Homogeneous functions of one or more variables

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.

Theorem

The function as in the definition of homogeneous function is:

 
Proof

From the definition of homogeneous function, for we have on one hand that:

but also:
and so:
If we now suppose to be differentiable[1], then differentiating with respect to this last equation we get:
where by we mean the derivative of with respect to its argument. Setting and defining we have:
which yields:
Now, , so since by definition we have and thus . Therefore:

 


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:

  1. In reality it would be sufficient the sole continuity of , but in this case the proof becomes longer.
 PreviousNext