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IEVref:103-02-05ID:
Language:enStatus: Standard
Term: harmonic mean value
Synonym1: harmonic average
[Preferred]
Synonym2:
Synonym3:
Symbol:
Definition: quantity representing the quantities in a finite set or in an interval,

  1. for n quantities x 1 , x 2 , x n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadIhadaWgaaWcba qcLboacaaIXaaaleqaaOGaaiilaiaaysW7caWG4bWaaSbaaSqaaKqz GdGaaGOmaaWcbeaakiaacYcacaaMe8UaaGPaVlablAciljaaykW7ca aMc8UaaGjbVlaadIhadaWgaaWcbaGaamOBaaqabaaaaa@49C5@ , by the reciprocal of the mean value of their reciprocals:

    1 X h = 1 n ( 1 x 1 + 1 x 2 +...+ 1 x n ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaamaalaaabaqcLbuaca aIXaaakeaacaWGybWaaSbaaSqaaKqzGdGaaeiAaaWcbeaaaaGccqGH 9aqpdaWcaaqaaKqzafGaaGymaaGcbaGaamOBaaaadaqadaqaamaala aabaqcLbuacaaIXaaakeaacaWG4bWaaSbaaSqaaKqzGdGaaGymaaWc beaaaaGccqGHRaWkdaWcaaqaaKqzafGaaGymaaGcbaGaamiEamaaBa aaleaajug4aiaaikdaaSqabaaaaOGaey4kaSIaaiOlaiaac6cacaGG UaGaey4kaSYaaSaaaeaajugqbiaaigdaaOqaaiaadIhadaWgaaWcba GaamOBaaqabaaaaaGccaGLOaGaayzkaaaaaa@512E@ if none of the n quantities is equal to zero;

    X h =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadIfadaWgaaWcbaqcLbqacaqGObaaleqaaOGaeyypa0tcLbuacaaIWaaaaa@3A34@ if at least one quantity is equal to zero;

  2. for a quantity x depending on a variable t, by the quantity X h MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadIfadaWgaaWcbaqcLboacaqGObaaleqaaaaa@389A@ defined by the reciprocal of the mean value of the reciprocal of the given quantity:

    1 X h = 1 T 0 T 1 x(t) dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaamaalaaabaqcLbuaca aIXaaakeaacaWGybWaaSbaaSqaaKqzGdGaaeiAaaWcbeaaaaGccqGH 9aqpdaWcaaqaaKqzafGaaGymaaGcbaGaamivaaaadaWdXaqaamaala aabaqcLbuacaaIXaaakeaacaWG4bGaaiikaiaadshacaGGPaaaaaWc baGaaGjcVNqzGdGaaGimaaWcbaGaaGjcVlaadsfaa0Gaey4kIipaju gqbiaacsgakiaadshaaaa@4D32@ if the value of the integral is finite;

    X h =0 MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadIfadaWgaaWcbaqcLbqacaqGObaaleqaaOGaeyypa0tcLbuacaaIWaaaaa@3A34@ in other cases

Note 1 to entry: The harmonic mean value of a periodic quantity is usually taken over an integration interval the range of which is the period multiplied by a natural number.

Note 2 to entry: The harmonic mean value of a quantity is denoted by adding the subscript h to the symbol of the quantity.


Publication date:2017-07
Source:
Replaces:103-02-05:2009-12
Internal notes:2017-08-25: Added <p> tag before list. LMO
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