3. (i) Prove that for
8 e-Ar II
I(A) = 1+12
dt = on(A) + Rn(A),
n(A) = 2xn + 1 ,
72 = 1,2,
Write down an expression for the remainder term Rn(A) in the form of an integral.
(ii) Show that On (A) is an asymptotic sequence as 1
(iii) Prove that Rn (A) = O(On+1 (A)) for A > 0.
(iv) Prove that Rn(A) = o(on(A)) as &
80. Hence deduce an asymptotic expansion for I(A) as
(v) For what values of 1 does the series
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.
[You may use without proof the result that the Mellin transform of
for 0 < R(s) < 1.
1 + I sin TS
(ii) Write down the Mellin transform of e-I. Hence using the result that
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
where L is a vertical line and
G(s) = -
Explain clearly where the line L needs to be located.
Use the result (2), stating clearly any assumptions that you make, to show that
dx~Apln(A) + A1 + O(A, Aln À) as X
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
where 72 = 0,1,2, and On, bn are non-zero. ]
I = 1 / (z) dz,
2i CN zu -
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
I = f(x) + E (-1)*
k: - I
(ii) Show also that
(-1) - k
f ( z) dz.
2mi Cx =( 2 - x)
(iii) Prove that f(x) is bounded on CN.
(iv) Using the result from (iii) show that
da = O
(v) Deduce that
cosec(x) - 1/1 - + (-1)*
T - k:
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