OffAxis
Axial Field of a Finite Solenoid 

The magnetic field in the central region of a solenoid can be approximated as a simple polynomial involving Legendre polynomials. Montgomery and Terrell state that "out to within a few percent of the inner radius of the coil, the field can be found quite accurately with only a few ... terms."  
Solenoid in cross section view. 

This
polynomial can be written as:
where, B_{O} is the central field strength of the solenoid, and the first four even numbered error coefficients, e _{n}, are: where: , , , , , r_{1}, r_{2}, and l are inner, and outer coil radii, and coil length, respectively. d is the distance from the coil center to the field measurement point, q is the angle between the measurement point position vector and the coil axis, and the corresponding Legendre polynomials, r_{n}, are: where: 

But What About OffAxis Radial Field Components?  According
to this
reference, there
is a corresponding series for obtaining the radial
component of the magnetic field in the central region of
a solenoid, which looks like this:
Where r ^{l}_{n} are derivatives of the even Legendre polynomials, such as: where It looks wonderful, except for the fact that the computed radial fields are wrong. I don't know why they're wrong, since the math is a little over my head. If you know the answer, please drop me a line. 

The
formulas on this page are adapted from: SOME USEFUL INFORMATION FOR THE DESIGN OF AIRCORE SOLENOIDS by D.Bruce Montgomery and J. Terrell., published November, 1961, under Air Force Contract AF19(604)7344. Montgomery and Terrell, in turn, credit: M.W. Garrett, J. Appl. Phys. 22, 9, Sept. 1951. 

