In physics, the Green's function (or fundamental solution) for the Laplacian (or Laplace operator) in three variables is used to describe the response of a particular type of physical system to a point source.
to a general system of this type can be written as an integral over a distribution of source given by
One physical system of this type is a charge distribution in electrostatics.
In such a system, the electric field is expressed as the negative gradient of the electric potential, and Gauss's law in differential form applies:
Combining these expressions gives us Poisson's equation:
will give the response of the system to the point charge
The free-space Green's function for the Laplace operator in three variables is given in terms of the reciprocal distance between two points and is known as the "Newton kernel" or "Newtonian potential".
are the standard Cartesian coordinates in a three-dimensional space, and
The algebraic expression of the Green's function for the three-variable Laplace operator, apart from the constant term
Many expansion formulas are possible, given the algebraic expression for the Green's function.
The free-space circular cylindrical Green's function (see below) is given in terms of the reciprocal distance between two points.
The expression is derived in Jackson's Classical Electrodynamics.
Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation.
In the case of a boundary put at infinity with the boundary condition setting the solution to zero at infinity, then one has an infinite-extent Green's function.
For the three-variable Laplace operator, one can for instance expand it in the rotationally invariant coordinate systems which allow separation of variables.
is the odd-half-integer degree Legendre function of the second kind, which is a toroidal harmonic.
Using one of the Whipple formulae for toroidal harmonics we can obtain an alternative form of the Green's function
This formula was used in 1999 for astrophysical applications in a paper published in The Astrophysical Journal, published by Howard Cohl and Joel Tohline.
For instance, a paper written in the Journal of Applied Physics in volume 18, 1947 pages 562-577 shows N.G.
In fact, virtually all the mathematics found in recent papers was already done by Chester Snow.
This is found in his book titled Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, National Bureau of Standards Applied Mathematics Series 19, 1952.
The article by Chester Snow, "Magnetic Fields of Cylindrical Coils and Annular Coils" (National Bureau of Standards, Applied Mathematical Series 38, December 30, 1953), clearly shows the relationship between the free-space Green's function in cylindrical coordinates and the Q-function expression.
Likewise, see another one of Snow's pieces of work, titled "Formulas for Computing Capacitance and Inductance", National Bureau of Standards Circular 544, September 10, 1954, pp 13–41.
Indeed, not much has been published recently on the subject of toroidal functions and their applications in engineering or physics.
One application was published; the article was written by J.P. Selvaggi, S. Salon, O. Kwon, and M.V.K.
They have solved numerous problems which exhibit circular cylindrical symmetry employing the toroidal functions.
The above expressions for the Green's function for the three-variable Laplace operator are examples of single summation expressions for this Green's function.
Examples of these can be seen to exist in rotational cylindrical coordinates as an integral Laplace transform in the difference of vertical heights whose kernel is given in terms of the order-zero Bessel function of the first kind as
Similarly, the Green's function for the three-variable Laplace equation can be given as a Fourier integral cosine transform of the difference of vertical heights whose kernel is given in terms of the order-zero modified Bessel function of the second kind as
Green's function expansions exist in all of the rotationally invariant coordinate systems which are known to yield solutions to the three-variable Laplace equation through the separation of variables technique.