8/3/2019 All Tutorials Complex Jan 12

1/4

8/3/2019 All Tutorials Complex Jan 12

2/4

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 ]

KVL/Jan12

8/3/2019 All Tutorials Complex Jan 12

3/4

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 ]

KVL/Jan12

8/3/2019 All Tutorials Complex Jan 12

4/4

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