I found the following sentence at the wikipedia page : Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, Iα and Kα(this is the mod. bessel function of the second kind) are exponentially growing and decaying functions, respectively.
Reference to this sentence Now, it would like to prove this or at least make this somehow plausible to me.
A "proof" of this statement for small arguments is also given on the page in this section by the asymptotic form Approximation given by Hankel
but just for small arguments and now I was wondering whether anybody here knows if such a great argument for the exponential behaviour of the modified bessel function of the second kind exists for large arguments too? Or is there another way how one could see this?