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Bessel functions of the second kind

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besselY(v, z)


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.

Environment Interactions

When called with floating-point arguments, these functions are sensitive to the environment variable DIGITS which determines the numerical working precision.


Example 1

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)

Example 2

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)

Example 3

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)

Example 4

The functions diff, float, limit, and series handle expressions involving the Bessel functions:

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)


v, z

arithmetical expressions

Return Values

Arithmetical expression.

Overloaded By



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:


See Also

MuPAD Functions

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