\[|x|=\sqrt{x^2}\]

\[|x+y|\le |x|+|y|\]

\[a^2-b^2=(a-b)(a+b)\]

\[\sum_{k=m}^{n}a_k=a_m+a_{m+1}+\cdots+a_n\]

\[\binom{n}{r}=\frac{n!}{r!(n-r)!}\]

\[(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\]

\[f:A\to B\]

\[x\in\operatorname{dom}f\]

\[f^{-1}(f(x))=x,\qquad f(f^{-1}(y))=y\]

\[f(x_1)=f(x_2)\Rightarrow x_1=x_2\]

\[a^2+b^2=c^2\]

\[\sin^2\theta+\cos^2\theta=1\]

\[\tan\theta=\frac{\sin\theta}{\cos\theta}\]

\[\sin(A+B)=\sin A\cos B+\cos A\sin B\]

\[\cos(A+B)=\cos A\cos B-\sin A\sin B\]

\[\sin2A=2\sin A\cos A,\qquad \cos2A=\cos^2A-\sin^2A\]

\[\theta=\arctan\left(\frac{y}{x}\right)\]

\[\lim_{x\to a}f(x)=L\]

\[\lim_{x\to a^-}f(x),\qquad \lim_{x\to a^+}f(x)\]

\[\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)\]

\[\lim_{x\to a}f(x)g(x)=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right)\]

\[\lim_{x\to0}\frac{\sin x}{x}=1\]

\[\lim_{x\to a}f(x)=f(a)\]

\[f'(x)=\frac{df}{dx}\]

\[f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]

\[\frac{d}{dx}x^n=nx^{n-1}\]

\[\frac{d}{dx}e^x=e^x\]

\[\frac{d}{dx}\sin x=\cos x,\qquad \frac{d}{dx}\cos x=-\sin x\]

\[(uv)'=u'v+uv'\]

\[\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}\]

\[\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}\]

\[f''(x)=\frac{d^2f}{dx^2}\]

\[\frac{d}{dy}f^{-1}(y)=\frac{1}{f'(x)},\qquad y=f(x)\]

\[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}\]

\[F'(x)=f(x)\]

\[\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\ne -1\]

\[\int\frac{1}{x}\,dx=\ln|x|+C\]

\[\int_a^b f(x)\,dx\]

\[\int_a^b f(x)\,dx=F(b)-F(a)\]

\[\int f(g(x))g'(x)\,dx=\int f(u)\,du\]

\[\int u\,dv=uv-\int v\,du\]

\[\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}\]

\[\sin^2x=\frac{1-\cos2x}{2},\qquad \cos^2x=\frac{1+\cos2x}{2}\]

\[z=a+bi\]

\[\operatorname{Re}(a+bi)=a\]

\[\operatorname{Im}(a+bi)=b\]

\[\overline{a+bi}=a-bi\]

\[|a+bi|=\sqrt{a^2+b^2}\]

\[\frac{z}{w}=\frac{z\overline w}{|w|^2},\qquad w\ne0\]

\[z=r(\cos\theta+i\sin\theta)\]

\[\theta=\arg z\]

\[e^{i\theta}=\cos\theta+i\sin\theta\]

\[z=re^{i\theta}\]

\[(re^{i\theta})^n=r^ne^{in\theta}\]

\[r_1e^{i\theta_1}r_2e^{i\theta_2}=r_1r_2e^{i(\theta_1+\theta_2)}\]

\[az+b=0\Rightarrow z=-\frac{b}{a},\qquad a\ne0\]

\[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

\[z_k=e^{2\pi ik/n},\qquad k=0,1,\ldots,n-1\]

\[z_k=\rho^{1/n}e^{i(\chi/n+2\pi k/n)},\qquad k=0,1,\ldots,n-1\]

\[p(z)=a_n\prod_{k=1}^{n}(z-z_k)\]

\[f:\mathbb C\to\mathbb C,\qquad f(z)=0\]

\[e^{z+2\pi i}=e^z\]

\[*:G\times G\to G\]

\[a*(b*c)=(a*b)*c\]

\[e*a=a*e=a\]

\[a*a^{-1}=a^{-1}*a=e\]

\[a,b\in H\Rightarrow ab^{-1}\in H\]

\[\phi(a*b)=\phi(a)\diamond\phi(b)\]

\[\ker\phi=\{g\in G:\phi(g)=e_K\}\]

\[\langle g\rangle=\{g^n:n\in\mathbb Z\}\]

\[|D_n|=2n\]

\[\mathrm{GL}_n(\mathbb R)=\{A\in\mathrm{Mat}_n(\mathbb R):\det A\ne0\}\]

\[\sum_{n=1}^{\infty}a_n=a_1+a_2+a_3+\cdots\]

\[S_N=\sum_{n=1}^{N}a_n\]

