CVEL8

From Example Problems
Jump to: navigation, search

Evaluate (-i)^{i}\, on the principal branch.

(-i)^{i}=e^{{i{\mathrm  {Log}}(-i)}}\, where capital functions are principal value.

And

{\mathrm  {Log}}(-i)={\mathrm  {Log}}|-i|+i{\mathrm  {Arg}}(-i)\,

={\mathrm  {Log}}(1)+i{\frac  {3\pi }{2}}\,

=i{\frac  {3\pi }{2}}\,

So

(-i)^{i}=e^{{i{\mathrm  {Log}}(-i)}}=e^{{ii{\frac  {3\pi }{2}}}}=e^{{-{\frac  {3\pi }{2}}}}=e^{{{\frac  {\pi }{2}}}}\,



Main Page : Complex Variables : Exponential and Log