物理极客铭

EPT

$x^{2}+x+t=0$

$x_{1}={\frac {-1+{\sqrt {1-2^{2}t}}}{1\cdot 2}}$

$x^{3}+x+t=0$

$x_{1}={\frac {-{\sqrt[{3}]{2^{2}\cdot 3(3^{2}t+{\sqrt {3(2^{2}+3^{3}t^{2})}})}}+{\sqrt[{3}]{2^{2}\cdot 3(-3^{2}t+{\sqrt {3(2^{2}+3^{3}t^{2})}}}})}{2\cdot 3}}$

$x^{4}+x+t=0$

$-{\frac {1}{2}}{\sqrt {-M+{\sqrt {(2M)^{2}-4\cdot 4t}}}}+{\frac {1}{2}}{\sqrt {M}}$

$M={\frac {1}{6}}(-{\sqrt[{3}]{2^{2}\cdot 3(-3^{2}+{\sqrt {3(3^{3}-4^{4}t^{3})}})}}+{\sqrt[{3}]{2^{2}\cdot 3(3^{2}+{\sqrt {3(3^{3}-4^{4}t^{3})}})}})$

exp

$e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+{x^{4} \over 4!}+\cdots$

$a^{x}=e^{x\ln a}=\sum _{n=0}^{\infty }{(x\ln a)^{n} \over n!}$

$\ln z=2\sum _{n=0}^{\infty }{\frac {1}{2n+1}}\left({\frac {z-1}{z+1}}\right)^{2n+1}$