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# converse of Euler’s homogeneous function theorem

Theorem. If the function $f$ of the real variables $x_{1},\,\ldots,\,x_{k}$ satisfies the identity

$\displaystyle x_{1}\frac{\partial f}{\partial x_{1}}+\ldots+x_{k}\frac{% \partial f}{\partial x_{k}}=nf,$ | (1) |

then $f$ is a homogeneous function of degree $n$.

Proof. Let $f(tx_{1},\,\ldots,\,tx_{k}):=\varphi(t)$. Differentiating with respect to $t$ we obtain

$\varphi^{{\prime}}(t)=x_{1}f^{{\prime}}_{{x_{1}}}(tx_{1},\,\ldots,\,tx_{k})+% \ldots+x_{k}f^{{\prime}}_{{x_{k}}}(tx_{1},\,\ldots,\,tx_{k})=\frac{1}{t}[tx_{1% }f^{{\prime}}_{{x_{1}}}(tx_{1},\,\ldots,\,tx_{k})+\ldots+tx_{k}f^{{\prime}}_{{% x_{k}}}(tx_{1},\,\ldots,\,tx_{k})],$ |

which by (1) may be written

$\varphi^{{\prime}}(t)=\frac{n}{t}f(tx_{1},\,\ldots,\,tx_{k})=\frac{n}{t}% \varphi(t).$ |

Accordingly,

$\frac{\varphi^{{\prime}}(t)}{\varphi(t)}=\frac{n}{t},$ |

which implies the integrated form

$\ln|\varphi(t)|=\ln{t^{n}}+\ln{C}$ |

for any positive $t$. Thus we have $\varphi(t)=Ct^{n}$, where $C$ is independent on $t$. Choosing $t=1$ we see that $C=\varphi(1)$, and therefore $\varphi(t)=t^{n}\varphi(1)$. This last equation means that

$f(tx_{1},\,\ldots,\,tx_{k})=t^{n}f(x_{1},\,\ldots,\,x_{k})$ |

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

# References

- 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset II. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).

Keywords:

Euler's theorem on homogeneous functions

Related:

ConverseTheorem, ChainRuleSeveralVariables, Logarithm

Synonym:

converse of Euler's theorem on homogeneous functions

Major Section:

Reference

Type of Math Object:

Theorem

## Mathematics Subject Classification

26B12*no label found*26A06

*no label found*15-00

*no label found*

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