Let us begin with the definition of homogeneous function.
Theorem (Homogeneous function)
A function is said to be homogeneous if:
where is, for now, an unspecified function (we will shortly see that it has a precise form).
An example of homogeneous function is ; in fact:
and so in this case
A very interesting property of homogeneous functions is that once its value in a point and the function are known, the entire can be reconstructed; in fact, any can be written in the form (of course with ), so that:
We now want to show that has a precise form.
The function as in the definition of homogeneous function is:
From the definition of homogeneous function, for we have on one hand that:
If we now suppose
to be differentiable
, then differentiating with respect to
this last equation we get:
we mean the derivative of
with respect to its argument. Setting
, so since
by definition we have
A homogeneous function such that is said to be homogeneous of degree .
In case is a function of more than one variable , the definition of homogeneous function changes to:
- ↑ In reality it would be sufficient the sole continuity of , but in this case the proof becomes longer.