(or a toddler) Why does “i^i” result in a real number “e^-0.5pi”? How did they figure those out in the first place?
In: 9
> Why does “i^i” result in a real number “e^-0.5pi”?
Technically i^i is multivalued: e^(-pi/2) is just one of the answers, and the only one that is real. You often have to deal with multivalued functions when working with complex numbers (it’s very similar to how 4 has two square roots: 2 and -2).
> How did they figure those out in the first place?
Well, first you have to define what is meant by raising a complex number to the power of another complex number. However, it turns out that there is only one way of doing this such that (a) it reduces to the usual definition of powers when you put real numbers in, and (b) the answers vary smoothly as you change the inputs.
Latest Answers