Usuari:Ferran Mir/proves/Sridhara

Símbols LaTex

Lletres gregues

$\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\varepsilon ,\zeta ,\eta ,\theta ,\vartheta ,\iota ,\kappa ,\lambda ,\mu ,\nu ,\xi ,o,\pi ,\varpi ,\rho ,\varrho ,\sigma ,\varsigma ,\tau ,\upsilon ,\phi ,\varphi ,\chi ,\psi ,\omega$

$\mathrm {A} ,\mathrm {B} ,\Gamma ,\Delta ,\mathrm {E} ,\mathrm {Z} ,\mathrm {H} ,\Theta ,\mathrm {I} ,\mathrm {K} ,\Lambda ,\mathrm {M} ,\mathrm {N} ,\Xi ,\Pi ,\mathrm {P} ,\Sigma ,\mathrm {T} ,\Upsilon ,\Phi ,\mathrm {X} ,\Psi ,\Omega$

Operadors binaris

$+,-,\times ,\div ,\pm ,\mp ,\ast ,\star ,\circ ,\bullet ,\cdot ,\cap ,\cup ,\uplus ,\sqcap ,\sqcup ,\vee ,\wedge ,\setminus ,\wr ,\diamond ,\bigtriangleup ,\bigtriangledown ,\triangleleft ,\triangleright ,\oplus ,\ominus ,\otimes ,\oslash ,\odot ,\bigcirc ,\dagger ,\ddagger ,\amalg$

$a!$ $3^{3}$

Relacions binàries

$=,<,>,\leq ,\geq ,\equiv ,\sim ,\simeq ,\prec ,\preceq ,\succ ,\succeq ,\ll ,\gg ,\approx ,\cong ,\neq ,\doteq$

Conjunts: $\subset ,\subseteq ,\sqsubset ,\sqsubseteq ,\in ,\ni ,\supset ,\supseteq ,\sqsupset ,\sqsupseteq$

Lògica:$\models ,\perp ,\mid ,\parallel ,\vdash ,\dashv ,\propto ,\smile ,\frown ,\asymp ,\bowtie$

Fletxes

$\leftarrow ,\Leftarrow ,\rightarrow ,\Rightarrow ,\leftrightarrow ,\Leftrightarrow ,\mapsto ,\longleftarrow ,\Longleftarrow ,\longrightarrow ,\Longrightarrow ,\longleftrightarrow ,\Longleftrightarrow ,\longmapsto ,\uparrow ,\Uparrow ,\downarrow ,\Downarrow ,\updownarrow ,\Updownarrow ,\nearrow ,\searrow ,\swarrow ,\nwarrow ,\hookleftarrow ,\hookrightarrow ,\leftharpoonup ,\leftharpoondown ,\rightharpoonup ,\rightharpoondown ,\rightleftharpoons$

Plantilles de matemàtiques

$e^{i\pi }+1=0$

$-1,-2,-3,...$

$1/n,1/n^{2},1/n^{3},...$

$ax^{2}+bx=c$

$4a^{2}x^{2}+4abx=4ac$

$4a^{2}x^{2}+4abx+b^{2}=4ac+b^{2}$

$2ax+b={\sqrt {4ac+b^{2}}}$

$x^{3}+ax=b$

$x^{3}=ax+b$

$<a,b>$

$x_{n}$

$x\subset n$

${\frac {d^{n}y}{dx^{n}}}-x^{m}y=0$

Arrels i Exponents

$x={\sqrt {a+{\sqrt {b}}}}+{\sqrt {a-{\sqrt {b}}}}$

$x^{2}=(2{\sqrt {a^{2}-b}})x+2a$,

$x={\sqrt[{3}]{a+{\sqrt {b}}}}+{\sqrt[{3}]{a-{\sqrt {b}}}}$ també resol l'equació $x^{3}=(3{\sqrt[{3}]{a^{2}-b}})x+2a$.

