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 BesselK

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:

All Bessel functions , , , and do not have periodicity.

All Bessel functions , , , and have mirror symmetry (ignoring the interval (-∞, 0)):

The two Bessel functions of the first kind have special parity (either odd or even) in each variable:

The two Bessel functions of the second kind have special parity (either odd or even) only in their parameter:

The Bessel functions , , , and have the following series expansions (which converge in the whole complex ‐plane):

The last four formulas have restrictions that do not allow their right sides to become indeterminate expressions for integer . In such cases, evaluation of the limit from the right sides leads to much more complicated representations, for example:

Interestingly, closed‐form expressions for the truncated version of the Taylor series at the origin can be expressed through the generalized hypergeometric function and the Meijer G function, for example:

The asymptotic behavior of the Bessel functions , , , and can be described by the following formulas (which show only the main terms):

The previous formulas are valid for any direction approaching the point to infinity (z∞). In particular cases, when or , the second and fourth formulas can be simplified to the following forms:

The Bessel functions , , , and have simple integral representations through the cosine (or the hyperbolic cosine or exponential function) and power functions in the integrand:

The argument of the Bessel functions , , , and sometimes can be simplified through formulas that remove square roots from the arguments. For the Bessel functions of the second kind and with integer index , this operation is realized by special formulas that include logarithms:

If the argument of a Bessel function includes an explicit minus sign, the following formulas produce Bessel functions without the minus sign argument:

If the arguments of the Bessel functions include sums, the following formulas hold:

If arguments of the Bessel functions include products, the following formulas hold:

The Bessel functions , , , and satisfy the following recurrence identities:

The last eight identities can be generalized to the following recurrence identities with jump length :

The derivatives of all the four Bessel functions , , , and have rather simple and symmetrical representations that can be expressed through other Bessel functions with different indices:

But these derivatives can be represented in other forms, for example:

The symbolic -order derivatives have more complicated representations through the regularized hypergeometric function or generalized Meijer G function:

The Bessel functions , , , and appeared as special solutions of two linear second-order differential equations (the so‐called Bessel equation):

where and are arbitrary constants.

When is real, the functions and each have an infinite number of real zeros, all of which are simple with the possible exception of the zero :

When , the zeros of are all real. If and is not an integer, the number of complex zeros of is ; if is odd, two of these zeros lie on the imaginary axis.

If , all zeros of are real.

The function has no zeros in the region for any real .

When is real, the functions and each have an infinite number of real zeros, all of which are simple with the possible exception of the zero :