Search results:
343.
$\arccos x = \dfrac{\pi}{2} - \arcsin x = \dfrac{\pi}{2} - \left(x + \dfrac{1}{2}\dfrac{x^3}{3} + \dfrac{1\cdot3}{2\cdot4}\frac{x^5}{5}+\cdots \right) \quad -1 < x < 1$
344.
$\ln(1+x) = \left(\dfrac{x-1}{x}\right) + \frac{1}{2}\left(\dfrac{x-1}{x}\right)^2 + \dfrac{1}{3}\left(\dfrac{x-1}{x}\right)^3 + \cdots \quad x \geq \frac{1}{2}$
345.
$\cot x = \dfrac{1}{x} - \dfrac{x}{3} - \dfrac{x^3}{45} - \cdots - \frac{2^{2n}B_nx^{2n-1}}{(2n)!} \quad 0 < x < \pi$
347.
$\dfrac{1}{\sqrt{1+x}} = 1 - \dfrac{1}{2}x + \dfrac{1\cdot 3}{2\cdot 4}x^2 - \dfrac{1\cdot 3 \cdot 5}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1$
348.
$\sqrt{1+x} = 1 + \dfrac{1}{2}x - \dfrac{1}{2\cdot 4}x^2 + \dfrac{1\cdot 3}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1$
353.
Multiplying complex numbers
$\left(a+bi\right)\left(c+di\right)=\left(ac-bd\right)+\left(ad+bc\right)i$
354.
Dividing complex numbers
$\dfrac{a+bi}{c+di}=\dfrac{ac+bd}{c^2+d^2}+\dfrac{bc-ad}{c^2+d^2}i$
356.
Polar representation of a complex number
$a+bi=r\left(\cos \psi + i \sin \psi \right)$
358.
Product in polar representation
$z_1 \cdot z_2 = r_1\left(\cos\psi_1+i\sin\psi_1\right) \cdot r_2\left(\cos\psi_2+i\sin\psi_2\right) = \\ = r_1r_2\left(\cos\left(\psi_1+\psi_2\right)+i\sin\left(\psi_1+\psi_2\right) \right)$
359.
Conjugate in polar form
$\overline{r\left(\cos\psi+i\sin\psi\right)}=r\left[\cos(-\psi)+i\sin(-\psi\right)]$
360.
Inverse in polar form
$\dfrac{1}{r\left(\cos\psi + i \sin \psi\right)}=\dfrac{1}{r}\left[ \cos(-\psi)+i\sin(-\psi)\right]$