Can you raise the imaginary number i to a power that is an irrational number?

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Raiden Williamson · Accepted Answer

(      e          i      (      π              /            2      +      2      k      π      )            )    x    =      e          i      x      (      π              /            2      +      2      k      π      )        .

Adelyn Rodriguez · Accepted Answer

Take Take c∈C arbitrarily. One might define       i    c    :=      e          ln      ⁡      (              i        c            )        =      e          c      ln      ⁡      (      i      )        =      e          c      ⋅                        i          π                2            , which works. There is a catch though: this makes use of the logarithm, which is not nicely defined on the entire complex plane. The set of all &#039;reasonable&#039; outcomes for       i    c   would be       e          i      c      ⋅              (                  π          2                +        2        k        π        )             for   k  ∈      Z  . Hence, it is not uniquely defined. This problem already arises when taking rational powers of i though, so nothing&#039;s really changed.

Can you raise the imaginary number i to a power

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