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A transformation of functions defined in domains of a Euclidean space  $  \mathbf R  ^ {n} $, 
$  n \geq  3 $, 
under which harmonic functions are transformed to harmonic functions. It was obtained by W. Thomson (Lord Kelvin, [[#References|[1]]]).

If  $  u $
is a [[Harmonic function|harmonic function]] in a domain  $  D \subset  \mathbf R  ^ {n} $, 
then its Kelvin transform is the function

$$ 
v ( y)  = \ 
\left (

\frac{R}{| y | }

\right )
 ^ {n-2}
u \left (

\frac{R  ^ {2} }{| y |  ^ {2} }
 y
\right ) ,\ \ 
v ( \infty )  =  0 ,
$$

which is harmonic in the domain  $  D  ^ {*} $
obtained from  $  D $
by inversion in the sphere  $  S _ {R} = \{ {x } : {| x | = R } \} $, 
that is, by the mapping of  $  \mathbf R  ^ {n} $
defined by

$$ 
x  \rightarrow  y  = \ 

\frac{R  ^ {2} }{| x |  ^ {2} }
 x ,\ \ 
0  \rightarrow  \infty ,
$$

where

$$ 
x  =  ( x _ {1} \dots x _ {n} ) ,\ \ 
| x |  =  ( x _ {1}  ^ {2} + \dots + x _ {n}  ^ {2} )  ^ {1/2} .
$$

Under the inversion, the point at infinity  $  \infty $
of the [[Aleksandrov compactification|Aleksandrov compactification]]  $  \overline{ {\mathbf R  ^ {n} }}\; $
is taken to the origin  $  0 $
and vice versa. Under the Kelvin transformation, harmonic functions  $  u $
in domains  $  D $
containing  $  \infty $
that are regular at  $  \infty $, 
that is, are such that  $  \lim\limits _ {| x | \rightarrow \infty }   u ( x) = 0 $, 
are transformed to harmonic functions  $  v $
in bounded domains  $  D  ^ {*} $
containing the origin  $  0 $, 
moreover,  $  v ( 0) = 0 $. 
Because of this property, the Kelvin transformation enables one to reduce exterior problems in potential theory to interior ones and vice versa (see [[#References|[2]]], [[#References|[3]]]).

Apart from under Kelvin transformation, harmonicity of functions in  $  \mathbf R  ^ {n} $, 
$  n \geq  3 $, 
is preserved under analytic transformations of the form  $  v ( y) = \phi ( y) u ( \psi ( y) ) $
only in the case when  $  \phi ( y) \equiv 1 $
and  $  \psi $
is a [[Homothety|homothety]], a translation or a symmetry with respect to a plane; for  $  n = 2 $
the large class of conformal mappings  $  \psi $
has this property.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Thomson,   "Extraits de deux letters adressées à M. Liouville"  ''J. Math. Pures Appl.'' , '''12'''  (1847)  pp. 256–264</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  pp. Chapt. 5  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR></table>

====Comments====
These results hold for  $  n = 2 $
as well. In this case, harmonicity of  $  u $
at infinity corresponds to boundedness of  $  u $
at  $  0 $. 
See, e.g., [[#References|[a1]]] or [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.L. Helms,   "Introduction to potential theory" , Wiley  (1969)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Wermer,   "Potential theory" , ''Lect. notes in math.'' , '''408''' , Springer  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O.D. Kellogg,   "Foundations of potential theory" , Dover, reprint  (1954)  (Re-issue: Springer, 1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Doob,   "Classical potential theory and its probabilistic counterpart" , Springer  (1984)  pp. 390</TD></TR></table>