Cylindrical multipole moments

Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as formula_1. Such potentials arise in the electric potential of long line charges, and the analogous sources for the magnetic potential and gravitational potential.
For clarity, we illustrate the expansion for a single line charge, then generalize to an arbitrary distribution of line charges. Through this article, the primed coordinates such 
as formula_2 refer to the position of the line charge(s), whereas the unprimed coordinates such as formula_3 refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector formula_4 has coordinates formula_5
where formula_6 is the radius from the formula_7 axis, formula_8 is the azimuthal angle and formula_7 is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the formula_7 axis.
Cylindrical multipole moments of a line charge.
The electric potential of a line charge formula_11 located at formula_2 is given by
where formula_14 is the shortest distance between the line charge and the observation point. 
By symmetry, the electric potential of an infinite linecharge has no formula_7-dependence. The line charge formula_11 is the charge per unit length in the 
formula_7-direction, and has units of (charge/length). If the radius formula_6 of the observation point is greater than the radius formula_19 of the line charge, we may factor out formula_20
and expand the logarithms in powers of formula_22
which may be written as
where the multipole moments are defined as <br>
formula_25<br>
formula_26<br>
and<br>
formula_27
Conversely, if the radius formula_6 of the observation point is less than the radius formula_19 of the line charge, we may factor out formula_30 and expand the logarithms in powers of formula_31
which may be written as
where the interior multipole moments are defined as<br>
formula_34<br>
formula_35<br>
and<br>
formula_36
General cylindrical multipole moments.
The generalization to an arbitrary distribution of line charges formula_37 is straightforward. The functional form is the same 
and the moments can be written
Note that the formula_37 represents the line charge per unit area in the formula_43 plane.
Interior cylindrical multipole moments.
Similarly, the interior cylindrical multipole expansion has the functional form
where the moments are defined
Interaction energies of cylindrical multipoles.
A simple formula for the interaction energy of cylindrical multipoles (charge density 1) with a second charge density can be derived. Let formula_48 be the second charge density, and define formula_49 as its integral over z
The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles
If the cylindrical multipoles are exterior, this equation becomes
where formula_55, formula_56 and formula_57 are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form
where formula_59 and formula_60 are the interior cylindrical multipoles of the second charge density. 
The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles
where formula_62 and formula_63 are the interior cylindrical multipole moments of charge distribution 1, and formula_64 and formula_65 are the 
exterior cylindrical multipoles of the second charge density.
As an example, these formulae could be used to determine the interaction energy of a small protein in the electrostatic field of a double-stranded DNA molecule; the latter is relatively straight and bears a constant linear charge density due to the phosphate 
groups of its backbone.