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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
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- Winter '17
- Math, Integral transforms, Fourier sine transform