Computing iⁱ

Complex Analysis · Axiom Academy

The imaginary unit raised to itself — and the real number that falls out. Evaluate , the imaginary unit raised to the imaginary power i . We will use the definition of a complex power, , and see what kind of number comes out. You just computed one of the most surprising results in complex analysis: an imaginary number raised to an imaginary power is a real number. Complex powers: , and is multi-valued, so z^ w generally has infinitely many values. Why it's real: the turns the imaginary exponent into a real one, so every value has the form — always a positive real number. The family: for ; consecutive values differ by the factor . Principal result: taking k = 0 gives . The same recipe handles any complex power: rewrite the base through e and , pick the branch you want, and simplify.

This is the written version of the interactive lesson above. See the full Complex Analysis course.