Bessel functions of the second kind
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besselJ(v, z) represent the Bessel functions
of the second kind:
Here Jν(z) are the 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:
besselY(2, 1 + I), besselY(0, x), besselY(v, x)
Floating point values are returned for floating-point arguments:
besselY(2, 5.0), besselY(3.2 + I, 10000.0)
Bessel functions can be expressed in terms of elementary functions if the index is an odd integer multiple of :
besselY(1/2, x), besselY(3/2, x)
besselY(5/2, x), besselY(-5/2, 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:
besselY(-3/4, -1.2), besselY(-3/4, -1.2 + I/10^10), besselY(-3/4, -1.2 - I/10^10)
diff(besselY(0, x), x, x), float(ln(3 + besselY(17, sqrt(PI))))
limit(besselY(2, x^2 + 1)*sqrt(x), x = infinity)
series(besselY(3, x), x = infinity, 3)
The Bessel functions are regular (holomorphic) functions of z throughout the z-plane cut along the negative real axis, and for fixed z ≠ 0, each is an entire (integral) function of v.
Jv (z) and Yv (z) satisfy Bessel's equation in w(v, z):
When the index v is an integer, the Bessel functions of the second kind are governed by reflection formulas: