Question

Considering only principal values separate into real and imaginary parts

`i^((log)(i+1))`

Answer 1

Let  `Z=i^(log(i+1))` 

`therefore logZ = log(1+i).logi`

But `log(i+1)=logsqrt2+itan^(-1)1 = logsqrt2+ipi/4`

and `logi=ipi/2`

`therefore log Z=(logsqrt2+ipi/4).ipi/2`
`=[1/2log2+ipi/4pi]/2` 

`(-pi^2)/8+ipi/4log2=e^(pi^2/8+itheta)= e^(pi^2/8itheta)`

where `theta=pi/4log2`

`=e^(pi^2/8)[costheta+isintheta]`

∴ Real part of Z `=e^(pi^2/8)costheta=e^(pi^2/8)cos[pi/4log2]`

∴ Imaginary part of Z `=e^(pi^2/8)sin[pi/4log2]`