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Article 11 — Appendix A.36I — modified Bessel function of the first kindCategory. Mathematics. Abstract. Modified Bessel function of the first kind of the real (fractional) order: definition, graph, properties and identities. Reference. This article is a part of scientific calculator LiXc product. See also. K_{ν} — modified Bessel function of the second kind, J_{ν} — Bessel function of the first 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, first of which has no singularity at 0 and is called modified Bessel function of the first kind — I_{ν}. Parameter ν is called order of the function. Second solution has singularity at 0 and is called modified Bessel function of the second kind — K_{ν}. 2. GraphModified 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 modified Bessel function family depicted below — fig. 1. Fig. 1. Graphs of the modified Bessel functions of the first kind y = I_{0}(x), y = I_{1}(x) and y = I_{2}(x).3. IdentitiesConnection to the ordinary Bessel function of the first kind: I_{ν}(z) = (−i)^{ν}J_{ν}(iz)Connection to the modified Bessel function of the second kind: I_{−ν}(z) = I_{ν}(z) + 2 /π sin(πν) K_{ν}(z)Next order recurrence: I_{ν+1}(z) = −2ν /z I_{ν}(z) + I_{ν−1}(z)Negative argument: I_{ν}(−z) = e^{iπν} I_{ν}(z) = [cos(πν) + i sin(πν)] I_{ν}(z)For the case of integer order ν=n the negative argument identity can be simplified down to: I_{n}(−z) = (−1)^{n} I_{n}(z)and for the case of halfinteger order ν=n+1/2 the identity can be simplified down to: I_{n+1/2}(−z) = i (−1)^{n} I_{n+1/2}(z)Reflection — negative order: I_{−ν}(z) = I_{ν}(z) + 2 /π sin(πν) K_{ν}(z)For the case of integer order ν=n reflection identity can be simplified down to: I_{−n}(z) = I_{n}(z)and for case of halfinteger order ν=n+1/2 the identity can be simplified down to: I_{−n−1/2}(z) = I_{n+1/2}(z) + (−1)^{n} 2 /π K_{n+1/2}(z)4. SupportModified Bessel function of the first kind of the real (fractional) order of the complex argument I_{ν} is supported in: 5. InterfaceModified Bessel function call looks like
or
where order is the function real order, and argument — function argument. 6. How to useTo calculate modified Bessel function of the first kind of the 0 order of the number:
or:
To calculate modified Bessel function of the first kind of the 1.2 order of the current result:
or:
To calculate modified Bessel function of the first kind of the 2.5 order of the number z in memory:
or:



