Measuring Antenna Capacitance

...not bad. And now its a small exercise for you to substitude k and a by terms with L und d. Perfect formula for this special case.  

Too few points to select realistic functional dependencies.

(for example: the behavior of parameter "a" is unclear on the interval L=0...20 cm, whether it tends to 1, or to another value)

For practical purposes (in order to obtain intermediate values of "a" and "k"), it is sufficient to have a piecewise linear interpolation or approximation by a 2-nd order polynomial (as anyone likes).

Assuming you mean instruments for measuring electronic systems... 

Basic electrical components (inductance, resistance and capacitance at various frequency ranges). 

Voltage, current, and power, also in various frequency ranges. 

Signal analysis. Time and frequency domain analysis of a large range of signals, both analogue and digital. The best known and perhaps most general is the oscilloscope. 

Specialised measurements, for example antennas and cables and fibres, semiconductor testing, testing for distortion, non linearity (amplitude, frequency, phase, group delay), bit error rates. 

Signal generators generating a wide range of test signals over various frequency ranges.