Monochromatic wave
The integral has the following form for a monochromatic wave:[2][3][4]
![{\displaystyle U(\mathbf {r} )={\frac {1}{4\pi }}\int _{S}\left[U{\frac {\partial }{\partial {\hat {\mathbf {n} }}}}\left({\frac {e^{iks}}{s}}\right)-{\frac {e^{iks}}{s}}{\frac {\partial U}{\partial {\hat {\mathbf {n} }}}}\right]dS,}](//wikimedia.org/api/rest_v1/media/math/render/svg/30cb22bc92192e0048753d8432ade8ce9886f120)
where the integration is performed over an arbitrary closed surface S enclosing the observation point
,
in
is the wavenumber,
in
is the distance from an (infinitesimally small) integral surface element to the point
,
is the spatial part of the solution of the homogeneous scalar wave equation (i.e.,
as the homogeneous scalar wave equation solution),
is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and
denotes differentiation along the surface normal (i.e., a normal derivative) i.e.,
for a scalar field
. Note that the surface normal is inward, i.e., it is toward the inside of the enclosed volume, in this integral; if the more usual outer-pointing normal is used, the integral will have the opposite sign.
This integral can be written in a more familiar form

where
.[3]
Non-monochromatic wave
A more general form can be derived for non-monochromatic waves. The complex amplitude of the wave can be represented by a Fourier integral of the form

where, by Fourier inversion, we have

The integral theorem (above) is applied to each Fourier component
, and the following expression is obtained:[2]
![{\displaystyle V(r,t)={\frac {1}{4\pi }}\int _{S}\left\{[V]{\frac {\partial }{\partial n}}\left({\frac {1}{s}}\right)-{\frac {1}{cs}}{\frac {\partial s}{\partial n}}\left[{\frac {\partial V}{\partial t}}\right]-{\frac {1}{s}}\left[{\frac {\partial V}{\partial n}}\right]\right\}dS,}](//wikimedia.org/api/rest_v1/media/math/render/svg/b2a820854da5a2358694a039272f017bd9b25963)
where the square brackets on V terms denote retarded values, i.e. the values at time t − s/c.
Kirchhoff showed that the above equation can be approximated to a simpler form in many cases, known as the Kirchhoff, or Fresnel–Kirchhoff diffraction formula, which is equivalent to the Huygens–Fresnel equation, except that it provides the inclination factor, which is not defined in the Huygens–Fresnel equation. The diffraction integral can be applied to a wide range of problems in optics.
Integral derivation
Here, the derivation of the Kirchhoff's integral theorem is introduced. First, the Green's second identity as the following is used.
where the integral surface normal unit vector
here is toward the volume
closed by an integral surface
. Scalar field functions
and
are set as solutions of the Helmholtz equation,
where
is the wavenumber (
is the wavelength), that gives the spatial part of a complex-valued monochromatic (single frequency in time) wave expression. (The product between the spatial part and the temporal part of the wave expression is a solution of the scalar wave equation.) Then, the volume part of the Green's second identity is zero, so only the surface integral is remained as
Now
is set as the solution of the Helmholtz equation to find and
is set as the spatial part of a complex-valued monochromatic spherical wave
where
is the distance from an observation point
in the closed volume
. Since there is a singularity for
at
where
(the value of
not defined at
), the integral surface must not include
. (Otherwise, the zero volume integral above is not justified.) A suggested integral surface is an inner sphere
centered at
with the radius of
and an outer arbitrary closed surface
.
Then the surface integral becomes
For the integral on the inner sphere
,
and by introducing the solid angle
in
,
due to
. (The spherical coordinate system which origin is at
can be used to derive this equality.)
By shrinking the sphere
toward the zero radius (but never touching
to avoid the singularity),
and the first and last terms in the
surface integral becomes zero, so the integral becomes
. As a result, denoting
, the location of
, and
by
, the position vector
, and
respectively,
