a complex complex root

[madeinmsia]

G'' = -kG
k is a constant
solving this ODE, r = +/- sqrt(-k)
if k > 0, then r = +/- sqrt(k)i
so G is in the form Acos(sqrt(k)x) + Bsin(sqrt(k)x)

so, what if k is a complex number, then
r = +/- sqrt(-ki)

then what is the form of G?

[antonantal]

so, what if k is a complex number, then
r = +/- sqrt(-ki)

then what is the form of G?

Don't you know how to calculate the square root of a complex number (http://www.mathpropress.com/stan/bibliography/complexSquareRoot.pdf)?