,,
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2521
COMPLEX ANALYSIS
2019 Term 3
(1) TIME ALLOWED - Tw. (2) h. or^
(2) TOTAL NUMBER OF QUESTIONS - 3
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUESTION IN A SEPARATE BOOK
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
(7) ONLY CALCULATORS HAVING AN AFFIXED "DNSW APPROVED"
STICKER MAY BE USED
(8) To OBTAIN FULL MARKS, YOUR ANSWERS MUST NOT ONLY BE
CORRECT, BUT ALSO ADEQUATELY EXPLAINED, CLEARLY WRIT-
TEN AND LOGICALLY SET OUT
All answers must be written in ink. Except where the^ are expressly required pencils
may only be used for drawing, sketching or graphical work
2019 Term 3
Use a separate book clearly marked Question I
I. a) Find all solutions of the equation e' = 3 + 4i, giving your answers in
Cartesian (rectangular) form
by Prove that if z = " + ill, where r and 11 are real, then
Ismzj =cosh r-cos 11^
c) Consider the complex function given by
I(^) = I(" + iu) = v' + it
Give reasons for your answers to the following questions
I) Where is I continuous?
10 \\/here is I differentiable?
ill) Where is I bolomorphic?
d) Consider the linear fractional transformation
T ^ C* + C* wh, re T(,) , 2^ + 4i
z+I
i) E^aluat. T(I) and T(-i)
10 Find a value of z sricl} that T(z) = o0
ill) If G is the unit circle Izj = I, is T(G) a line or a circle? Explain
Iv) Sketch the linage under T of the set A - { z e C : Izj < I }
e) Evaluate the integral
MATH2521 Page 2
where G is the arc of the parabola 11 = r' from r = -I to r = I
,
,
I ' I 'd, ,
Please see over
2019 Term 3
Use a separate book clearly marked Question 2
2 . a) Consider the integral
.
where G is the unit circle, traced once anticlockwise, and n is a ITon-
negative Integer
I) write down the Taylor series of I(z) = e"' in powers of z. State the
region of COILvergence of this series
it) Explain why lit = O if n is even
ill) Evaluate lit if n is odd and it = 2m + I
by Let
MATH2521
'it ~ I, , ,, ,
,, _ 30(' ~ 2)
I) Determine all (maximal) regions in which 9 has a convergent Laurent
series in powers of z - I
it) Find the Laurent series for 9(z) in powers of z - I which converges
at z - 2 - i
ill) Write down an expression for the coefficient of (z - I)2521 in the
Laurent series fronT part (11), and an expressioiT for the coefficient of
(, _ I)-2521. Do not simplify your answers
Page 3
c) Let
9(^) -
I) Find all singularities of h, and determine the type of each singularity
10 Find the residue of h at each singularity
ill) E\, aluate
where G is the circle Izj = 2, traced once anticlockwise
h(,) = xih(it/4)
^(^ - I)^
sinh(72/4)
c ,(^ - ip dz ,
Please see over
2019 Term 3
Use a separate book clearly marked Question 3
3. a) I) Find the roots of the quadratic I(z) = 2' + 4iz - I. Show that one
of these roots lies outside the unit circle in the complex plane, and
the other lies inside
10 Use complex methods to evaluate the integral
MATH2521
by The aim of this question is to evaluate the real improper integral
where a is a real positive constant. Consider the complex function given
by
I
9 (^) ' T~zz4 + q4
Let R be a real number greater thaiT a; let 01 be the straight line segment
from O to a; let 02 be the quarter cirde (centred at the origin) from a to
at; and let 03 be the straight line segment from at to O. Let G be the
Closed contour 01 + 02 + 03
I) SketclT the COLTtour C, labelling all important features
11) Find all the singularities of 9 inside G, and evaluate the residue of 9
at each singularity
in) Show that
2r dB
o 2 + sino
Page 4
cod
r4 + o4 I
Iv) ExplaiiT clearly why
*
v) Evaluate the real integral I
;
11m I 9(') d^ ~ oR+o0 C
fu' ' "" " ' _,,R~o0 C
END OF EXAMINATION
,
*

欢迎咨询51作业君