Question

Escreva na forma algebrica os numerous complexos:

A) ln(1-i);
B)Sin($/pi$*i);
C)sinh($/pi$.i); Sinh- Seno hiperbolico

Answer 1

a)
$1-i=\sqrt2\left(\dfrac{1}{\sqrt2}-\dfrac{1}{\sqrt2}i\right)=\sqrt2\cdot cis\left(-\dfrac{\pi}{4}\right)\\\\\Longrightarrow1-i=\sqrt2\cdot e^{-\frac{\pi}{4}i}\\\\\\ \ln(1-i)=\ln(\sqrt2\cdot e^{-\frac{\pi}{4}i})=\ln(\sqrt2)+\ln(e^{-\frac{\pi}{4}i})\\\\ \boxed{\ln(1-i)=\ln(\sqrt2)-\dfrac{\pi}{4}i}$

b)
$\sin(\pi i)=i\sinh(\pi)\\\\ \boxed{\sin(\pi i)=\dfrac{e^\pi-e^{-\pi}}{2}i}$

c)
$\sinh(\pi i)=\dfrac{e^{\pi i}-e^{-\pi i}}{2}=\dfrac{cis(\pi)-cis(-\pi)}{2}\\\\\sinh(\pi i)=\dfrac{(\cos(\pi)+i\sin(\pi))-(\cos(-\pi)+i\sin(-\pi))}{2}\\\\\sinh(\pi i)=\dfrac{(\cos(\pi)+i\sin(\pi))-(\cos(\pi)+i\sin(\pi))}{2}\\\\ \boxed{\sinh(\pi i)=0}$