CVEL9

From Example Problems
Jump to: navigation, search

(1+i)^{i}\, on the principal branch.

(1+i)^{i}=e^{{i\,{{\rm {Ln}}}(1+i)}}

{{\rm {Ln}}}(1+i)=\ln |1+i|+i\,{{\rm {Arg}}}(1+i)=\ln {\sqrt  {2}}+i\,{\frac  {\pi }{4}}

Therefore,

(1+i)^{i}=e^{{i\,{{\rm {Ln}}}(1+i)}}=e^{{i\left(\ln({\sqrt  {2}})+i\,\pi /4\right)}}=e^{{-\pi /4}}\left(\cos(\ln {\sqrt  {2}})+i\,\sin(\ln {\sqrt  {2}})\right)

Complex Variables
Main Page