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Article 11 — Appendix A.34J — Bessel function of the first kindCategory. 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 LiX and scientific calculator LiXc products. See also. Y_{ν} — Bessel function of the second kind, I_{ν} — modified Bessel function of the first kind. 1. DefinitionBy definition Bessel function is solution of the Bessel equation z^{2} w′′ + z w′ + (z^{2} − ν^{2}) w = 0As second order equation it has two solutions, first of which has no singularity at 0 and is called Bessel function of the first kind — J_{ν}. Parameter ν is called order of the function. Second solution has singularity at 0 and is called Bessel function of the second kind — Y_{ν}. 2. GraphBessel 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 functions of the first kind y = J_{0}(x), y = J_{1}(x) and y = J_{2}(x).3. IdentitiesNext order recurrence: J_{ν+1}(x) = 2ν /x J_{ν}(x) − J_{ν−1}(x)Negative argument: J_{ν}(−z) = e^{iπν} J_{ν}(z) = [cos(πν) + i sin(πν)] J_{ν}(z)For the case of integer order ν=n the negative argument identity can be simplified down to: J_{n}(−z) = (−1)^{n} J_{n}(z)and for the case of halfinteger order ν=n+1/2 the identity can be simplified down to: J_{n+1/2}(−z) = i (−1)^{n} J_{n+1/2}(z)Reflection — negative order: J_{−ν}(z) = cos(πν) J_{ν}(z) − sin(πν) Y_{ν}(z)For the case of integer order ν=n reflection identity can be simplified down to: J_{−n}(z) = (−1)^{n} J_{n}(z)and for case of halfinteger order ν=n+1/2 the identity can be simplified down to: J_{−n−1/2}(z) = (−1)^{n+1} Y_{n+1/2}(z)4. SupportBessel function of the first kind of the integer order J_{n} is supported in: Bessel function of the first kind of the real (fractional) order of the complex argument J_{ν} is supported in: 5. InterfaceBessel function call looks like
or
where order is the function real order, and argument — function argument. 6. How to useTo calculate Bessel function of the first kind of the 0 order of the number:
or:
To calculate Bessel function of the first kind of the 1.2 order of the current result:
or:
To calculate Bessel function of the first kind of the 2.5 order of the number z in memory:
or:



