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Article 11 — Appendix A.35Y — Bessel function of the second kindCategory. 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 LiX and scientific calculator LiXc products. See also. J_{ν} — Bessel function of the first kind, K_{ν} — modified Bessel function of the second kind. 1. DefinitionBy definition Bessel function is solution of the Besssel equation z^{2} w′′ + z w′ + (z^{2} − ν^{2}) w = 0As second order equation it has two solutions, second of which has singularity at 0 and is called Bessel function of the second kind — Y_{ν}. Parameter ν is called order of the function. First solution has no singularity at 0 and is called Bessel function of the first kind — J_{ν}. 2. GraphBessel 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 = Y_{0}(x), y = Y_{1}(x) and y = Y_{2}(x).3. IdentitiesNext order recurrence: Y_{ν+1}(z) = 2ν /z Y_{ν}(z) − Y_{ν−1}(z)Negative argument: Y_{ν}(−z) = e^{−iπν} Y_{ν}(z) + i 2 cos(πν) J_{ν}(z) = cos(πν) Y_{ν}(z) + i [2 cos(πν) J_{ν}(z) − sin(πν) Y_{ν}(z)]For the case of integer order ν=n the negative argument identity can be simplified down to: Y_{n}(−z) = (−1)^{n} Y_{n}(z) + i (−1)^{n} 2 J_{n}(z)and for the case of halfinteger order ν=n+1/2 the identity can be simplified down to: Y_{n+1/2}(−z) = i (−1)^{n+1} Y_{n+1/2}(z)Reflection — negative order: Y_{−ν}(z) = cos(πν) Y_{ν}(z) + sin(πν) J_{ν}(z)For the case of integer order ν=n the reflection identity can be simplified down to: Y_{−n}(z) = (−1)^{n} Y_{n}(z)and for the case of halfinteger order ν=n+1/2 the identity can be simplified down to: Y_{−n−1/2}(z) = (−1)^{n} J_{n+1/2}(z)4. SupportBessel function of the second kind of the integer order Y_{n} is supported in: Bessel function of the second kind of the real (fractional) order of the complex argument Y_{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 second kind of the 0 order of the number:
or:
To calculate Bessel function of the second kind of the 1.2 order of the current result:
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To calculate Bessel function of the second kind of the 2.5 order of the number z in memory:
or:



