2. The Bessel function J0(x) of the first kind of order 0 can be expressed in terms of the integral J0(x) = Re 1 i Z C e ix cosh t dt, where C is any contour which ranges from i/2 to + i/2. This contour is called the Sommerfeld contour. (a) Show that there is a saddle point at t = 0 and that the steepest descent path can be approximated in the neighbourhood of t = 0 by the straight line t = (1 + i)s and cosh t can be approximated by 1 + is2 , where s is real. Note: the i is important in calculating the steepest descent contour. (b) Hence show that the leading order asymptotic expansion of J0(x) as x is J0(x) 2 x cos(x /4). (c) Use the result of part (a) to suggest a suitable change of variable to apply Watsons Lemma. Use Watsons Lemma to find the first two terms in the asymptotic expansion of J0(x) as x . 1