Download Polar Form of Complex Numbers

Polar Form of Complex Numbers
I. Exponential Form
The Cartesian form, z=x+iy, can be plotted on the x and iy coordinates. Then as a vector, z=1+i*Sqrt(3)
has magnitude 2 at angle 60 degrees and we could write z =2*cos(60) +i*2*sin(60); this is abbreviated
as z = 2cis( where cis stands for cos(x) plus i times sin(x). Euler’s formula writes this with exponential
notation, z=2*exp(i*Pi/3). In general, exp(i*x)=cos(x)+i*sin(x). Here are some examples:
z1  2  i 2 3  4cis (60)  4e i / 3
z 2  5  5i  5 2cis (135)  5 2e i 3 / 4
z1 * z 2  20 2cis (195)  20 2e i13 /12
z 2 / z1 
5 2 i 5 /12
e
4
Multiplication of z by exp(i*t) rotates z by t. If z=1, the vector z is rotated by exp(i*Pi) to -1:
e i   1
Multiplication of z1=3cis(15) by z2=2cis(45) results in a vector with length 6 at angle 60.
3 * cis (15) * 2 * cis (45)  6cis (60)  3  i3 3 . Here is a full example, check with your calculator:
z1  cis (30), z 2  cis (45), z1 * z 2  cis (75)  0.2588  i 0.9659
To add we must use the cartesian form :
3 i
1 i
 , z2 
, z1 * z 2  0.2588  i 0.9659
2 2
2
3
3i
z1  z 2 
1
2
2
z1 
If we are given z=-1+i*sqrt(2) we must determine the angle, t, using the arctan(-sqrt(2)). However, the
arctan returns only values between -90 and 90. If z is in the II or III quadrant we need to add 180 to the
arctan. Here is an example with z in the II quadrant.
z  1  i 2 , z  3, t  arctan(  2 )  54.736  180  125.264
z  3cis (125.264)
May 4, 2011
Dr. G. Boyd Swartz
HeroesGifted.org
Polar Form of Complex Numbers
Problems 1:
Here are some problems, find both Cartesian and polar forms in decimal and degrees:
z1  1  i 3 , z 2  1  i 2
z1 * z 2, z 2 / z1, z1  z 2
Repeat the operations for :
z1 
I.
2
(1  i ), z 2  2cis (225)
2
Powers of z:
De Moivre’s Theorem is: cis(t)^n=cis(n*t) since exp(it)^n=exp(itn). We can use this to compute powers:
z1=3cis(15), z^3=27cis(45) and z2=(1+i)^3=2*sqrt(2)cis(135). Let’s check by expanding z2:
(1  i)3  2(1  i)
2 2 (cos(135)  i sin( 135))  2  2i
We can also use the theorem to derive trig identities:
e it e ir  e i (t r )
(cos(t )  i sin( t ))(cos( r )  i sin( r ))  cos(t  r )  i sin( t  r )
cos(t ) cos( r )  sin( t ) sin( r )  i (sin( t ) cos( r )  sin( r ) cos(t ))  cos(t  r )  i sin( t  r )
cos(t ) cos( r )  sin( t ) sin( r )  cos(t  r )
sin( t ) cos( r )  sin( r ) cos(t )  sin( t  r )
Problems 2:
Use De Moivre to find an identity for sin3x and cos3x.
II.
Roots of a complex number
Functions of complex numbers can be multivalued, thus we have sqrt(4) = -2,2 and the
Sqrt(i) ={ exp(i*(Pi/4 + Pi*i*k))}^(1/2) for k=0, 1= +/-(sqrt(2)/2)*(1 +i ).
The polar form z = exp(i*z + 2*Pi*i*k) for k = 0, 1, 2, …. Has the same value for each k
The complex number w = z ^ (m/n) will have multiple values as k varies. This is central to the
fundamental theorem of algebra that z^n has n roots. Guass invented the theory of complex numbers in
order to prove it.
Problems 3: Find the four roots of unity.
May 4, 2011
Dr. G. Boyd Swartz
HeroesGifted.org
Polar Form of Complex Numbers
May 4, 2011
Dr. G. Boyd Swartz
HeroesGifted.org