differential equations - helmholtz zero in R^3

If $u$ is a solution to the equation $triangle u +k^2 u=0$ in a 3D domain $Omega$, then
for any $xinOmega$ and any $r>0$ such that ${yinmathbb R^3: |x-y|leq r }subsetOmega$, we have
$$u(x)=frac {p(r)}{4pi r^2}int_{|x-y|=r} u(y)dS_y,qquadqquadqquad(1)$$
where
$$p(r)=frac{rk}{sin rk}.$$

Formula (1) is an analogue of the mean value theorem for harmonic functions (in the case of spherical means).

Edit added: relation (1) is valid for all $r_1leq r$. If we multiply it by $4pi r^2/p(r)$ and integrate between $0$ and $r$ we will obtain that
$$u(x)=frac{k^3}{4pi(sin rk-rkcos rk)}int_{|x-y|leq r} u(y)dy.$$
The latter formula generalizes the property that the value of a harmonic function at $xinOmega$ is equal to function's average value over a ball with the center at $x$.

A short derivation of formula (1) can be found in chapter IV of Methods of Mathematical Physics (Vol. 2)
by Courant and Hilbert (or see Harald's comment below).