A function I_n(x) which is one of the solutions to the modified Bessel differential equation and is closely related to the Bessel function of the first kind J_n(x). The above plot shows I_n(x) for n = 1, 2, ..., 5. The modified Bessel function of the first kind is implemented in the Wolfram Language as BesselI[nu, z]. The modified Bessel function of the first kind I_n(z) can be defined by the contour integral I_n(z) = 1/(2π i) ∮e^((z/2)(t + 1/t)) t^(-n - 1) d t, where the contour encloses the origin and is traversed in a counterclockwise direction.

Bessel function of the first kind | continued fraction constants | modified Bessel function of the second kind | Weber's formula

BesselI

Friedrich Wilhelm Bessel