i^ln(1 + i)

I'll presume natural logarithms

1 + i =>

sqrt(2) * (sqrt(2)/2 + i * sqrt(2)/2) =>

sqrt(2) * (cos(pi/4) + i * sin(pi/4)) =>

sqrt(2) * e^((pi/4) * i)

i^ln(1 + i) =>

i^(ln(sqrt(2) * e^((pi/4) * i)) =>

i^(ln(sqrt(2)) + ln(e^((pi/4) * i))) =>

i^((1/2) * ln(2) + (pi/4) * i * ln(e)) =>

i^((1/2) * ln(2) + (pi/4) * i) =>

i^((1/2) * ln(2)) * i^((pi/4) * i)

i = 0 + i

i = cos(pi/2) + i * sin(pi/2)

i = e^((pi/2) * i)

i^((1/2) * ln(2)) * e^((pi/2) * i * (pi/4) * i) =>

i^((1/2) * ln(2)) * e^((pi^2 / 8) * i^2) =>

i^((1/2) * ln(2)) * e^(-pi^2 / 8)

i = (cos(pi/2) + i * sin(pi/2))

We know that (cos(t) + i * sin(t))^n = cos(n * t) + i * sin(n * t). That's DeMoivre's Theorem.

i^((1/2) * ln(2)) =>

(cos(pi/2) + i * sin(pi/2))^((1/2) * ln(2)) =>

cos((1/2) * ln(2) * pi/2) + i * sin((1/2) * ln(2) * pi/2)) =>

cos(ln(2) * pi / 4) + i * sin(ln(2) * pi / 4)

Putting it all together:

i^((1/2) * ln(2)) * e^(-pi^2 / 8) =>

e^(-pi^2 / 8) * (cos(ln(2) * pi / 4) + i * sin(ln(2) * pi / 4))

The real part:

e^(-pi^2 / 8) * cos(ln(2) * pi/4)

The imaginary part

e^(-pi^2 / 8) * sin(ln(2) * pi/4)

Assuming log(base 10):

http://www.wolframalpha.com/input/?i=simplify+i%5E...