Ian Jackson wrote:
"What is the impedance at the centre of an infinitely long dipole in
free space?

It is the antenna`s Zo. This depends on the size of the conductor used
to make the dipoole.

Arnold B. Bailey has already worked all this out and presents it in his
1950 edition from Rider`s of "TV and Other Receiving Antennas".

Like the Zo of a transmission line, antenna Zo has nothing to do with
reflections and terminations. When you first apply power, energy must
flow into the antenna at some definite voltage to current ratio. This is
the surge impedance or Zo. If the antenna or line is uniform and
infinitely long, the energy sent away is never heard from again. Zo is
the only impedance anywhere.

Page 345 gives the surge impedance in ohms for a balanced antenna as:

Zo = 276 log 1/P

P is the circumference of the antenna rod, or periphery, expressed as a
fraction of the free-space wavelength (see page 342) This may sound
goofy but Bailey has his reasons.

Bailey`s graph on page 345 gives dipole impedances from 70 ohms to 680
ohms for rod peripheries from 1 wavelength down to 0.00001 wavelength

If you have no reflections or standing waves, the impedance you
calculate should be the Zo.

Best regards, Richard Harrison, KB5WZI