Complex Tetration!

Algebra Level 4

Tetration is defined as

${^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n$

Find the value of

${^{2}i}.$

Notation : $i =\sqrt{-1}$ denotes the imaginary unit .

2 solutions

Ronak Agarwal
Aug 18, 2014

We have $^{ 2 }{ i }={ i }^{ i }$ .

Now we will use polar form of writing the complex numbers.

$i={e}^{\frac{i\pi}{2}}$

Using this we get :

${i}^{i}={e}^{\frac{{i}^{2}\pi}{2}}={e}^{\frac{-\pi}{2}}$

Math Man
Aug 18, 2014

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

How did you get this result?

Sharky Kesa - 6 years, 9 months ago

ln i can be defined as ln i =ln|i|+i arg(i)=(pi/2) i

Wanchun Shen - 6 years, 9 months ago