converse of Euler’s homogeneous function theorem

Theorem.  If the function f of the real variables x1,,xk satisfies the identityPlanetmathPlanetmathPlanetmath

x1fx1++xkfxk=nf, (1)

then f is a homogeneous function of degree n.

Proof.  Let  f(tx1,,txk):=φ(t).  Differentiating with respect to t we obtain


which by (1) may be written




which implies the integrated form


for any positive t.  Thus we have  φ(t)=Ctn,  where C is independent on t.  Choosing  t=1  we see that  C=φ(1),  and therefore  φ(t)=tnφ(1).  This last equation means that


saying that f is a (positively) homogeneous function of degree n.


  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset II.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).
Title converse of Euler’s homogeneous function theorem
Canonical name ConverseOfEulersHomogeneousFunctionTheorem
Date of creation 2013-03-22 18:07:56
Last modified on 2013-03-22 18:07:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 26B12
Classification msc 26A06
Classification msc 15-00
Synonym converse of Euler’s theorem on homogeneous functions
Related topic ConverseTheorem
Related topic ChainRuleSeveralVariables
Related topic Logarithm