Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

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

Taking ${log}_e$ on both sides

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

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

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

= (1-i) $[\frac{1}{2}log2+\frac{i\pi }{4}]$

= $\frac{1}{2}{log}_{e}2+\frac{i\pi }{4}-\frac{i}{2}{log}_{e}2+\frac{\pi }{4}$

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

Imaginary part of Z {Im(z)} = $\frac{\pi }{4}-log\sqrt{2}$