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Transcript
Complex Numbers: A Brief Review
• Any complex number z can be written in the Cartesian form z = x + iy, where x and
y are real and i is one of the square roots of −1 (the other being −i). Here, x ≡ Re z is
the real part of z and y ≡ Im z is its imaginary part. Re z and Im z can be interpreted
as the x and y components of a point representing z on the complex plane.
• By comparing Taylor series expansions, one can verify Euler’s formula: eiθ = cos θ +
i sin θ for any real θ (expressed in radians). This leads to an alternative polar form
z = reiθ for any complex number in terms of two reals: r ≡ |z| ≥ 0 is the magnitude
or modulus of z, and θ ≡ arg z is its argument or phase.
• We can convert between Cartesian and polar forms for
complex number z using
p
x = r cos θ
r = x2 + y 2
θ = atan(y/x)
y = r sin θ
Im z
x
r
z
y
θ
Arguments θ and θ+2πn for integer n describe the same
point on the complex plane. The principal argument
Arg z is defined to satisfy −π < Arg z ≤ π.
Re z
• The real and imaginary parts of complex numbers are added/subtracted separately:
z1 ± z2 = (x1 + iy1 ) ± (x2 + iy2 ) = (x1 ± x2 ) + i(y1 ± y2 ).
• Complex numbers are multiplied/divided by multiplying/dividing their magnitudes
and adding/subtracting their arguments:
z1 /z2 = (r1 /r2 ) ei(θ1 −θ2 ) .
z1 z2 = r1 eiθ1 r2 eiθ2 = (r1 r2 ) ei(θ1 +θ2 ) ≡ z2 z1 ,
• One can also raise a complex number to a complex power:
x2 +iy2
= r1x2 +iy2 ei(x2 +iy2 )θ1 = r1x2 e−y2 θ1 ei(x2 θ1 +y2 ln r1 ) ,
z1z2 = r1 eiθ1
where the last step uses an extension to complex b of the standard result ab = eb ln a .
• The complex conjugate of a complex number z = x + iy = reiθ is z ∗ = x − iy = re−iθ .
The point representing z ∗ on the complex plane is obtained by reflecting the point
representing z about the horizontal axis. Note that z +z ∗ = 2 Re z and z −z ∗ = 2i Im z.
• Using the multiplication property above,
zz ∗ = reiθ re−iθ = r2 = |z|2 .
It is sometimes useful to use this to rewrite
z1
z1 z2∗
(x1 + iy1 )(x2 − iy2 )
(x1 x2 + y1 y2 ) + i(y1 x2 − x1 y2 )
=
=
=
.
2
2
z2
|z2 |2
x2 + y2
x22 + y22