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Article 11 — Appendix A.37K — modified Bessel function of the second kindCategory. Mathematics. Abstract. Modified Bessel function of the second kind of the real (fractional) order: definition, graph, properties and identities. Reference. This article is a part of scientific calculator LiXc product. See also. I_{ν} — modified Bessel function of the first kind, Y_{ν} — Bessel function of the second kind. 1. DefinitionBy definition modified Bessel function is solution of the modified Bessel 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 modified Bessel function of the second kind — K_{ν}. Parameter ν is called order of the function. First solution has no singularity at 0 and is called modified Bessel function of the first kind — I_{ν}. 2. GraphModified Bessel functions of the second kind defined everywhere on the real axis, at 0 functions have singularity, so, their domain is (−∞, 0)∪(0, +∞). Graphs of the first three representatives of the second kind modified Bessel function family depicted below — fig. 1. Fig. 1. Graphs of the modified Bessel functions of the second kind y = K_{0}(x), y = K_{1}(x) and y = K_{2}(x).3. IdentitiesConnection to the ordinary Bessel functions: K_{ν}(z) = π /2 i^{ν+1}[J_{ν}(iz) + iY_{ν}(iz)]Connection to the modified Bessel function of the first kind: K_{ν}(z) = π / [2 sin(πν)] [I_{−ν}(z) − I_{ν}(z)]Next order recurrence: K_{ν+1}(z) = 2ν /z K_{ν}(z) + K_{ν−1}(z)Negative argument: K_{ν}(−z) = e^{−iπν} K_{ν}(z) − i π I_{ν}(z) = cos(πν) K_{ν}(z) − i [sin(πν) K_{ν}(z) + π I_{ν}(z)]For the case of integer order ν=n the negative argument identity can be simplified down to: K_{n}(−z) = (−1)^{n} K_{n}(z) − i π I_{n}(z)and for the case of halfinteger order ν=n+1/2 the identity can be simplified down to: K_{n+1/2}(−z) = −i [(−1)^{n} K_{n+1/2}(z) + π I_{n+1/2}(z)]Reflection — negative order: K_{−ν}(z) = K_{ν}(z)4. SupportModified Bessel function of the second kind of the real (fractional) order of the complex argument K_{ν} is supported in: 5. InterfaceModified Bessel function call looks like
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where order is the function real order, and argument — function argument. 6. How to useTo calculate modified Bessel function of the second kind of the 0 order of the number:
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To calculate modified Bessel function of the second kind of the 1.2 order of the current result:
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To calculate modified Bessel function of the second kind of the 2.5 order of the number z in memory:
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