The best-known properties and formulas for Bessel functions
For real values of parameter and positive argument , the values of all four Bessel functions , , , and are real.
The Bessel functions , , , and have rather simple values for the argument :
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.
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 .
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: