v = y\\ J(u,v) = 1 \\ u=r\cos\theta\\v=r\sin\theta\\0 \leq r \leq 1 \\ 0 \leq \theta \leq \pi  \end{array}$


$d(x,T)=d(S,T)$


$P(X \leq x \mid H_{0})$


$\lim_{x\rightarrow a}f(x)g(x)=\lim_{x\rightarrow a}f(x)\times\lim_{x\rightarrow a}g(x)$


$ 1/x + 4/y + 9/z $


$K\subset X-Â$


$\lim_{x \rightarrow c} f(x) := L$


$ F(x) = \int_1^x f(t)dt $


$\lfloor \frac{\lfloor x \rfloor}{n} \rfloor =  \lfloor \frac{ bn + r_2}{n} \rfloor = b$


$\textrm{coefficient of x^2 in } \frac{(1+x)^n}{x} + \frac{((1+x)^{n-3}-1).(1+x)^n}{x^2(1+x)^{n-3}} $


$\gamma(Tx) = \gamma(ax) = \gamma(a)\gamma(x) = \gamma(x)$


$c=(a+b,a-b)$


$\tan t = \frac{x}{3} $


$ \pi^{-\frac s2} \Gamma (\frac s2) \zeta(s) = \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \zeta(1-s)$


$A_1 \times A_2\times A_3\times \cdots $


$E^{(p)}=E^p$


$ \begin{align} \color{#C00000}{\frac{(1+x)^{n+1}-1}{x}} &=\sum_{j=0}^n\color{#00A000}{\binom{n+1}{j+1}}x^j \end{align} $


$B(p,q)=\Big\{\langle x,y\rangle\in\Bbb R^2:a<x\le b\text{ and }c<y\le d\Big\}\;,$


$\lim_{\gamma\to0}\frac{x^\gamma-1}\gamma=\left.\frac{\mathrm d}{\mathrm d \gamma}x^\gamma\right|_{\gamma=0}=\ln x.\tag{III}$


$(x*y)*z = x+y+z-[x+y]-[x+y+z-[x+y]]$