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The same as [[Poisson integral|Poisson integral]].

A formula giving an integral representation for the solution of the [[Cauchy problem|Cauchy problem]] for the [[Wave equation|wave equation]] in  $  \mathbf R  ^ {3} $:

$$ 

\frac{\partial   ^ {2} u }{\partial  t  ^ {2} }

- a  ^ {2} \Delta u  =  0 ,\ \ 
t > 0 ,\  M = ( x , y , z ) ,
$$

$$ 
- \infty  <  x , y , z  <  \infty ,
$$

$$ 
u ( M , 0 )  =  \phi ( M) ,\  
\frac{\partial  u ( M , 0 ) }{\partial  t }
  =  \psi ( M) .
$$

This solution has the form

$$ \tag{1 }
u ( M , t )  = \ 

\frac \partial {\partial  t }

\{ t \Gamma _ {at} ( \phi ) \} + t \Gamma _ {at} ( \psi ) ,
$$

where

$$ 
\Gamma _ {at} ( \phi )  = \ 

\frac{1}{4 \pi }

\int\limits _ {S _ {at} } \phi ( P)  d \Omega
$$

is the mean value of the function  $  \phi $
on the sphere  $  S _ {at} $
in the  $  ( x , y , z ) $-
space of radius  $  a t $
and centre at the point  $  M $, 
and  $  d \Omega $
is the area element on the unit sphere. In the case of the inhomogeneous wave equation a third term is added to formula (1) (see ).

From formula (1), by the method of descent (cf. [[Descent, method of|Descent, method of]]) formulas are obtained for solving the Cauchy problem in two- (Poisson's formula) and one- (d'Alembert formula) dimensional space (see ). See also [[Kirchhoff formula|Kirchhoff formula]].

Sometimes the phrase  "Poisson formula"  is used for the integral representation of the solution to the Cauchy problem for the [[Heat equation|heat equation]] in the space  $  \mathbf R  ^ {3} $:

$$ 

\frac{\partial  u }{\partial  t }
 - a  ^ {2} \Delta u  =  0 ,\ \ 
t > 0 ,\  M = ( x , y , z ) ,
$$

$$ 
- \infty  <  x , y , z  <  \infty ,
$$

$$ 
u ( M , 0 )  =  \phi ( M) .
$$

This solution has the form

$$ \tag{2 }
u ( M , t )  = \ 

\frac{1}{( 2 \sqrt {\pi a  ^ {2} t } )  ^ {3} }

\int\limits _ {\mathbf R  ^ {3} }
\phi ( P) e ^ {| M P |  ^ {2} / 4 a  ^ {2} t }  d \sigma ( P) .
$$

Formula (2) immediately generalizes to an  $  n $-
dimensional space,  $  n \geq  1 $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.D. Poisson,   ''Mém. Acad. Sci. Paris'' , '''3'''  (1818)  pp. 121–176</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Tikhonov,   A.A. Samarskii,   "Equations of mathematical physics" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.B. Whitham,   "Linear and non-linear waves" , Wiley (Interscience)  (1974)  pp. 229ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.V. Bitsadze,   "Equations of mathematical physics" , MIR  (1980)  pp. 179  (Translated from Russian)</TD></TR></table>

{{SourceattributionEOM|Poisson formula}}
[[Category:Mathematics]]