STD 11 - 4.1 complex nubers — JEE कक्षा 11 साइन्स — Question

$ \Rightarrow \,\log \,z$$ =  - i\,\,\log (1 - i)$$ =  - i\,\log \sqrt 2 \,\left( {\cos \frac{\pi }{4} - i\sin \frac{\pi }{4}} \right)$

$ =  - \,i\,\log \left( {\sqrt 2 {e^{ - \,i\,\pi /4}}} \right)$

$ =  - i\,\left[ {\frac{1}{2}\log 2 + \log \,{e^{ - i\,\pi /4}}} \right]$

$ =  - i\,\left[ {\frac{1}{2}\log 2 - \frac{{i\pi }}{4}} \right]$ $ =  - \frac{i}{2}\log \,2\, - \frac{\pi }{4}$

$z = {e^{ - \pi /4}}\,\,{e^{ - i/2\,\log 2}}$(केवल वास्तविक भाग लेने पर)

$ \Rightarrow \,\,{\mathop{\rm Re}\nolimits} (z) = \,{e^{ - \pi /4}}\,\cos \,\left( {\frac{1}{2}\log 2} \right)$.