1. The modified Bessel function of the second kind of order n, Kn(x), is given by the integral expression Kn(x) = Z 0 e x cosh t cosh(nt) dt. (a) Show that the first term in the asymptotic expansion of Kn(x) as x is Kn(x) 2x e x . (b) Use Watsons Lemma and your result in part (a) to find the first two terms in the asymptotic expansion of Kn(x) as x . Note that cosh1 z = log z + p z 2 1 and the Taylor series about the origin of cosh z is 1 + 1 2 z 2 + 1 24 z 4 + . . . of log(1 + z) is z 1 2 z 2 + 1 3 z 3 . . .