Root of a simple equation with imaginary exponent

Question

Let $i=\sqrt{-1}$. What is the solution of $(x-1)^i = 0$?

EDIT: I try to solve the following boundary value problem for $a>0$:

with $\lambda>0$ and $y(a)=0$. It is corresponding to the radial part of the Klein-Gordon equation of a scalar field inside its own Schwarzschild radius $a$. The solution of this equation is (from Maple)

I try to find the (spectrum) values of $\lambda$, for which the boundary condition $y(a)=0$ is satisfied. Since $HeunC(...)$, for $x=a$, is constant ($\neq 0$), I have to ask for a solution of $(a-x)^{i\times const}=0$.

I have not expressed this context in the initial question because I don't think that somebody finds it interesting...

Answer 1

Hint: If $z, \alpha \in {\Bbb C}$, define $z^\alpha= \exp(\alpha \log z)$, where $\exp$ is defined in some independent manner such as by a power series.