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Unformatted text preview: MATH34001 Two Hours THE UNIVERSITY OF MANCHESTER APPLIED COMPLEX ANALYSIS AND INTEGRAL TRANSFORMS Wednesday, 18th January 2017 14:00 — 16:00 Answer ALL FOUR questions in SECTION A (45 marks in total) and Answer TWO of the THREE questions in SECTION B (40 marks in total). If more than two questions from Section B are attempted, then credit will be given for the best two answers. Electronic calculators are permitted, provided they cannot store text. page 1 of 5 P.T.O. MATH34001 SECTION A Answer ALL four questions A1. By considering the contour integral around a ‘D—contour’ of the function eipz f1(z) = m, where p > 0 and q > 0, show that /°° qcospa: — msinpm dcc = «6—qu —00 (x2 + (12? 2(12 [Hint: you should first show that (q cos pa: — Isinpx) can be written as I m{U (flat-pm} for some complex U (an) you should find] [11 marks] A2. The Gamma function F(z) is defined by F(z)=/ e_ttz_1dt. 0 State the domain in which this integral defines a regular function of z, and derive the values of I‘(1) and I‘(1/2) (for which you may find the formula given in B7 useful). Show that F(z +1) = zF(z) in the above domain of regularity, and explain how this recurrence formula is used to produce an analytic continuation of l‘(z) into the Whole complex plane except for simple poles at the points z = 0, —1, —2, —3, Find the residues of F(z) at these poles. [11 marks] A3. Use the strong form of Schwarz’s reflection principle to find the function f3(z) that has the following properties, stating carefully any results you use:- (i) f3(z) is regular in I m(z) > 0 , and continuous in Im(z) 2 0 , except for a simple pole at z = 2 +73 with residue 6 — i; (ii) f3(z) is real for real 2; (iii) f3(z) —> 3 as |z| —> oo in Im(z) Z 0. [12 marks] A4. Define the Laplace Transform fi4(p) of the function f4(t), and state the formula for the inverse transform (Bromwich’s Integral). Find f4(t) by contour integration for the particular case in which A 9 F = —_ 4(1)) p (p + 3)2 (Describe the case if > 0 fully, but also give brief details in the case t < 0.) [11 marks] page 2 of 5 P.T.O. MATH34001 SECTION B Answer TWO of the three questions B5. The function f5(z) is given by _ (z + 1)“ f5(z) — (z — 1)b where a and b are real constants. The branch of this function is chosen such that 0 S arg(z :I: 1) < 27r. For each of the component functions in f5(z), draw a clearly labelled diagram showing the branch cut and the polar coordinates used to evaluate the function. Show that a branch cut is n_ot needed for f5(z) on the section (1,00) of the real axis provided a — b = integer, and find f5(m + 20) and f5(m — 20) for real 35 6 (—1,1). Now consider the contour integral 1 2/3i (Z + ) dz, C Z — 1 where C is a closed loop that encircles the real interval [—1, 1]. Use this to show that /1 (1 + 3:)2/3(1— $)_2/3dx — 8—” —1 3V3 I [Hint: You will need to find the behaviour of the integrand as 2 —> 00 and for this you may make use of the expansion (1+M)"=1Iaul a(a 1M2 I 0013) for I’ll/l < 1 and any 04.] [20 marks] page 3 of 5 P.T.O. MATH34001 B6. For the Gamma function I‘(z) as in question A2, show that the function f6 (2) given by f5(z) = sin(7rz) I‘(z) I‘(1 — z) is an entire function. Under the additional assumption that f6(z) is bounded as |z| —> oo, derive the Reflection Formula 2 Z . sin(7rz) ( ¢ ) [Hintz consider z = 1/2.] Use this formula to deduce (i) that Hz) never vanishes, and (ii) that 1 ‘F(§+w) for any real y. Now consider the Psi or Digamma function 111(2), which is defined by the formula 71' F(z)F(1— z) = 2 7T cosh (7ry) _ F’(z) _ d — I‘(z) — Elnmz). 1W) Show that 1M2) has simple poles at the points z = 0,—1,—2,—3,..., and find the corresponding residues. Now use the recurrence relation for F(z) (as in question A2) to obtain a corresponding recurrence relation for 1/1(z), and finally use the above Reflection Formula for F(z) to show that «“1 — z) = 1/)(2) + 7rcot(7rz) (z ¢ Z). [20 marks] page 4 of 5 P.T.O. MATH34001 B7. (a) Integrate the function f7(z) = 6"“2 around the rectangular contour with vertices at the points z = :|:X and :|:X + ia, where m, X and a are all real and positive. Hence derive the result, in the limit as X —> oo, / e‘mm2 cos (2mam) d3: = ]l% (3—7“2 (m > 0) (*). [You may assume without proof that / 6"”2 d3: = E for m > 0.] [7 marks] _00 m (b) The real function f (1:) exists for 0 < a: < 00; define the Fourier Cosine Transform F009) and the Fourier Sine Transform F5(k) of f (3:), and write down formulae for the respective inverse transforms. Consider the integral 00 [(k) 2/ f"(m) 6““ dx; 0 show by integration by parts, and with assumptions you should state, that 109) = —f’(0) + Z'I’Cfm) — k2{Fc(k) + iFs(/€)}- (**) If you were required to solve a second—order differential equation for f (3:) over (0, 00) using a Fourier Transform, and if you knew f (m) but not f’ (m) at the end a: = 0, explain why (**) means you should use a Fourier Sine Transform. [6 marks] (0) Now it is required to solve for the temperature u($, t) which satisfies the heat equation 8a 8211 together with the end condition u(0, t) = 0 (t > 0), and the initial condition _ no if 0<r<b ”(mm—{0 if b<x<oo where no and b are real, positive constants. Write down the Fourier Sine Transform U5(k, t) of u(a:, t), and show that it satisfies 8U 6—: = —ak2U5 together with U5(k, 0) = %(1 — cos kb). Solve these for Us(k, t), and hence show that u(:r, t) satisfies Bu uo °° 2 — Z — 2 _ _ _ —0ttk 8:1: 27f /_00{ cos km cos k(x b) cos k(a: + b)} e dk, and use the identity (*) to evaluate this integral. [7 marks] [B7 total :- 20 marks] END OF EXAMINATION PAPER page 5 of 5 ...
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  • Winter '17
  • Math, Integral transforms, Fourier sine transform

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