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##m\ddot {x}(t)=-kx(t).##

The solutions are##x(t)=Ae^{i\omega t}+Be^{-i\omega t},##

where ##\omega=\sqrt{\frac{k}{m}}##, and constants ##A## and ##B##. Physically, what does it mean for a solution to be complex? Is it only the real part that is of interest or also the imaginary part? Of course, using Euler's formula, the solution can be rewritten as##(A+B)\cos{(\omega t)}+i(A-B)\sin{(\omega t)},##

and one can introduce new, complex constants. However, there is still the issue of a real and imaginary part with that representation.