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单词 ConverseOfEulersHomogeneousFunctionTheorem
释义

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

φ(t)=x1fx1(tx1,,txk)++xkfxk(tx1,,txk)=1t[tx1fx1(tx1,,txk)++txkfxk(tx1,,txk)],

which by (1) may be written

φ(t)=ntf(tx1,,txk)=ntφ(t).

Accordingly,

φ(t)φ(t)=nt,

which implies the integrated form

ln|φ(t)|=lntn+lnC

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

f(tx1,,txk)=tnf(x1,,xk)

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

References

  • 1 Ernst Lindelöf: Differentiali- ja integralilaskuja sen sovellutukset II.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).
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更新时间:2025/5/4 10:50:57