What exactly is modified about a modified Bessel function?

Special functions often have arcane names that not very helpful without some context. The previous post goes into some reasons for this. This post will expand on a point at the end of the post about “modified” functions. Things are given their names for a reason. Discovering that reason helps you understand their motivation and use. Pure math perspective For each integer n, the modified Bessel function In is essentially the Bessel function Jn evaluated along the imaginary axis. Specifically, From a certain shallow perspective, that’s the end of the story: modified Bessel functions are modified in the sense that the argument is multiplied by i. And there’s a fiddly constant term up front for no apparent reason. But of course that’s not the end of the story or else this wouldn’t be worth an entire post. The equation above is analogous to the relationships between circular and hyperbolic functions These relationships are interesting because the circular and hyperbolic functions are…