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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, graph, properties and identities.

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

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. Graph

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

Fig. 1. Graphs of the Bessel function of the first kind y = J0(x), y = J1(x), y = J2(x) Fig. 1. Graphs 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 of the integer order Jn is supported in:

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

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]);