Xmas Maths 2015 Simplify the expression below into a seasonal greeting using commonly-used symbols in commonly-used formulas in maths and physics. Colours are purely ornamental!
$$
\begin{align}
\frac{
\color{green}{(x+iy)}
\color{red}{(y^3-x^3)}
\color{orange}{(v^2-u^2)}
\color{red}{(3V_{\text{sphere}})^{\frac 13}}
\color{orange}{E\cdot} 
\color{green}{\text{KE}}
}
{
\color{orange}{2^{\frac 23}}
\color{green}{c^2}
\color{red}{e^{i\theta}}
\color{orange}{v^2}
\color{green}{(x^2+xy+y^2)}}
\color{red}{\sum_{n=0}^{\infty}\frac 1{n!}}
\color{orange}{\bigg/}
\color{orange}{\left(\int_{-\infty}^\infty e^{-x^2} dx\right)^{\frac 23}}
\end{align}$$
NB: Knowledge of the following would be helpful:  
Basic Maths:  


*

*Taylor series expansion  

*Normalizing factor for the integral of a normal distribution  

*Rectangular and polar forms for complex variables  

*Volume of a sphere   


Basic Physics:  


*

*Kinematics formulae for motion under constant acceleration   

*Einstein's equation   

*One of the energy equations  

 A: $$
\begin{align}
&\frac{
\color{green}{(x+iy)}
\color{red}{(y^3-x^3)}
\color{orange}{(v^2-u^2)}
\color{red}{(3V_{\text{sphere}})^{\frac 13}}
\color{orange}{E\cdot} 
\color{green}{\text{KE}}
}
{
\color{orange}{2^{\frac 23}}
\color{green}{c^2}
\color{red}{e^{i\theta}}
\color{orange}{v^2}
\color{green}{(x^2+xy+y^2)}}
\color{red}{\sum_{n=0}^{\infty}\frac 1{n!}}
\color{orange}{\bigg/}
\color{orange}{\left(\int_{-\infty}^\infty e^{-x^2} dx\right)^{\frac 23}}\\
&=
\frac{
\color{green}{(x+iy)}
\color{red}{(y^3-x^3)}
\color{orange}{(v^2-u^2)}
\color{red}{(3V_{\text{sphere}})^{\frac 13}}
\color{orange}{E\cdot} 
\color{green}{\text{KE}}
}
{
\color{red}{e^{i\theta}}
\color{green}{(x^2+xy+y^2)}
\color{orange}{\cdot2^{\frac 23}}
\color{green}{c^2}
\color{orange}{v^2}
}
\color{red}{\sum_{n=0}^{\infty}\frac 1{n!}}
\color{orange}{\bigg/}
\color{orange}{\left(\sqrt{\pi}\right)^{\frac 23}}\\
&=
\color{green}{\left(\frac{x+iy}{e^{i\theta}}\right)}
\color{red}{\left(\frac{y^3-x^3}{x^2+xy+y^2}\right)}
\color{orange}{(v^2-u^2)}
\color{red}{\left(\frac {(3V_\text{sphere})^\frac 13}{\left(2\sqrt{\pi}\right)^{\frac 23}}\right)}
\color{orange}{\left(\frac{E}{c^2}\right)}
\color{green}{\left(\frac{\text{KE}}{v^2}\right)}
\color{red}{\sum_{n=0}^{\infty}\frac 1{n!}}
\\
&=
\color{green}{\left(\frac{re^{i\theta}}{e^{i\theta}}\right)}
\color{red}{\left(\frac{(y-x)(y^2+xy+x^2)}{x^2+xy+y^2}\right)}
\color{orange}{(v^2-u^2)}
\color{red}{\left(\frac {3\cdot \frac 43 \pi r^3}{4\pi}\right)^\frac 13}
\color{orange}{\left(\frac{mc^2}{c^2}\right)}
\color{green}{\left(\frac{\frac 12 mv^2}{v^2}\right)}
\color{red}{(e)}
\\
&=
\color{green}{\left(r\right)}
\color{red}{\left(y-x\right)}
\color{orange}{(2as)}
\color{red}{\left(r^3\right)^\frac 13}
\color{orange}{\left(m\right)}
\color{green}{\left(\frac 12m\right)}
\color{red}{(e)}
\\
&=
\color{green}{\left(r\right)}
\color{red}{\left(y-x\right)}
\color{orange}{(as)}
\color{red}{\left(r\right)}
\color{orange}{\left(m\right)}
\color{green}{\left( m\right)}
\color{red}{(e)}
\\
&=
\color{orange}{\left(m\right)}
\color{red}{(e)}
\color{green}{\left(r\right)}
\color{red}{\left(r\right)}
\color{red}{\left(y-x\right)}
\color{green}{\left(m\right)}
\color{orange}{(as)}
\end{align}$$
Merry Christmas, everyone!!

The following links might be helpful.
- Complex numbers and polar coordinates
- Difference of two cubes
- Kinematics formulae for constant acceleration in a straight line
- Volume of a sphere
- Einstein's mass-energy equivalence
- Kinetic energy
- Taylor/Maclaurin series expansion of $e$
- Gaussian integral (normalizing factor for the normal distribution)
A: Notice:


*

*$$\sum_{n=0}^{\infty}\frac{1}{n!}=\lim_{m\to\infty}\sum_{n=0}^{m}\frac{1}{n!}=\lim_{m\to\infty}\left(1+\frac{1}{m}\right)^m=e$$

*$$\int_{-\infty}^{\infty}e^{-x^2}\space\text{d}x=\lim_{a\to\infty}\int_{-a}^{a}e^{-x^2}\space\text{d}x=\lim_{a\to\infty}\left[\frac{\text{erf}(x)\sqrt{\pi}}{2}\right]_{-a}^{a}=\sqrt{\pi}$$

*$$\text{V}_{sphere}=\frac{4\pi r^3}{3}$$

*$$\text{E}=mc^2$$

*$$\text{EK}=\frac{mv^2}{2}$$

