a mobile antenna using a
Hamstick or Bugcatcher with shunt capacitor feed or an L match. Can someone
please explain, including the math, on how a ~500pf capacitor transforms 10
ohms to 50 ohms for 40 meters.

Hi Hank,
One of the eight possible matching solutions using L networks is a shunt
capacitor and series inductor. If you have an antenna with some inductive
loading as a Hamstick or Bugcatcher one can use the inductive loading as the
series inductance, and a shunt capacitor to realize an L matching network.
My older antenna handbooks talks about this type matching in the mobile
antenna section. If you can measure the complex impedance at the base of the
mobile antenna with an analyzer, the Smith Chart will quickly tell you if a
shunt Cap. will realize a 50 ohm match.

73 Gary N4AST