THE UNIVERSITY OF NEW SOUTH WALES, SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS NOVEMBER 2018 MATH2521 COMPLEX ANALYSIS (1) TIME ALLOWED - 2 HOURS (2) TOTAL NUMBER OF QUESTIONS - 4 (3) ANSWER ALL QUESTIONS (4) THE QUESTIONS ARE OF EQUAL VALUE (5) THIS PAPER MAY BE RETAINED B Y THE CANDIDATE (6) ONLY CALCULATORS WITH AN AFFIXED "UNSW APPROVED" STICKER MAY BE USED (7) SOME RESULTS, FORMULAE AND MACLAURIN SERIES ARE SHOWN ON THE LAST TWO PAGES. All answers must be written in ink. Except where they are expressly required pencils may only be used for drawing, sketching or graphical work. November 2018 MATH2521 Answer this question in a separate booklet labelled "Ql". 1) (i) Consider a = 2 - 2i. a) Write a in polar form. b) Compute a5 and express it in cartesian form. c) Give all the solutions of the equation z5 = a, in polar form. (ii) Give an example of a set that is closed, not open and unbounded. (iii) Consider the function Find the image of the set under the function f. f(z) = --, defined on IC\ {-2}. z+2 S={zEIC: iz-11>1} (iv) Consider the function u : JR2 -+ JR, defined by u(x, y) = x3 - 3xy2 + 2xy. a) Show that u is harmonic. b) Find a harmonic conjugate v of u. ( v) Consider the function f(z) = (z + z)(z - z). Find the set of points where f is differentiable. Page Please see over . .. 1 wember 2018 MATH2521 Answer this question in a separate booklet labelled "Q2". 2) (i) Find all the solutions of sin(z) = 3i. (ii) Use De Moivre's Theorem, to express cos(3x) in terms of cos(x) and sin(x). (iii) Evaluate Log(i3), where Log is the principal logarithm. (iv) Give all the solutions of ez = i -3. (v) Evaluate A:= l cos(2z)dz, where "I is any contour from 1 + i to 1 - i. Give the answer in cartesian form. (vi)· Evaluate B := l zdz, Page 2 where "I is the contour equal to the circle center O with radius 2, taken once anti-clockwise. Please see over ... November 2018 MATH2521 Page 3----------------------------------- - Answer this question in a separate booklet labelled "Q3". 3) (i) Evaluate the integral C := i (z+ l;�z+2) dz, where 'Y is the contour equal to { z EC : izl = 3}, oriented anti-clockwise. (ii) Consider f(z) = sin(z) . z(z - 3)2 a) Find all singularities off and classify their type (as removable, pole or essential). b) Find the residue at each singularity. c) Evaluate the integral (using the Residue Theorem) D := l f(z)dz, ·7 where 'Y is the contour equal to {z EC: lzl = 4}, oriented anti-clockwise. (iii) Consider the function 1 g(z) = (4-z)(l-z)· a) State the largest open annulus, centered at O and containing the point 2, in which g is analytic. b) Find the Laurent series for g about O (in powers of z) which converges tog in that annulus above. c) Let 'Y be a simple closed contour anti-clockwise around O in the annulus of question b). Using Laurent's theorem and your answer to b), write down the value of E ·= 1 g(z)d . 3 z. 7 z Please see over ... ovember 2018 MATH2521 Answer this question in a separate booklet labelled "Q4". 4) (i) Consider a Mi:ibius transformation T(z) = az+b, cz+ d where a, b, c, d are real numbers satisfying ad-be> 0. Denote by H+ the set of z E (C such that Im(z) > 0. a) Show that if x is real, then T(x) is real. Page 4 b) Show that if z is in H+, then T(z) has a strictly positive imaginary part. (ii) Find the image of the unit disc D(O, 1) := {z E (C: [z[ < 1} under the Mi:ibius transformation (iii) Consider the function z+2 T(z) = -1. z- 1 j(z) = -1 4· +z Using residues, we want to calculate the improper integral l+oo 1 F:= - 4 dx. _00 1 + X Given any R > l, we define the simple closed contour rR = [ -R, R] + CR where CR is the semicircle z = Rei@ (0 :::; e '.': 7r) of radius R and [-R, R] is the segment on the real axis z = x (-R '.': x '.': R). The contour rR is shown below: a) Find all the singularities of f inside the contour rR and calculate the residue at each of those singularities. b) Show that c) Deduce the numerical value of lim { j(z)dz = 0. R---+oo}cR 1+00 1 G: = - -4 dx. _00 1+ X Please see over ... November 2018 MATH2521 Some Results and Formulae Cauchy Integral Formula If f is analytic on and inside a simple closed once-anticlockwise contour I and a is inside I then j(n\(a) = 2, 1 ( f(zin+l dz for n = O, 1, 2, . . . n. 2m 'Y z -a Taylor series Theorem If f is analytic in the open disc D( a, R) then the Taylor series for f about a oo j(n)( ) � a (z-ar � n! n=O converges to f(z) in this disc. It is the unique power series about a convergent to f in the disc. See over page for a list of Maclaurin series (Taylor series about 0). Laurent's Theorem If f is analytic in the annulus r < I z - a I < R there is a m1ique series of the form convergent to f(z) in this annulus. If I is simple once-anticlockwise contour in the annulus with a inside 1, then Residues 1 1 f(z) an = -. ( ) +l dz for n = 0, ±1, ±2, ... 2?Ti z -a n 'Y 1. If ¢(z) is analytic at a and m E z+ then for f(z) = ¢(z) , (z - a)m ¢ (m-l)(a) res(!, a) = (m - l)! . 2. If .f has a pole of order m at a then res(!, a) = lim z-ta 1 d
3. If p and q are analytic at a and q has a simple zero at a and p( a) # 0 then f(z) = �/;( has a simple pole at a p(a) and res(.f, a) = q' ( a) Page 5 Please see over ... ovember 2018 MATH2521 ome Maclaurin series z2 z3 ez=l+z+-+-+··· 2! 3! �Z n = L, 1 for n. n=O z E IC z3 z5 sinz=z--'+--··· . 3! 5! 00 z2n+l L(-1r (2n + 1)! for z E IC n=O z2 z4 cosz = 1--+- - · · · 2! 4! z3 zs sinhz = z+-+-+··· 3! 5! z2 z4 coshz=l+-+-+··· 2! 4! 00 2n = L(-lt-( z )I for z E IC 2n. n=O 00 z2n+l = L ( l)I for z E IC 2n+ n=O 00 2n = L (�n)! for z E IC n=O z2 z3 z4 Log(l + z) = z --+ - - -+ . . · 2 3 4 1 2 3 --=l+z+z +z +· .. 1-z 00 LZn for lzl < 1 n=O lzl < 1 1 2 3 - -=1-z+z -z +·" l+z L(-1rzn for 1z1 < 1 n=O END OF EXAMINATION Page 6 欢迎咨询51作业君
