CIVE2602 - Engineering Mathematics 2.2

Lecturer: Dr Duncan Borman

• Intro to Complex Numbers (does not fit into Limits and Sequences,

but important you have an overview)

- Real and imaginary numbers- Working with complex numbers- Different complex number representations

Lecture 4

Limits, Sequences and Partial differentiation

What is ? 9

What is ? 1

What two numbers multiply together to give -1?

What is ? 1

i 1

Complex Numbers

A Complex number (z) has Real and Imaginary part:

For example:biaz

12 ior

Test i2 i3 i4 etc

What is ? 2i

3i

4i

5i

6i

12i 1

Adding Complex Numbers Add real parts

Add imaginary parts

iz 321 iz 42

Example

21 zz

iz 3

31 zz

2312 zzz

Multiplying Complex Numbers

Multiplying by a real number

Multiplying by an imaginary number

)32(5 i

)32(3 ii

Multiplying by a Complex number

)32()3( ii

Remember 12 i

Complex Conjugate

If we have a Complex number :

Its Complex Conjugate is:

bia

bia

When a complex number is multiplied by its Conjugate, the imaginary parts cancel out e.g.:

)25)(25( ii

Dividing by a Complex number

)51(

)21(

i

i

)51()21( ii

This is a bit trickier. We need to “get rid” of the imaginary part from the bottom line.

Multiply top and bottom by the Complex Conjugate

)51(

)51(

)51(

)21(

i

i

i

i

)25551(

)10521(

ii

ii

)311(26

1i

Try these:

)1(6 i

)2)(34( ii

1)

2)

3)

4)

5)

6)

7)

)2()4( ii

)43(6 ii

)52(432 iii

)33)(1( iii

2)1( i

3 +10i

)1(6 i

)2)(34( ii

1)

2)

3)

4)

5)

6)

7)

)2()4( ii

)43(6 ii

)52(432 iii

)33)(1( iii

2)1( i

Try these:

3 -2i

-6 +6i

8 + 3 +6i -4i = 11+2i

i(3 +3 -3i +3i) = 6i

)2)(2(

)2)(4(

ii

ii

14

4218 ii

1/5 (7+6i)

1 -1 +i +i = 2i

Why should we care about complex numbers?

They allow us to describe real physical effects and phenomena.

In fact there are a huge range of applications. -They turn up all over the place in physics or engineering.

For example:

-to describe phase differences in electrical circuits -fluid flow (2D potential flow)-stress analysis -signal processing, -image processing,

We show complex numbers on an Argand diagram

Imaginary

Real

iz 25

Complex Roots of Equations

Quickly Solve

0232 xx

Complex Roots of Equations

Now Solve

012 xx

Multiple choice Choose A,B,C or D for each of these:

What is 1)

i43 B

i23D

A i3

C i4

)1()32( ii

Multiple choice Choose A,B,C or D for each of these:

What is 2)

8B

i68D

A 10

C i610

)3)(3( ii

Multiple choice Choose A,B,C or D for each of these:

What is 3)

i22B

2D

A i31

C i

)2)(1( iii

Multiple choice

Estimate which number is represented on the Argand diagram

4)

iz 44 B

iz 22D

A iz 44

C iz 33

Imaginary

Real

z

Multiple choice

Estimate which number is represented on the Argand diagram

5)

iz 4B

iiz 26 D

A iz 25

C iz 55

Imaginary

Real

z

Other representations of complex numbers

Modulus and Argument form iz 43Imaginary

Real

r4

3

22 43 r

3

4tan

=Modulus of Z or |Z|)mod(zr

)arg(Z =Argument Z

yixz Imaginary

Real

ry

x22 yxr

x

ytan

Other representations of complex numbers

Modulus and Argument form

sinry

also:

andcosrx

yixz so:

)sin(cos irz

yixz

Modulus and Argument form

)sin(cos irz

Q) Covert z=1+i to mod and arg format

)mod(z

)arg(z

z

22 yxr

x

ytan

Other representations of complex numbers

Exponential form

)sin(cos iriyxz

irez

We need to cover Taylor series to see proof of this - we do this in next 2 lectures

Q) Covert z= (3+2i)(1-i) to both modulus and argument form and exponential form

The angle must be in radians!

Mathlab week 1 task

Week 2 task is due for a week today: Use “James” this week

Multiple choice Choose A,B,C or D for each of these:

Differentiate the following wrt x:

1) xxf 3sin)(

xxf sin3)(' B

xxf 3cos3)(' D

A xxf 3cos3

1)('

C xxf 3sin3

1)('

Multiple choice Choose A,B,C or D for each of these:

Differentiate the following:

2) xxf 10ln)(

10ln)(' xfA B

0)(' xfC D

10

1)('x

xf

xxf

10

1)('

Multiple choice Choose A,B,C or D for each of these:

Differentiating more complex functions

3)

A B

C D

xxf

2

1)('

2

1)(x

xf

xxf

2)('

3

2)('

xxf

3

1)('

xxf

Multiple choice Choose A,B,C or D for each of these:

Differentiating more complex functions

4) xxexf )(

A B

C D

xxexf x )('

xx exexf )('

)1()(' xx eexf

xexf x )('

Multiple choice Choose A,B,C or D for each of these:

Differentiate the following wrt x:

5) xxxf sin)(

xxxxf sincos)(' A xxxxxf cossin)(' B

xxxf sin)(' C xxxf sin)(' D

Multiple choice Choose A,B,C or D for each of these:

Differentiate the following wrt x:

6)

)3cos()3sin(9)(' xxxf B

)3cos()3sin(3)(' xxxf D

A xxf 3cos6)('

C )3sin().3cos(6)(' xxxf

xxf 3cos)( 2

Multiple choice Choose A,B,C or D for each of these:

Differentiating more complex functions

7)

A B

C Dxx exexf )('

x

exf

x

)(

2)('

x

exexf

xx 2

)('x

xeexf

xx

2)('

x

xeexf

xx

2

..

v

dvuduv

Examples sheet – attempt Q1 and Q2 for tomorrow

Examples class 11am (Tuesday)

Task will be available today

Problem sheet 1 available on VLE (5%)

Hand in 27/10/08

MathLab problems –please see me at the end