I get the desired result if I inverse the Conjugate and ComplexExpand commands:

PiecewiseExpand[Conjugate@ComplexExpand[h[x]], Element[x, Reals]]

It is like the ComplexExpand is ineffective... but change the result if it is done after the Conjugate.

The function for you is probably ComplexExpand. Unfortunately neither ComplexExpand nor Conjugate threads nicely over Piecewise (an oversight by Wofram?). Even Composition ignores Piecewise. You can force a function inside Piecewise with this utility:

piecewiseThread[comd_] := 
 HoldPattern[Piecewise[stuff_, dflt_]] :> 
  Piecewise[Map[{comd[#[[1]]], #[[2]]} &, stuff], comd[dflt]]

Then you can type

Wronskian[{f[x], f[x] /. piecewiseThread[Conjugate]} /. 
  piecewiseThread[ComplexExpand], x]

and get a nice formula.

Thank you for the functional programming tip! A month ago, I had no clue about Mathematica. Now I can't stop using it! Here is another problem I encountered (it is about plane waves in physics):

f[x_] := Piecewise[{{E^(I K x) + r E^(-I K x), x <= -a/2}, {t E^(I K x) , x >= a/2}}]

and I want to calculate the wronskian between the function f[a/2] and its conjugate f*[a/2]. I want the same also at x=-a/2. How would I do that with the Piecewise definition? The only way I could do the calculation is by typing it long explicitly (so I could not use the built-in Wronskian[] function):

wronskian=Refine[f[x], Assumptions -> {a > 0, x >= a/2}] D[Refine[Conjugate[f[x]], Assumptions -> {a > 0, x >= a/2}], x] -
Refine[Conjugate@f[x], Assumptions -> {a > 0, x >= a/2}] D[Refine[f[x], Assumptions -> {a > 0, x >= a/2}], x]

All my attempts in defining another function h[x]=Refine[...] do not work because I loose the x dependence of the function (all the parameters become equivalent and the derivative is not performed).

JGuy

Thank you very much! I have tried:

w[f_[x_], g_[x_]] := D[f[x], x] g[x] - f[x] D[g[x], x]
derivativewronskian = D[w[h[x], k[x]], x]

Reduce[derivativewronskian ==  d[x] /. {D[h[x], {x,2}] -> (v - e0) h[x]/a, D[k[x], {x, 2}] -> (v - e0) k[x]/a}]

It nicely returned d[x]==0. Amazing! Thanks again!

JGuy