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Area Mathematics - Functions / Means

IEV ref103-02-05

en
harmonic mean value
harmonic average
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.


fr
valeur moyenne harmonique, f
moyenne harmonique, f
grandeur représentant les grandeurs d’un ensemble fini ou d’un intervalle,

  1. pour n grandeurs x 1 , x 2 , x n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadIhadaWgaaWcba qcLboacaaIXaaaleqaaOGaaiilaiaaysW7caWG4bWaaSbaaSqaaKqz GdGaaGOmaaWcbeaakiaacYcacaaMe8UaaGPaVlablAciljaaykW7ca aMc8UaaGjbVlaadIhadaWgaaWcbaGaamOBaaqabaaaaa@49C5@ , par l'inverse de la valeur moyenne de leurs inverses:

    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@ si aucune des n grandeurs n'est égale à zéro;

    X h =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadIfadaWgaaWcba qcLbqacaqGObaaleqaaOGaeyypa0tcLbuacaaIWaaaaa@3A37@ si au moins une des grandeurs est égale à zéro;

  2. pour une grandeur x fonction de la variable t, par la grandeur X h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadIfadaWgaaWcba qcLboacaqGObaaleqaaaaa@389D@ définie comme l'inverse de la valeur moyenne de l'inverse de la grandeur donnée:

    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@ si la valeur de l'intégrale est finie;

    X h =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadIfadaWgaaWcba qcLbqacaqGObaaleqaaOGaeyypa0tcLbuacaaIWaaaaa@3A37@ dans les autres cas

Note 1 à l'article: La valeur moyenne harmonique d'une grandeur périodique est généralement prise sur un intervalle d'intégration dont l’étendue est le produit de la période par un entier naturel.

Note 2 à l'article: La valeur moyenne harmonique d'une grandeur est notée en ajoutant l'indice h au symbole de la grandeur.


ar
القيمة المتوسطة التوافقية ( الهارمونية )
المتوسط الهارمونى

de
harmonischer Mittelwert, m
inverser Mittelwert, m

es
valor medio armónico

it
valore medio armonico
media armonica

ja
調和平均値
調和平均

pl
średnia harmoniczna
wartość średnia harmoniczna

pt
valor médio harmónico
média harmónica

sr
хармонијска средња вредност, ж јд
хармонијски просек, м јд

sv
harmoniskt medelvärde

zh
调和平均值

Publication date: 2017-07
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