The Art of Interface

Article 11 — Appendix A.35

Y — Bessel function of the second kind

Category. Mathematics.

Abstract. Bessel function of the second kind of the real (fractional) order: definition, plot, properties and identities.

Reference. This article is a part of Librow professional formula calculator project.

See also. Jν — Bessel function of the first kind, Kν — modified Bessel function of the second kind.

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1. Definition

By definition Bessel function is solution of the Besssel equation

z2 w′′ + z w′ + (z2 − ν2) w = 0

As second order equation it has two solutions, second of which has singularity at 0 and is called Bessel function of the second kind — Yν. Parameter ν is called order of the function.

First solution has no singularity at 0 and is called Bessel function of the first kind — Jν.

2. Plot

Bessel functions of the second kind defined on positive part of the real axis, at 0 functions have singularity, so, their domain is (0, +∞). Plots of the first three representatives of the second kind Bessel function family depicted below — fig. 1.

Fig. 1. Plot of the Bessel functions of the second kind y = Y0(x), y = Y1(x), y = Y2(x). Fig. 1. Plot of the Bessel functions of the second kind y = Y0(x), y = Y1(x) and y = Y2(x).

3. Identities

Next order recurrence:

Yν+1(z) = 2ν /z Yν(z) − Yν−1(z)

Negative argument:

Yν(−z) = e−iπν Yν(z) + i 2 cos(πν) Jν(z) = cos(πν) Yν(z) + i [2 cos(πν) Jν(z) − sin(πν) Yν(z)]

For the case of integer order ν=n the negative argument identity can be simplified down to:

Yn(−z) = (−1)n Yn(z) + i (−1)n 2 Jn(z)

and for the case of half-integer order ν=n+1/2 the identity can be simplified down to:

Yn+1/2(−z) = i (−1)n+1 Yn+1/2(z)

Reflection — negative order:

Y−ν(z) = cos(πν) Yν(z) + sin(πν) Jν(z)

For the case of integer order ν=n the reflection identity can be simplified down to:

Y−n(z) = (−1)n Yn(z)

and for the case of half-integer order ν=n+1/2 the identity can be simplified down to:

Y−n−1/2(z) = (−1)n Jn+1/2(z)

4. Support

Bessel function of the second kind Yν of the real (fractional) order and complex argument is supported by professional version of the Librow calculator.

5. Interface

Bessel function call looks like

Y(order, argument);

or

BesselY(order, argument);

where order is the function real order, and argument — function argument.

6. How to use

To calculate Bessel function of the second kind of the 0 order of the number:

Y(0, 1.5);

or:

BesselY(0, 1.5);

To calculate Bessel function of the second kind of the 1.2 order of the current result:

Y(1.2, rslt);

or:

BesselY(1.2, rslt);

To calculate Bessel function of the second kind of the 2.5 order of the number z in memory:

Y(2.5, mem[z]);

or:

BesselY(2.5, mem[z]);