The Art of Interface

Article 11 — Appendix A.34

J — Bessel function of the first kind

Category. Mathematics.

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

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

See also. Yν — Bessel function of the second kind, Iν — modified Bessel function of the first kind.

1. Definition

By definition Bessel function is solution of the Bessel equation

z2 w′′ + z w′ + (z2 − ν2) w = 0

As second order equation it has two solutions, first of which has no singularity at 0 and is called Bessel function of the first kind — Jν. Parameter ν is called order of the function.

Second solution has singularity at 0 and is called Bessel function of the second kind — Yν.

2. Plot

Bessel functions of the first kind defined everywhere on the real axis, so, their domain is (−∞, +∞). Plots of the first three representatives of the first kind Bessel function family depicted below — fig. 1.

Fig. 1. Plots of the Bessel function of the first kind y = J0(x), y = J1(x), y = J2(x) Fig. 1. Plots of the Bessel functions of the first kind y = J0(x), y = J1(x) and y = J2(x).

3. Identities

Next order recurrence:

Jν+1(x) = 2ν /x Jν(x) − Jν−1(x)

Negative argument:

Jν(−z) = eiπν Jν(z) = [cos(πν) + i sin(πν)] Jν(z)

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

Jn(−z) = (−1)n Jn(z)

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

Jn+1/2(−z) = i (−1)n Jn+1/2(z)

Reflection — negative order:

J−ν(z) = cos(πν) Jν(z) − sin(πν) Yν(z)

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

J−n(z) = (−1)n Jn(z)

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

J−n−1/2(z) = (−1)n+1 Yn+1/2(z)

4. Support

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

5. Interface

Bessel function call looks like

J(order, argument);

or

BesselJ(order, argument);

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

6. How to use

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

J(0, -1.5);

or:

BesselJ(0, -1.5);

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

J(1.2, rslt);

or:

BesselJ(1.2, rslt);

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

J(2.5, mem[z]);

or:

BesselJ(2.5, mem[z]);