General

The Bessel functions have been known since the 18th century when mathematicians and scientists started to describe physical processes through differential equations. Many different‐looking processes satisfy the same partial differential equations. These equations were named Laplace, d`Alembert (wave), Poisson, Helmholtz, and heat (diffusion) equations. Different methods were used to investigate these equations. The most powerful was the separation of variables method, which in polar coordinates often leads to ordinary differential equations of special structure:

This equation with concrete values of the parameter appeared in the articles by F. W. Bessel (1816, 1824) who built two partial solutions and of the previous equation in the form of series:

Substituting the series into the differential equation produces the following solutions:

O. Schlömilch (1857) used the name Bessel functions for these solutions, E. Lommel (1868) considered as an arbitrary real parameter, and H. Hankel (1869) considered complex values for . The two independent solutions of the differential equation were notated as and .

For integer index , the functions and coincide or have different signs. In such cases, the second linear independent solution of the previous differential equation was introduced by C. G. Neumann (1867) as the limit case of the following special linear combination of the functions and :

J. Watson (1867) introduced the notation for this function. Other authors (H. Hankel (1869), H. Weber (1873), and L. Schläfli (1875)) investigated its properties. In particular, the general solution of the previous differential equation for all values of the parameter can be presented by the formula:

where and are arbitrary complex constants.

In a similar way, A. B. Basset (1888) and H. M. MacDonald (1899) introduced the modified Bessel functions and , which satisfy the modified Bessel differential equation:

The first differential equation can be converted into the last one by changing the independent variable to .