\[S_N=\frac{a(1-r^N)}{1-r},\qquad r\ne1\]

\[\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r},\qquad |r|\lt1\]

\[\sum_{n=1}^{\infty}\frac{1}{n^p}\text{ converges for }p\gt1\]

\[L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\]

\[L=\lim_{n\to\infty}\sqrt[n]{|a_n|}\]

\[\sum |a_n|\text{ converges }\Rightarrow\sum a_n\text{ converges}\]

\[\sum_{n=1}^{\infty}a_n=S\quad\Longleftrightarrow\quad \lim_{N\to\infty}S_N=S\]

\[\sum_{k=1}^{n}(ca_k+b_k)=c\sum_{k=1}^{n}a_k+\sum_{k=1}^{n}b_k\]

\[\sum_{k=1}^{n}(b_{k+1}-b_k)=b_{n+1}-b_1\]

\[\sum a_n\text{ alternating},\quad |a_n|\to0,\quad |a_{n+1}|\le |a_n|\Rightarrow\sum a_n\text{ converges}\]

\[\sum a_n\text{ converges but }\sum |a_n|\text{ diverges}\]

\[\sum_{n=0}^{\infty}a_n(x-c)^n\]

\[a_n=\frac{f^{(n)}(c)}{n!}\]

\[R=\frac{1}{\lim_{n\to\infty}|a_{n+1}/a_n|}\]

\[|x-c|\lt R\]

\[T_N(x)=\sum_{n=0}^{N}\frac{f^{(n)}(c)}{n!}(x-c)^n\]

\[f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n\]

\[|R_N(x)|\le\frac{M|x-c|^{N+1}}{(N+1)!}\]

\[\sin x=x+O(x^3),\qquad \cos x=1-\frac{x^2}{2}+O(x^4)\]

\[\frac{d}{dx}\sum_{n=0}^{\infty}a_n(x-c)^n=\sum_{n=1}^{\infty}na_n(x-c)^{n-1}\]

\[\int\sum_{n=0}^{\infty}a_n(x-c)^n\,dx=C+\sum_{n=0}^{\infty}\frac{a_n}{n+1}(x-c)^{n+1}\]

\[e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}\]

\[\sin x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!},\qquad \cos x=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}\]

\[A\mathbf x=\mathbf b\]

\[A=(a_{ij})\]

\[(AB)_{ij}=\sum_k a_{ik}b_{kj}\]

\[\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\]

\[\det(AB)=\det(A)\det(B)\]

\[A^{-1}A=AA^{-1}=I\]

\[\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\]

\[A=LU\]

\[\det(A)=\sum_{j=1}^{n}(-1)^{i+j}a_{ij}M_{ij}\]

\[A\text{ is invertible}\quad\Longleftrightarrow\quad \det(A)\ne0\]

\[A\mathbf x=\mathbf b\quad\Rightarrow\quad \mathbf x=A^{-1}\mathbf b\]

\[(AB)^T=B^TA^T\]

\[a_1\mathbf v_1+\cdots+a_n\mathbf v_n\]

\[\mathbf u,\mathbf v\in W\Rightarrow a\mathbf u+b\mathbf v\in W\]

\[a_1\mathbf v_1+\cdots+a_n\mathbf v_n=\mathbf0\Rightarrow a_1=\cdots=a_n=0\]

\[\operatorname{span}\{\mathbf v_1,\ldots,\mathbf v_n\}=\{a_1\mathbf v_1+\cdots+a_n\mathbf v_n\}\]

\[\mathbf v=a_1\mathbf b_1+\cdots+a_n\mathbf b_n\]

\[\dim V=\text{number of vectors in a basis}\]

\[\operatorname{rank}A=\dim\operatorname{col}(A)\]

\[\operatorname{rank}A+\operatorname{nullity}A=n\]

\[\mathbf v=c_1\mathbf b_1+\cdots+c_n\mathbf b_n\]

\[\operatorname{Im}(A)=\{\mathbf w:\mathbf w=A\mathbf v\text{ for some }\mathbf v\}\]

\[\ker(A)=\{\mathbf v:A\mathbf v=\mathbf0\}\]

\[T(a\mathbf u+b\mathbf v)=aT(\mathbf u)+bT(\mathbf v)\]

\[[T(\mathbf v)]_{\beta}=A[\mathbf v]_{\alpha}\]

\[Q^TQ=I\]

\[A\mathbf v=\lambda\mathbf v\]

\[\det(A-\lambda I)=0\]

\[E_{\lambda}=\ker(A-\lambda I)\]

\[A=PDP^{-1}\]

\[A^n=PD^nP^{-1}\]

\[A^{\dagger}=A\]

\[A^{\dagger}A=AA^{\dagger}=I\]

\[A^{\dagger}A=AA^{\dagger}\]

\[A=PDP^{-1}\quad\Rightarrow\quad e^A=Pe^DP^{-1}\]