Prove that all values of 〖(1-i) 〗^(i√2) lie on a straight line. Were i is the imaginary number √(-1).

Question

Prove that all values of  〖(1-i) 〗^(i√2)  lie on a straight line.  Were i is the imaginary number √(-1).

anonymous · Answer

is that$$\large (1-i)^{i\sqrt{2}}$$?

anonymous · Answer

Yes

experimentx · Answer

upon simplification, we get
$$ \huge \left( \sqrt{2} e^{i\pi\over 4}\right )^{i\sqrt2}$$ 
using this probably you will find the principle root.

experimentx · Answer

do you need to find all the roots?

anonymous · Answer

Yes please

experimentx · Answer

I don't know right now ... let me ask some other people. most likely I'm thinking ... it has only one root.

anonymous · Answer

Thank you. I will post another problem in complex analysis.

experimentx · Answer

$$ \huge \left(  e^{\ln \sqrt2+{i\pi\over 4}}\right )^{i\sqrt2} = e^{{-\sqrt 2 \over 4} + i\sqrt 2 \ln 2}$$

anonymous · Answer

Many Thanks

experimentx · Answer

no probs ...hardly i put any effort.