●
H o m e
●
R a n d o m
●
N e a r b y
●
L o g i n
●
S e t t i n g s
●
D o n a t e
●
A b o u t W i k i p e d i a
●
D i s c l a i m e r s
Search
I d e n t r i c m e a n ●
A r t i c l e ●
T a l k ●
L a n g u a g e
●
●
The identric mean of two positive real numbers x y [1 ]
I (
x ,
y )
=
1 e
⋅
lim
(
ξ
,
η
)
→
(
x ,
y )
ξ
ξ
η
η
ξ
−
η
=
lim
(
ξ
,
η
)
→
(
x ,
y )
exp
(
ξ
⋅
ln
ξ
−
η
⋅
ln
η
ξ
−
η
−
1
)
=
{
x
if
x =
y
1 e
x
x
y
y
x −
y
else
{\displaystyle {\begin{aligned}I(x,y)&={\frac {1}{e}}\cdot \lim _{(\xi ,\eta )\to (x,y)}{\sqrt[{\xi -\eta }]{\frac {\xi ^{\xi }}{\eta ^{\eta }}}}\\[8pt]&=\lim _{(\xi ,\eta )\to (x,y)}\exp \left({\frac {\xi \cdot \ln \xi -\eta \cdot \ln \eta }{\xi -\eta }}-1\right)\\[8pt]&={\begin{cases}x&{\text{if }}x=y\\[8pt]{\frac {1}{e}}{\sqrt[{x-y}]{\frac {x^{x}}{y^{y}}}}&{\text{else}}\end{cases}}\end{aligned}}}
It can be derived from the mean value theorem by considering the secant of the graph of the function
x ↦
x ⋅
ln
x
{\displaystyle x\mapsto x\cdot \ln x}
mean value theorem for divided differences . The identric mean is a special case of the Stolarsky mean .
See also
edit
References
edit
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Identric_mean&oldid=1178186609 "
L a s t e d i t e d o n 2 O c t o b e r 2 0 2 3 , a t 0 2 : 2 2 L a n g u a g e s
● F r a n ç a i s ● ע ב ר י ת ● ភ ា ស ា ខ ្ ម ែ រ
● T h i s p a g e w a s l a s t e d i t e d o n 2 O c t o b e r 2 0 2 3 , a t 0 2 : 2 2 ( U T C ) . ● C o n t e n t i s a v a i l a b l e u n d e r C C B Y - S A 4 . 0 u n l e s s o t h e r w i s e n o t e d . ● P r i v a c y p o l i c y ● A b o u t W i k i p e d i a ● D i s c l a i m e r s ● C o n t a c t W i k i p e d i a ● C o d e o f C o n d u c t ● D e v e l o p e r s ● S t a t i s t i c s ● C o o k i e s t a t e m e n t ● T e r m s o f U s e ● D e s k t o p
![{\displaystyle {\begin{aligned}I(x,y)&={\frac {1}{e}}\cdot \lim _{(\xi ,\eta )\to (x,y)}{\sqrt[{\xi -\eta }]{\frac {\xi ^{\xi }}{\eta ^{\eta }}}}\\[8pt]&=\lim _{(\xi ,\eta )\to (x,y)}\exp \left({\frac {\xi \cdot \ln \xi -\eta \cdot \ln \eta }{\xi -\eta }}-1\right)\\[8pt]&={\begin{cases}x&{\text{if }}x=y\\[8pt]{\frac {1}{e}}{\sqrt[{x-y}]{\frac {x^{x}}{y^{y}}}}&{\text{else}}\end{cases}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/173d3e35b69f2dee0f1b883c42b1c521b57051f6)
