Question

Find the real part of the complex number $(1-i{)}^{(1+i)}$

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

z = $(1-i{)}^{(1+i)}$

Taking log on both sides

log z = (1+i) log(1-i)

= (1+i) log $\sqrt{2}[cos\left(\frac{\pi }{4}\right)+isin(-\frac{\pi }{4})]$

= (1+i) ${log}_e\left(e^{-i\frac{\pi }{4}}\right).\sqrt{2}$

= (1+i) $(-\frac{i\pi }{4})+(1+i){log}_e\sqrt{2}$

= $-\frac{i\pi }{4}+\frac{\pi }{4}+{log}_e\sqrt{2}+i{log}_e\sqrt{2}$

= $(\frac{\pi }{4}+{log}_e\sqrt{2})+i({log}_e\sqrt{2}-\frac{\pi }{4})$

Re(z) = ${log}_e\sqrt{2}+\frac{\pi }{4}$