I'm late to the party. But let me offer first the finite-differences linear equation A[n]=A[n-1]*k For A[0] = 1, the solution is the sequence 1, k, k

I'm late to the party. But let me offer first the finite-differences linear equation A[n]=A[n-1]*k For A[0] = 1, the solution is the sequence 1, k, k^2, k^3, k^4, k^5, k^6.... If we let k be the complex number i, this becomes 1, i, -1, -i, 1, i, -1, -i, 1... As you can see, the introduction of imaginary parts extends the original idea of powers-of-k (which if real go exponentially to infinity if k>1 and to 0 if k<1) to an alternating oscillatory pattern both in the real parts 1, 0, -1, 0, 1, 0, -1, 0, 1 and the imaginary parts. What you're seeing there is almost the celebrated relationship between imaginary exponentials and sine functions. Now: that was the finite-differences linear equation. If instead we take the differential linear equation f'(x) = kf(x) we get the exponential function. Again, if k=i, you get oscillatory behavior.