Cylindric function

Cylindric function (or cylinder finction or cylindrical function) is class of special functions \(f\) satisfying equation
 * \(\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ z^2 f''(z) + z f'(z) + (z^2-\nu^2) f(z)=0\)

for some fixed value of \(\nu\) . Often, it is assumed that \(\nu\) is complex, of real or integer.

This article competes with Bessel function referring to the same topic; and, for summer of 2012, the Bessel function seems to be more advanced. Tho articles on the same subject are kept in TORI in order to elaborate the best terminology: should the term Bessel function refer to only BesselJ, or also to other solutions of the similar equation and include the Neumann function BesselY?

The function with modified argument (multiplied by \(\mathrm i\) is called Modified cylindric function. The most used are BesseLi and BesselK. In the name BesseLi, the capitalization of the last two letters is swapped because many font designers are lazy and did not provide different images for characlters I, l and | ; the notation suggested allows to avoid confusions: L and i are easier to distinguish than l and I.

Several cylindric functions have special names: BessleJ = J, BesselY=Y, BesselH=H, or, correspondently, Bessel function, Neumann function and Hankel function. Short names with single letters are also used; the corresponding value of parameter \(\nu\) in equation (1) is indicated as subscript. These functions are related with equation
 * \(\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ H_\nu(z)=J_\nu(z)+\mathrm i Y_\nu(z)\)

The term Bessel function is often applied also to all cylindric functions . In this sense, for example, BesselY is also Bessel function, although it has more specific name Neumann function. In order to keep the name Bessel function for BessleJ, the term Cylindric function is used in this article. The modified cylindric functions also are considered as cylindric functions.

Equatoin (1) and cylindric functions often appear in the consideration of waves in a system with circular symmetry; in particular, functions BesselJ0 and BesselK0 are used to build the principal mode of an idealized circular waveguide.

Keywords
Bessel function, BesselJ0, BesselJ1, BesselY0, BesselK0