REDEEMER'S UNIVERSITY
Lagos-Ibadan
Expressway, Redemption
km 46/48
City, Ogun State
COLLEGE OF NATURAL SCIENCES
DEPARTMENT OF MATHEMATICAL SCIENCES
SECOND SEMESTER EXAMINATIONS
2012/2013
SESSION
COURSE CODE: MAT 214
COURSE TITLE: Complex Analysis
INSTRUCTIONS:
Answer THREE questions in all choosing ONE from each Section
TIME ALLOWED: 2 hours
SECTION
A
Question 1
(a). To
any
complex
z:= (x
number
+ iy)
E CC,
there
corresponds
another
2 := (x - iy) E CC called the conjugate of z E CC. Using the above definition,
complex
number
show that for any.
Z,ZPZ2 E CC, the following hold:
(i).
z, + Z2
= Z;
+ 22
(ii)
Z, . Z2
= 2, . 22
(iii)
(z, / Z2)
.
(b) Hence or otherwise show that for any z E CC, Re
(c). Define a (complex) Cauchy sequence (zJ
equivalent: (i). (z,,) is a Cauchy sequence
Z
= Z;/22
Z+2
= --
2
(iv).
2
=
iff
Z
Z-2
and lm Z = --
2i
z
E
IR .
.
in CC and hence show that the following statements are
(ii) (Rez,,) and (Im z;') are both Cauchy sequences.
(iii) (Re a,) and (Im zJ are both convergent
(iv) (z,,) is convergent.
SECTION B
Question 2
(a). Given a function I: CC
--+
CC. Suppose zo' Zo + 6.z (where 6.z = 6.x
two distinct points in the interior of the domain D of
I.
+ i6. Y , with
x, Y E IR ) are
Write down the meaning of the following
expressions:
(i). the derivative I' (zo) of I at
(b). Using the definition of
(i). I(z)
= Z2 +3z
zo'
(ii). I is analytic at the point Zo .(iii) I is harmonic at Zo
I' , compute
at the point
(c). Show that the function I(z)
Zo
the derivatives of the following functions:
= (2,3)
(ii) I(z)
= exp(2z)
at the point
Zo
= (0,0)
= 2 has no derivative at any point of the complex plane.
Question 3.
(a). Let
f :U
--+
CC be a complex valued function and let Zo E U C CC be any given point. When is Zo
said to be (i). a singularity of
I
(ii). an ordinary point (iii) a pole of order n .
(b). (i) Locate and name all the singularities of the function
f :C
-
C defined by:
+Z4 +2
f(z) = (z-I)3(3z+2)2
Z8
(ii). Determine whether the above function has singularity at z = 00
(c). Evaluate the following integral:
:f ~z-zo ,
.
where C is the circle z = Zo
+ reil,
O:S; t
:s; 2n , r > o.
c
SECTIONC
Question 4
(a). State and prove the Taylor's theorem
(b). Find the first four terms of the Taylor's series expansion of the complex variable function
I(z) =
z+1
(z -3)(z -4)
about z = 2. Find the region of convergence.
(c). State Laurent's theorem.
Question 5
(a). (i) What is conformal transformation? (ii) Find the image of
.(b). Show that the transformation
Iz -
3il = 3 under the mapping w =
w = 3 - z transforms the circle with centre (5/2,0)
z-2
and radius
1. .
1.
2
z
in
the z -plane into the imaginary axis in the w -plane and the interior or the circle into the right half of the
plane.
(c). Evaluate the complex integral
Jc cosh(z)
dz
where C is
Izl = 2