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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, graph, properties and identities.

References. This article is a part of scientific calculator Li-X and scientific calculator Li-Xc products.

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

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. Graph

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

Fig. 1. Graphs of the Bessel functions of the second kind y = Y0(x), y = Y1(x), y = Y2(x) Fig. 1. Graphs 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) = eiπν 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 of the integer order Yn is supported in:

Bessel function of the second kind of the real (fractional) order of the complex argument Yν is supported in:

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]);