A) \[{{e}^{-\pi /4}}\cos \left( \frac{1}{2}\log 2 \right)\]
B) \[-{{e}^{-\pi /4}}\sin \left( \frac{1}{2}\log 2 \right)\]
C) \[{{e}^{\pi /4}}\cos \left( \frac{1}{2}\log 2 \right)\]
D) \[{{e}^{-\pi /4}}\sin \left( \frac{1}{2}\log 2 \right)\]
Correct Answer: A
Solution :
Let\[z={{(1-i)}^{-i}}\]. Taking log on both sides, \[\Rightarrow \,\log \,z\]\[=-i\,\,\log (1-i)\]\[=-i\,\log \sqrt{2}\,\left( \cos \frac{\pi }{4}-i\sin \frac{\pi }{4} \right)\] \[=-\,i\,\log \left( \sqrt{2}{{e}^{-\,i\,\pi /4}} \right)\]\[=-i\,\left[ \frac{1}{2}\log 2+\log \,{{e}^{-i\,\pi /4}} \right]\] \[=-i\,\left[ \frac{1}{2}\log 2-\frac{i\pi }{4} \right]\] \[=-\frac{i}{2}\log \,2\,-\frac{\pi }{4}\] Þ \[z={{e}^{-\pi /4}}\,\,{{e}^{-i/2\,\log 2}}\]. Taking real part only, \[\Rightarrow \,\,\operatorname{Re}(z)=\,{{e}^{-\pi /4}}\,\cos \,\left( \frac{1}{2}\log 2 \right)\].
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