Does anybody have any idea on how to define a linearly homogeneous function? The only solution I have come up with involves using rules and 'ReplaceAll'. This solution is not adequate, as it requires a lot of fine tuning. An example: Evaluating the following code, for a suitably defined function f (linearly homogeneous) should return 'True' f[tau x,tau y]==tau f[x,y] Alternatively, evaluating In[1]:= f[mu x1,mu x2] should return Out[1]:= mu f[x1,x2] Optionaly, it would really be an added bonus if somehow the definitions would persist, in the following sense: The derivative of a linearly homogeneous function should be a homogeneous function of degree 0, i.e. evaluating In[2]:= Derivative[1,0][f][k x,k y] should return Out[2]:= Derivative[1,0][f][x,y] I would be really gratefull if someone could offer a hint towards the right direction

Hi Jarque, I think the problem with the derivate ist just that D[f[x , y], x] /. {x -> a x, y -> a y} expands to the somewhat strange expression: D[f[a x , a y], a x] What you probably really want is: D[f[a x , a y], x] So use instead: D[f[x, y] /. {x -> a x, y -> a y}, x] and you get a*Derivative[1,0][f][x,y] as expected.

yes,evaluating In[1]:= f[x,x] should produce Out[1]:= x f[1,1] so I did make use of your later suggestion (i.e. f[t_,t_]:= t f[1,1] ) I also replaced the 'Orderless' attribute with the definition f[t_ x_,t_]:= t f[x,1] because, for the context which these definitions are made, f[a,1] == f[1,a] does not hold always. So far and for a limited but sensible range of inputs it seems to be working as it should *Unfortunately, the behavior for the derivatives so far seems not the desired one (which makes perfect sense since no defitions involve the derivatives). D[f[x , y], x] /. {x -> a x, y -> a y} evaluates to Derivative[1,0][f][a x,a y] when it would be desirable to evaluate to Derivative[1,0][f][x,y]

What should happen with f[x,x] ? Should it return x f[1,1] ? if so, than add another rule: f[\[Tau]_, \[Tau]_ ] := \[Tau] f[1, 1] Regards.

How to define a homogeneous function?