The wikipedia article states that this approximate impedance is calculated assuming a sinusoidal current distribution on a centre-fed dipole.

There are no fields in the conductors as this analysis assumes perfect conductors.

This formula is an approximation, and the term 

\$Ci(\frac{2ka^2}{L}) \$

which goes as \$log(\frac{2ka^2}{L}) \$  as \$a \rightarrow 0\$

and will dominate the reactance as \$a \rightarrow 0\$ for all element lengths not \$ \lambda/2\$, and the fact that it doesn't have any affect at precisely \$ \lambda/2\$ I think should be seen as a limitation of the approximation.

Edit:
1. **This is an approximation.** Read the references as to the approximations used in both the assumed current distribution and in constructing the integral equation.
2. In most antenna work with good conductors there are minimal fields in the conductors.  The reactance that seems to concern you is still there with perfect conductors, which have no fields in the conductors. 
3. The generally accepted intuitive answer as to why a half-wave dipole resonates slightly shorter than half a wavelength is the capacitive effect of the wire ends.

Apart from that I'm going to leave you to the guys over at amateur radio stack exchange where you have posted the identical question.

[amateur radio stack exchange question][1]


  [1]: https://ham.stackexchange.com/questions/18806/does-the-self-inductance-and-velocity-factor-of-the-elements-of-a-dipole-antenna