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EE2007 Tutorial 1
Complex Numbers, Complex Functions,Limits and Continuity
Exercise 1.
(a) [October/November 2003 G264/Q3]
Find all the values of ln(i1/2).
[ i(14
+ n), n = 0,1,2, . . . ]
(b) [October/November 2002 G264/Q5, (Rephrased)]
Find all the values of ii and show that they are all real.Hence, or otherwise, find all the points z such that zi is real.
[ e(/22n); enei, n = 0, 1, . . . ]
(c) [April/May 2005 EE2007/Q3]
Find all the values of e(ii). [ ee
(2 +2n) ]
Exercise 2. Find out whether f(z) is continuous at the origin if f(0) = 0 and for z = 0,the function f(z) is equal to
(a) Re z|z|
(b) Im z1+|z|
[ No; Yes ]
Exercise 3. The function f(z) is equal to Im z|z|
for z = 0 and f(z) = 0 for z = 0.
Determine if f(z) is continuous at z = 0, z = 5 and z = 5 + i.[ No;Yes;Yes ]
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EE2007 Tutorial 2
Differentiabiliy and Analyticity of ComplexFunctions, Cauchy-Riemann Equations and
Complex Line Integration
Exercise 1. Find the constants a and b such that f(z) = ( 2x y) + i(ax + by) is
differentiable for all z. Hence, find f
(z). [ a = 1 and b = 2, f(z) = 2 + i ]
Exercise 2. Are the following functions analytic?
(a) f(z) = Re(z2)
(b) f(z) = iz4
(c) f(z) = z z[ No; Yes, for z = 0; No ]
Exercise 3.
(a) [EE2007, April/May 2005, Q1]
Find f(z), the derivative of f(z) = 2xy ix2. State clearly the point (or points)where f(z) exists. [ on x-axis, f(z) = i2x ]
(b) [G264, April/May 2004, Q3]
Using the Cauchy-Riemann equations, determine the analyticity of the function f(z) =z2 2z + 3 and find its derivative.
[ z, f
(z) = 2z 2 ]
Exercise 4. EvaluateC
f(z) dz where
(a) f(z) = Re z, C the parabola y = x2 from 0 to 1 + i.
(b) f(z) = ez, C the boundary of the square with vertices 0, 1, 1+i, and i (clockwise).
(c) f(z) = Im(z2), C as in problem (b).[ 12
+ i23
, 0, 1 i ]
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EE2007 Tutorial 3
Cauchy Integral Theorem, Cauchy IntegralFormula
and Real Integrals
Exercise 1. Integrate 1z41
counterclockwise around the circle
(a) |z 1| = 1, (b) |z 3| = 1
[ i2
, 0 ]
Exercise 2. Evaluate the following integrals where C is any simple closed path such thatall the singularities lie inside C (CCW).
(a)C
z+ez
z3zdz
(b) C
30z223z+5
(2z1)2
(3z1)dz
[ 2i(1 + e+e1
2 ), 5i ]
Exercise 3. Evaluate the following real integrals using the complex integration method.
(a)20
d
53sin
(b)20
cos 1312cos2
d
[ /2, 0 ]
Exercise 4. Evaluate the following improper integrals using the complex integrationmethod.
(a)
x
(x22x+2)2dx
(b)
1(4+x2)2
dx
[ /2, /16 ]
KVL/Jan12