Introduction to the Bessel functions
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 .
Definitions of Bessel functions
The Bessel functions of the first kind and are defined as sums of the following infinite series:
These sums are convergent everywhere in the complex ‐plane. The Bessel functions of the second kind and for noninteger parameter are defined as special linear combinations of the last two functions:
In the case of integer index , the right‐hand sides of the previous expressions give removable indeterminate values of the type . In this case, the Bessel functions and are defined through the following limits:
A quick look at the Bessel functions
Here is a quick look at the graphics for the Bessel functions along the real axis.
Connections within the group of Bessel functions and with other function groups
Representations through more general functions
The Bessel functions , , , and are particular cases of more general functions: hypergeometric and Meijer G functions.
In particular, the functions and can be represented through the regularized hypergeometric functions (without any restrictions on the parameter ):
Similar formulas, but with restrictions on the parameter , represent and through the classical hypergeometric function :
The functions and can also be represented through the hypergeometric functions by the following formulas:
Similar formulas for other Bessel functions and always include restrictions on the parameter, namely :
In the case of integer , the right‐hand sides of the preceding six formulas evaluate to removable indeterminate expressions of the type , . The limit of the right‐hand sides exists and produces complicated series expansions including logarithmic and polygamma functions. These difficulties can be removed by using the generalized Meijer G function. The generalized Meijer G function allows represention of all four Bessel functions for all values of the parameter by the following simple formulas:
The classical Meijer G function is less convenient because it can lead to additional restrictions:
Representations through other Bessel functions
Each of the Bessel functions can be represented through other Bessel functions:
The best-known properties and formulas for Bessel functions
Real values for real arguments
For real values of parameter and positive argument , the values of all four Bessel functions , , , and are real.
Simple values at zero
The Bessel functions , , , and have rather simple values for the argument :
Specific values for specialized parameters
In the case of half‐integer (ν= ) all Bessel functions , , and can be expressed through sine, cosine, or exponential functions multiplied by rational and square root functions. Modulo simple factors, these are the so‐called spherical Bessel functions, for example:
The previous formulas are particular cases of the following, more general formulas:
It can be shown that for other values of the parameters , the Bessel functions cannot be represented through elementary functions. But for values equal to , and , all Bessel functions can be converted into other known special functions, the Airy functions and their derivatives, for example:
All four Bessel functions , , , and are defined for all complex values of the parameter and variable , and they are analytical functions of and over the whole complex ‐ and ‐planes.
Poles and essential singularities
For fixed , the functions , , , and have an essential singularity at . At the same time, the point is a branch point (except in the case of integer for the two functions ). For fixed integer , the functions and are entire functions of .
For fixed , the functions , , , and are entire functions of and have only one essential singular point at .
Branch points and branch cuts
For fixed noninteger , the functions , , , and have two branch points: , , and one straight line branch cut between them. For fixed integer , only the functions and have two branch points: , , and one straight line branch cut between them.
For cases where the functions , , , and have branch cuts, the branch cuts are single‐valued functions on the ‐plane cut along the interval , where they are continuous from above:
These functions have discontinuities that are described by the following formulas: