To the simplest possible degree, an inductor is a component were Z ~= jwL is a good fit over a useful frequency range.

If you go the opposite direction, saying that an inductor is *exactly* Z = jwL, you must now allow that L varies, i.e., L(w), and it takes on complex values.

In general, we express Z as a polynomial -- a formula using whole powers of w, because those powers come from ideal resistance, inductance and capacitance, added and multiplied and divided together according to circuit rules.

If instead of expressing Z as a polynomial, we express it as the single formula Z = jwL, the polynomial has to get sucked into L instead. In that case, apparent inductance varies, and you get frequency-dependent, complex valued L. It's simply taking a complex number and putting it on the other side of the equation.

This is normally used with core materials, where permeability is expressed as mu = mu' + j mu''. Typically, mu' stays flat, then drops off; meanwhile, mu'' rises to a peak, then drops off.

You can take the apparent inductance of anything; a resistor has Z = R, so we can solve for its inductance as L = Z / (jw) = -j R / w. That is, negative-imaginary inductance is resistance, and it will be inversely proportional to frequency.

Tim