User:Dennis J au/sandbox

Quadratic Polynomial

${\displaystyle ax^{2}+bx+c=0}$

<math>ax^2 + bx + c = 0</math>

Quadratic Polynomial (Force PNG Rendering)

${\displaystyle ax^{2}+bx+c=0\,}$

<math>ax^2 + bx + c = 0\,</math>

Quadratic Formula

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$

<math>x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}</math>

Tall Parentheses and Fractions

${\displaystyle 2=\left({\frac {\left(3-x\right)\times 2}{3-x}}\right)}$

<math>2 = \left(
 \frac{\left(3-x\right) \times 2}{3-x}
 \right)</math>

${\displaystyle S_{\text{new}}=S_{\text{old}}-{\frac {\left(5-T\right)^{2}}{2}}}$

 <math>S_{\text{new}} = S_{\text{old}} - \frac{ \left( 5-T \right) ^2} {2}</math>
 

Integrals

${\displaystyle \int _{a}^{x}\!\!\!\int _{a}^{s}f(y)\,dy\,ds=\int _{a}^{x}f(y)(x-y)\,dy}$

<math>\int_a^x \!\!\!\int_a^s f(y)\,dy\,ds
 = \int_a^x f(y)(x-y)\,dy</math>

Summation

${\displaystyle \sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {m^{2}\,n}{3^{m}\left(m\,3^{n}+n\,3^{m}\right)}}}$

<math>\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{m^2\,n}
 {3^m\left(m\,3^n+n\,3^m\right)}</math>

Differential Equation

${\displaystyle u''+p(x)u'+q(x)u=f(x),\quad x>a}$

<math>u'' + p(x)u' + q(x)u=f(x),\quad x>a</math>

Complex numbers

${\displaystyle |{\bar {z}}|=|z|,|({\bar {z}})^{n}|=|z|^{n},\arg(z^{n})=n\arg(z)}$

<math>|\bar{z}| = |z|,
 |(\bar{z})^n| = |z|^n,
 \arg(z^n) = n \arg(z)</math>

Limits

${\displaystyle \lim _{z\rightarrow z_{0}}f(z)=f(z_{0})}$

<math>\lim_{z\rightarrow z_0} f(z)=f(z_0)</math>

Integral Equation

${\displaystyle \phi _{n}(\kappa )={\frac {1}{4\pi ^{2}\kappa ^{2}}}\int _{0}^{\infty }{\frac {\sin(\kappa R)}{\kappa R}}{\frac {\partial }{\partial R}}\left[R^{2}{\frac {\partial D_{n}(R)}{\partial R}}\right]\,dR}$

<math>\phi_n(\kappa) =
 \frac{1}{4\pi^2\kappa^2} \int_0^\infty
 \frac{\sin(\kappa R)}{\kappa R}
 \frac{\partial}{\partial R}
 \left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR</math>

Example

${\displaystyle \phi _{n}(\kappa )=0.033C_{n}^{2}\kappa ^{-11/3},\quad {\frac {1}{L_{0}}}\ll \kappa \ll {\frac {1}{l_{0}}}}$

<math>\phi_n(\kappa) = 
 0.033C_n^2\kappa^{-11/3},\quad
 \frac{1}{L_0}\ll\kappa\ll\frac{1}{l_0}</math>

Continuation and cases

${\displaystyle f(x)={\begin{cases}1&-1\leq x<0\\{\frac {1}{2}}&x=0\\1-x^{2}&{\mbox{otherwise}}\end{cases}}}$

<math>
 f(x) =
 \begin{cases}
 1 & -1 \le x < 0 \\
 \frac{1}{2} & x = 0 \\
 1 - x^2 & \mbox{otherwise}
 \end{cases}
 </math>

Prefixed subscript

${\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};c_{1},\dots ,c_{q};z)=\sum _{n=0}^{\infty }{\frac {(a_{1})_{n}\cdots (a_{p})_{n}}{(c_{1})_{n}\cdots (c_{q})_{n}}}{\frac {z^{n}}{n!}}}$

 <math>{}_pF_q(a_1,\dots,a_p;c_1,\dots,c_q;z)
 = \sum_{n=0}^\infty
 \frac{(a_1)_n\cdots(a_p)_n}{(c_1)_n\cdots(c_q)_n}
 \frac{z^n}{n!}</math>

Fraction and small fraction

${\displaystyle {\frac {a}{b}}}$   ${\displaystyle {\tfrac {a}{b}}}$
<math> \frac {a}{b}\  \tfrac {a}{b} </math>