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Lecture

# L05_supplementary.pdf

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University of Toronto St. George

Physics

PHY354H1

Erich Poppitz

Winter

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Lecture 5: Supplementary notes
Elementary algebra of complex numbers
We will list the basic properties of complex numbers you need to know for
the course:
1. Every complex number can be written in the form z = x + iy, where
2
i = ▯1. The real part of z is x, and the imaginary part of z is y. We
write Refzg = x and Imfzg = y.
▯ Example: Show that
Imfzg = Ref▯izg: (1)
▯ Answer: This follows because z = x + iy and ▯iz = y ▯ ix.
2. To add and multiply complex numbers, we use the following rule: if
z1= x +1iy a1d z = x2+ iy2, the2
z = z1+ z 2 (x + 1 ) +2i(y + y1) 2 (2)
and
z1 2= x x1▯2y y 1 2(x y +1 2y ) 2 1 (3)
▯ Example: If z = 2+3i, and z = 3▯6i, ▯nd z +z and z ▯z .
1 2 1 2 1 2
▯ Answer: z1+ z 2 (2 + 3) + i(3 ▯ 6) = 5 ▯ 3i; and z ▯ z 1 2
2 ▯ 3 ▯ 3 ▯ (▯6) + i(2(▯6) + 3(3)) = 24 ▯ 3i.
3. Equality of two complex numbers z and1z (as 2e▯ned above) means
equality of their real and imaginary parts:
z1= z ,2x = x1; y 2 y 1 2 (4)
▯
4. The complex conjugate of z is z = x ▯ iy, and the magnitude of z is
q
p 2 2
jzj = zz▯ = x + y :
▯ pxample: Find the complex conjugate and magnitude of z =
2 + 3i.
1 p p p
▯ Answer: z = 2 ▯ 3i, jzj =2 + 9 = 11.
5. The polar form of a complex number is related to polar coordinates in
the Cartesian plane. You can use Taylor series expansion of sin▯, cos▯,
and ei▯to prove De Moivre’s theorem:
i▯
cos▯ + isin▯ = e (5)
This lets us write z in polar coordinates:
z = x + iy
= (r cos▯) + i(r sin▯)
= r(cos▯ + isin▯)
= re i▯(from De Moivre’s theorem)
The transformation into polar coordinate form is very useful in solving dif-
ferential equations, as we’ll learn, so we should get used to going back and
forth between Cartesian and polar representations.
▯ Example: Express z = ▯i in polar form.
▯ Answer: First, remember z = x+iy where x = Re(z) and y = Im(z),
In our case:
x = 0 y = ▯1
Next, ▯nd the magnitude of the complex number to determine r:
q q
r = x + y = 0 + (▯1) = 1
Note: using property 4 above, I could have also found the magnitude
using: q q
p p 2
r = zz▯ = (▯i)(i) = ▯i = ▯(▯1) = 1
Finally, determine ▯ by using the fact that
▯ = arctan(y=x) = arctan(▯1=0) = ▯▯=2
Now plugging everything into polar coordinate form and using De
Moivre’s theorem:
▯i = (1)[cos(▯▯=2) + isin(▯▯=2)] = e▯=2 (6)
2 ▯ Example: Express z = (5 + 6i)=(2 + i) in Cartesian form and then in
polar form.
▯ Answer: To put in standard cartesian x+iy form, we will need to mul-
tiply the numerator and denominator by the complex conjugate of the
denominator (standard trick to get rid of all complex denominators):
5 + 6i
z =
2 + i
5 + 6i 2 ▯ i
= ▯
2 + i 2 ▯ i
(5 + 6i)(2 ▯ i)
= 4 + 1 + 2i ▯ 2i
= 10 + 6 + 12i ▯ 5i
5
16 7
= + i (Cartesian)
p 5
16 + 52 i arctan(5=16)
= e (Polar)
5
▯ 3:35e i0:30
You can use Python to reduce complex numbers to Cartesian form,
using the complex function that is part of numpy, which is included
in pylab. The Python expression complex(a,b) corresponds to a + ib.
Here’s how to do the Cartesian reduction numerically for the previous
example:
from numpy import * #or from pylab import *
complex(5,6)/complex(2,1)
This gives the answer
(3.2000000000000002+1.3999999999999999j)
Notice that python

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