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F. Useful mathematical formulas

		$\Omega$	$\displaystyle \frac{df}{dz}$
$arccos / acos / cos^{-1}$	$\displaystyle \frac{1}{2} \pi + i \ln\left( iz+\sqrt{1-z^2} \right)$		$\displaystyle\frac{-1}{\sqrt{1-z^2}}$
$arcsin / asin / sin^{-1}$	$\displaystyle \ln\left( z+\sqrt{1+z^2} \right)$		$\displaystyle\frac{1}{\sqrt{1+z^2}}$
$arctan / atan / tan^{-1}$	$\displaystyle \frac{i}{2}\left( \ln\left( 1 - iz\right)-\ln\left(1+iz\right) \right)$		$\displaystyle\frac{1}{1+z^2}$
$arccotg / acotg / cotg^{-1}$	$\displaystyle \frac{i}{2}\left( \ln\left(\frac{z-i}{z}\right)-\ln\left(\frac{z+i}{z}\right) \right)$		$\displaystyle\frac{-1}{1+z^2}$
$argch / acosh / cosh^{-1}$	$\displaystyle\ln\left( z+\sqrt{z+1}\sqrt{z-1}\right)$		$\displaystyle\frac{1}{\sqrt{z^2-1}}$
$argcoth / acoth / coth^{-1}$	$\displaystyle \frac{1}{2}\left( \ln\left( 1+\frac{1}{z}\right)-\ln\left(1-\frac{1}{z}\right) \right)$	$z\neq 0$	$\displaystyle\frac{1}{1-z^2}$

$\displaystyle \arctan(z) = \frac{pi}{2} - \arccos\left(\frac{z}{\sqrt{z^2+1}}\right)$

(F.1)

$\displaystyle \arctan(x) = sign(x)\arccos\left(\frac{1}{\sqrt{x^2+1}}\right)$

(F.2)

$\displaystyle \arccos(z) = \frac{pi}{2} - \arctan\left(\frac{z}{\sqrt{1-z^2}}\right)$

(F.3)

$\displaystyle \arccos(x) = \arctan\left(\frac{\sqrt{1-x^2} }{x}\right)$

(F.4)

$\displaystyle \ln z = \ln\vert z\vert + i \arg(z)$

(F.5)

Integrals of rationnal functions, from the free mathematical handbook [1]:
$r = \sqrt{x^2+a^2}$

$\displaystyle \int r \;dx = \frac{1}{2}\left(x r +a^2\,\ln\left(x+r\right)\right)$

$\displaystyle \int\frac{r\;dx}{x} = r-a\ln\left\vert\frac{a+r}{x}\right\vert = r - a\, \operatorname{arsinh}\frac{a}{x}$

$s = \sqrt{x^2-a^2}$

$\displaystyle \int\frac{s\;dx}{x} = s - a\arccos\left\vert\frac{a}{x}\right\vert$

$\displaystyle \int\frac{dx}{s} = \int\frac{dx}{\sqrt{x^2-a^2}} =\ln\left\vert\frac{x+s}{a}\right\vert$

$R = \sqrt{ax^2+bx+c}$

$\displaystyle \int R dx=\frac{2ax+b}{4a}R+\frac{4ac-b^2}{8a}\int\frac{dx}{R}$

$\displaystyle \int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln\left\vert 2\sqrt{a}R+2ax+b\right\vert$ (for $\displaystyle a>0$ )

$\displaystyle \int\frac{dx}{R} = \frac{1}{\sqrt{a}}\,\operatorname{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}}$ (for $\displaystyle a>0$ , $\displaystyle 4ac-b^2>0$ )

$\displaystyle \int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln\vert 2ax+b\vert$ (for $\displaystyle a>0$ , $\displaystyle 4ac-b^2=0$ ) $\displaystyle$

Emmanuel Branlard
http://emmanuel.branlard.free.fr/