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3. (i) Prove that for 8 e-Ar II n I(A) = 1+12 dt = on(A) + Rn(A), k-0 where n(A) = 2xn + 1 , 72 = 1,2, Write down an expression for the remainder term Rn(A) in the form of an integral. [9 marks] (ii) Show that On (A) is an asymptotic sequence as 1 00. [2 marks] (iii) Prove that Rn (A) = O(On+1 (A)) for A > 0. [4 marks] (iv) Prove that Rn(A) = o(on(A)) as & 80. Hence deduce an asymptotic expansion for I(A) as X 00. [3 marks] do (v) For what values of 1 does the series E Pt converge? k-1 4. (i) Show that the Mellin transform M[h(x); 8] defined by M/h (x): 8]= = ... x° - of the function = + 1 I with X > 0, is given by = , 0 < Re(s) < 1. 1 7T [You may use without proof the result that the Mellin transform of is for 0 < R(s) < 1. 1 + I sin TS ] (ii) Write down the Mellin transform of e-I. Hence using the result that [2 marks] Fioo - where C lies in the common strip of analyticity of M[h; 8] and M[f; 1 - s] , show that 0 A+r e-= dx = 1 ds (2) where L is a vertical line and G(s) = - sin TTS Explain clearly where the line L needs to be located. [5 marks] (iii) Use the result (2), stating clearly any assumptions that you make, to show that e-I dx~Apln(A) + A1 + O(A, Aln À) as X 0+ 0 X + I where the constant terms Ao.A, need to be calculated explicitly. A formal proof of the O(A, AIn A) term is not required. [ You may also use without proof the results - 12 + as Z 0, where 72 = 0,1,2, and On, bn are non-zero. ] 5. Consider I = 1 / (z) dz, 2i CN zu - 1 where f(z) = cosec( 2) - - and I is not a pole of f (z) and you may assume that I does not lie on CN. The contour CN is formed by the square (taken counter clockwise) with vertices at the locations ze - I(N + 1 i (N + 1/2 ) 7T in the complex -plane and N is a positive integer. (i) Show that N 1 I = f(x) + E (-1)* k: - I k=-N kyo [4 marks] (ii) Show also that N (-1) - k I I - + f ( z) dz. k: k=-N 2mi Cx =( 2 - x) kyo [4 marks] (iii) Prove that f(x) is bounded on CN. [5 marks] (iv) Using the result from (iii) show that / (z) da = O Cx 20 - I) ( N. 1 as N 80. [5 marks] (v) Deduce that 80 cosec(x) - 1/1 - + (-1)* 1 1 + T - k: ) in k: k=-do

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