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Complex Numbers Notes

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Mathematics for NST Part IA Cambridge University, 2012-2013

Notes for Complex Numbers The fundamental theorem of algebra: Gauss (1799) proved in his 'Fundamental Theorem of Algebra' that the field of complex numbers is algebraically closed. That is, it encloses and completes the field of numbers as a whole. A direct result of his proof is that the following n

P( z)=an z + an-1 z Has

n-1

n th order polynomial:

+...+ a1 z +a 0=0, an [?] 0

n possibly-repeated complex roots for all possible complex coefficients a0 , ... an .

Equivalently, the equation as at least one complex root. The imaginary part of any root may also be zero such that the root is purely real. Non-rigorous proof of the fundamental theorem of algebra: The fundamental theorem of algebra can be 'proven' as follows:

*

Generate a locus of points in the complex plane corresponding to given

P( z)

for a

|z|=R

*

Continuously deform the locus of points from large

*

Crossing the origin indicates the presence of a value

R to

R=0

|z|=R at which the value of

the polynomial is zero. Thus each time the locus crosses the origin as it is deformed, the presence of a root is confirmed

a0 [?] 0 , since a0 =0 gives a trivial root

Assuming that that

P( z) is simply divided by

*

n times

P( z) is almost a single circle of radius a1 z

a0

Thus, during the continual deformation of one to the other, the locus must cross the origin

*

n-1 :

P( z) is almost a circle centred on the origin, which

R , the locus of

At very small centred on

*

to form a polynomial of order

R , the locus of

For large

wraps around the origin

*

z

z=0 in which case

Hence

n times

P(z)

must have

n complex roots

The imaginary exponential: The most important result in complex number analysis, which underpins the algebraic rules used to manipulate such numbers, concerns the exponential of an imaginary number:

e

ith

Mathematics for NST Part IA Cambridge University, 2012-2013 The value of this exponential is derived using the Taylor Series for

e x =1+x +

x :

x 2 x3 x 4 x 5
+ + + +...
2! 3 ! 4 ! 5 !

[?] e ith =1+ith-

(

e

[?] e ith = 1-

th2 i th3 th 4 ith 5
- + +
+...
2! 3 ! 4 ! 5 !

th2 th 4
th3 th 5
+ + ... +i th- + +...
2! 4 !
3! 5 !

) (

)

ith

[?] e =cos th+i sin th
[?] r e ith =r ( cos th+i sinth )

Algebraic laws derived from the imaginary exponential: The algebraic laws which allow for the manipulation of complex numbers, including de Moivre's Theorem, are derived from the definition of the imaginary exponential above: i th1

Let

z 1=r 1 e =r 1 ( cos th1 +isin th1 )

Let

z 2=r 2 ei th =r 2(cos th2 +isin th2 ) 2

Multiplication:

z 1 z 2=( r 1 e i th )( r 2 ei th ) =r 1 r 2 e 1

2 i (th 1+th2 )

[?] z 1 z 2=r 1 r 2 (cos ( th 1+ th2 ) +isin ( th1 +th 2) )

th
( 1+th 2)
[?] r 1 cis th1 x r 2 cisth 2=r 1 r 2 cis
Division:

z 1 r 1 e i th r 1 i ( th -th )
=
= e z2 r2 ei th r 2 1

1 2

2 [?]

z1 r 1
= (cos ( th1-th 2 ) +i sin ( th1 -th2 ) ) z2 r 2

Mathematics for NST Part IA Cambridge University, 2012-2013

th
( 1-th2 ) r cisth 1 r 1
[?] 1
= cis
r 2 cisth 2 r 2 De Moivre's Theorem: n

( r e ith ) =r n e nith
n

[?] [ r ( cos th+i sin th ) ] =r n ( cos ( nth ) +i sin ( nth )) nth
() n
[?] ( r cis th ) =r n cis

Manipulating numbers in the complex plane: Im Re

z 2(c , d ) z 1(a , b) z 3(a+c , b+d ) The manipulation of complex numbers may be represented by geometric operations in the complex plane: Addition/subtraction:

z 1=a+ib

Addition or subtraction of

z 2=c +id

*

to give

z3

to or from

:

Corresponds to a shift in co-ordinates of

(+- a ,+- b) in the Argand diagram

*

Geometrically equivalent to the

addition/subtraction of position vectors

z2

r1 r2

z 1 and

from the origin of the Argand diagram.

Mathematics for NST Part IA Cambridge University, 2012-2013

z2 Im Re

th2 z1 z3 r2 r1
th1
th1 +th2

Mathematics for NST Part IA Cambridge University, 2012-2013 Multiplication:

z 1=r 1 e

Multiplication of give

*

i th1

with

i th2

z 2=r 2 e

to

z3 : Corresponds to a rotation by angle

+-th 2 about the origin, coupled with a change in

distance from the origin by factor

r2

:

[?] z 1 z 2=r 1 r 2 ei th ei th =r 1 r 2 e i(th +th ) 1

*

If

|z 2|=1

2 1

2 , the complex number is

simply rotated about the origin by angle

th2

Curves in the Argand Plane: Since the real and imaginary axes of the Argand plane are geometrically equivalent to the

x and

y

axes of a Cartesian plane, the locus of points

|z +a+ib|=r in the Argand

plane are equivalent to the circle in the Cartesian plane given by the equation

( x+a )2 + ( y +b )2=r 2 :

|z +a+ib|=r , z=x + yi
[?]|x +a+i ( y +b )|=r

[?] ( x +a )2 + ( y +b )2=r 2 Transformations of circles in the Argand Plane:

R

I Translation: Complex number

u+ vi is added to

z , such that:

|z|=r With centre at to:

|z +u+iv|=r

(0,0) is transformed

Mathematics for NST Part IA Cambridge University, 2012-2013

Dilation:

R

I For a dilation in the real direction, the real part of real number

z

is multiplied by some

u , such that:

|z|=|x + yi|=r -|ux + yi|=r

R

I For a dilation in the imaginary direction, the imaginary part of multiplied by some real number

z

is

v ,

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