${\sqrt[{3}]{{\frac {b}{2}}+{\sqrt {{\frac {b^{2}}{4}}+{\frac {a^{3}}{27}}}}}}+{\sqrt[{3}]{{\frac {b}{2}}-{\sqrt {{\frac {b^{2}}{4}}+{\frac {a^{3}}{27}}}}}}$

$x={\sqrt[{3}]{2+{\sqrt {-121}}}}+{\sqrt[{3}]{2-{\sqrt {-121}}}}$

$x^{1/3}$

Fraccions

${\frac {1}{2}}\pi ={\frac {2\times {2}\times {4}\times {4}\times {6}\times {6}\times {8}...}{1\times {3}\times {3}\times {5}\times {5}\times {7}\times {7}...}}$

$z=t\div {\sqrt {n-1}}$

Limits

$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

${\frac {d}{dx}}a^{x}=\lim _{h\to 0}{\frac {a^{x+h}-a^{x}}{h}}=\lim _{h\to 0}{\frac {a^{x}a^{h}-a^{x}}{h}}=a^{x}\left(\lim _{h\to 0}{\frac {a^{h}-1}{h}}\right).$

Sumatoris

$e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots$

$n!$

$\prod \limits _{n=0}^{\infty }={\frac {\sqrt {2\pi }}{\Gamma (x)}}$

Suma de Kloosterman: $K(a,b,n)=\sum \limits _{k(n)}e\left({\frac {ak+b{\bar {k}}}{n}}\right)$ , o també: $K(a,b;m)=\sum _{\stackrel {0\leq x\leq m-1}{\gcd(x,m)=1}}e^{{\frac {2\pi i}{m}}(ax+bx^{*})}$ , o també: $K(u,v,n)\equiv \sum _{h}{exp}\left[{\frac {2\pi i(uh+v{\bar {h}})}{n}}\right]$

Identitat del producte quíntuple

$\prod \limits _{n}{\frac {(1-x)^{2n}(1-a^{2}x^{2n-2})(1-a^{-2}x^{}2n)}{(1+ax^{2n-1})(1+a^{-1}x^{2n-1}}}=\displaystyle \sum \limits _{n}(a^{-3n}-a^{3n+2})x^{n(3n+2)},\mid x\mid <1,a\neq 0.$

$\prod \limits _{n\geq 1}(1-s^{n})(1-s^{n}t)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^{2})(1-s^{2n-1}t^{-2})=\displaystyle \sum \limits _{n\in Z}s^{(3n^{2}+n)/2}(t^{3n}-t^{-3n-1})$

Trigonometria

$\arctan {\frac {1}{x}}={\frac {1}{x}}-{\frac {1}{3x^{3}}}+{\frac {1}{5x^{5}}}-...$

$\cos x$

${\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{5}}+\arctan {\frac {1}{8}}$

Integrals

$\int _{a}^{b}f(x)\,dx$

$\int x^{n}\sin x\,dx$

$\int _{0}^{\infty }J_{(x)}^{a}e^{-bx}x^{c-1}dx$

$\int _{S_{0}}^{S_{1}}F(x,y,\theta ,\kappa )ds$

Matrius

${\begin{bmatrix}1&1\\0&1\\\end{bmatrix}}$

${\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\\\end{bmatrix}}$

${\begin{bmatrix}1&2&3&4\\5&6&7&8\\9&0&1&2\\3&4&5&6\\\end{bmatrix}}$

${\begin{matrix}1&1\\0&1\\\end{matrix}}$

${\begin{vmatrix}D_{1}t&-a_{12}t_{2}&\dots &-a_{1n}t_{n}\\-a_{21}t_{1}&D_{2}t&\dots &-a_{2n}t_{n}\\\dots &\dots &\dots &\dots \\-a_{n1}t_{1}&-a_{n2}t_{2}&\dots &D_{n}t\end{vmatrix}}$

Altres

Fórmula de Crofton: $\delta P[H]=n(P[H\xi _{1}\in \delta S]-P[H]s^{-1}\delta s$

Equacions de Mathisson:

1 ${\frac {D}{ds}}\left(mu^{\lambda }+u_{\mu }{\frac {DS^{\lambda \mu }}{ds}}\right)=-{\frac {1}{2}}u^{\pi }S^{\rho \sigma }R_{\pi \rho \sigma }^{\lambda }$

2 ${\frac {DS^{\mu \nu }}{ds}}+u^{\mu }u_{\sigma }{\frac {DS^{\nu \sigma }}{ds}}-u^{\nu }u_{\sigma }{\frac {DS^{\mu \sigma }}{ds}}=0$

Teorema de Calugareanu: $Lk(K)=Lw(K)+Wr(K)$

I de Moran: ${\frac {\sum \limits _{i=1}^{n}\sum \limits _{j=1}^{n}W_{ij}(x_{i}-{\bar {x}})(x_{j}-{\bar {x}})}{S(\sum \limits _{i=1}^{n}\sum \limits _{j=1}^{n}W_{ij})}}$

${\ddot {x}}+kf(x){\dot {x}}+g(x)=kp(t)$