Modified Bessel functions of the second kind
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besselK(v, z) represents the modified Bessel
functions of the second kind:
Here Iν(z) are the modified Bessel functions of the first kind:
The Bessel functions are defined for complex arguments v and z.
A floating-point value is returned if either of the arguments is a floating-point number and the other argument is numerical. For most exact arguments the Bessel functions return an unevaluated function call. Special values at index v = 0 and/or argument z = 0 are implemented. Explicit symbolic expressions are returned, when the index v is a half integer. See Example 2.
For nonnegative integer indices v some of the Bessel functions have a branch cut along the negative real axis. A jump occurs when crossing this cut. See Example 3.
When called with floating-point arguments, these functions are
sensitive to the environment variable
DIGITS which determines
the numerical working precision.
Unevaluated calls are returned for exact or symbolic arguments:
besselK(2, 1 + I), besselK(0, x), besselK(v, x)
Floating point values are returned for floating-point arguments:
besselK(2, 5.0), besselK(3.2 + I, 10000.0)
Bessel functions can be expressed in terms of elementary functions if the index is an odd integer multiple of :
besselK(1/2, x), besselK(3/2, x)
besselK(7/2, x), besselK(-7/2, x)
The negative real axis is a branch cut of the Bessel functions for non-integer indices v. A jump occurs when crossing this cut:
besselK(-3/4, -1.2), besselK(-3/4, -1.2 + I/10^10), besselK(-3/4, -1.2 - I/10^10)
diff(besselK(0, x), x, x), float(ln(3 + besselK(17, sqrt(PI))))
limit(besselK(2, x^2 + 1)*sqrt(x), x = infinity)
series(besselK(3, x), x = infinity, 3)
The modified Bessel functions Iv(z) and Kv(z) satisfy the modified Bessel equation:
When the index v is an integer, the modified Bessel functions of the second kind are governed by reflection formula: