If the feedpoint resistance of the 1/2-wave vertical is
R ohms, then the input impedance of the coil tapping
point is R*(K squared) where K is the fraction of total
coil turns below the tap.

It requires only that L and C are resonant at the same
resonant frequency of the 1/2-wave antenna.

A roller inductor makes a nice job of it.
Disadvantage - the roller inductance L being fixed,
with C, limits the matching range to only 1 or 2 bands
before things become inefficient. For highest
efficiency L should be big and C should be quite small
like 10 to 50pF.

Solution - prune the antenna to an off-resonance
frequency. I have a program for it but never published
it because its too simple.
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Reg G4FGQ.