Complex Numbers

Addition	z1 + z2 = x1 + x2 + i (y1 + y2 )
Multiplication	z1 z2 = x1 x2 - y1 y2 + i (x1 y2 + y1 x2 )
Complex Conjugation	z* = x - i y
	No matter how complicated the form of a complex expression, its complex conjugate can be found by replacing i by -i everywhere i occurs in the expression.

Powers of i
i = sqrt(-1) i 2 = -1 i 3 = -i i 4 = 1 i -1 = -i i -2 = -1 i -3 = i i -4 = 1

1. The unit vector 1_y is equivalent to i

2. tan(f) = y / x = Im(z) / Re(z)

3. |z| = sqrt(x² + y²) = sqrt(z z*) = sqrt(z* z)

4. Taking the complex conjugate takes f to - f

Draw the sum of two complex numbers:

Using

Re ( z₁ + z₂ ) = x₁ + x₂

Im ( z₁ + z₂ ) = y₁ + y₂

You can easily see how to represent complex addtion graphically.

Draw z + z*:

and using

Re ( z + z* ) = 2 x

Im ( z + z* ) = 0

You can easily see  z + z* = 2 Re (z) graphically.

Draw z - z*:

Similarly, using

Re ( z - z* ) = 0

Im ( z - z* ) = 2 i Im(z)

You can easily see  z - z* = 2 i Im(z) graphically.
 

Drawing of z, Re z, and Im z

Returning to the first figure, we see that we can write

      z = |z| [cos(f) + i sin(f) ]

where we remember that

      |z| = sqrt( Re(z)² + Im(z)² ).

Interestingly, it can be shown that

      cos(f) + i sin(f) = e ^{i f}

so that we can write the exceedingly simple form

      z = |z| e ^{i f}

Tabulating all the cos, sin, and exp relations:

e i f = cos(f) + i sin(f)	2 cos(f) = e i f + e - i f
e - i f = cos(f) - i sin(f)	2 i sin(f) = e i f - e - i f

     1    z*
     - = ----
     z   |z|2

     d               dz
     -- Re(z) = Re ( -- )      [etc.]
     dt              dt

         (z₁ z₂ )* = z₁* z₂*

         |z₁ z₂| = |z₁| |z₂|   but   Re(z₁ z₂) = Re(z₁)Re(z₂) - Im(z₁)Im(z₂)

         Re (I₀ e ^{i w t})     =     Re [ |I₀| e^{i f} e ^{i w t} ]     =     |I₀| cos(w t + f)