converse of Euler’s homogeneous function theorem
Theorem. If the function of the real variables satisfies the identity
(1) |
then is a homogeneous function of degree .
Proof. Let . Differentiating with respect to we obtain
which by (1) may be written
Accordingly,
which implies the integrated form
for any positive . Thus we have , where is independent on . Choosing we see that , and therefore . This last equation means that
saying that is a (positively) homogeneous function of degree .
References
- 1 Ernst Lindelöf: Differentiali- ja integralilaskuja sen sovellutukset II. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).