Some Useful Properties of Complex Numbers
√
Complex numbers take the general form z = x + iy where i = −1 and where x and
y are both real numbers. There are a few rules associated with the manipulation of
complex numbers which are worthwhile being thoroughly familiar with. They are
summarized below.
• Real and imaginary parts The real and imaginary parts of the complex
number z = x + iy are given by
Real part Re z = x
Imaginary part Im z = y.
(1)
with
Re(az1 + bz2 ) = aRe(z1 ) + bRe(z2 ) and Im(az1 + bz2 ) = aIm(z1 ) + bIm(z2 ) (2)
where a and b are both real numbers.
• Complex conjugate The complex conjugate of a complex number z, written
z ∗ (or sometimes, in mathematical texts, z̄) is obtained by the replacement
i → −i, so that z ∗ = x − iy.
• The modulus of a complex number The product of a complex number
with its complex conjugate is a real, positive number:
zz ∗ = (x + iy)(x − iy) = x2 + y 2
(3)
zz ∗ = |z|2 = x2 + y 2
(4)
and is often written
where
|z| =
p
x2 + y 2
(5)
is known as the modulus of z.
• Euler’s theorem The complex number eix can be written
eix = cos x + i sin x
from which follows:
(a) cos x = Re eix
(6)
sin x = Im eix
(b) The complex conjugate of eix is e−ix so that
e−ix = cos x − i sin x.
(7)
(c) which leads us to the following important results, the first by adding
Eq. (6) and Eq. (7), the second by finding their difference:
eix + e−ix
cos x =
2
eix − e−ix
sin x =
.
2i
The last two results are well worth trying to commit to memory.
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(8)
(9)
• Polar form A complex number z can be written in the form:
z = reiθ
where
r=
Thus
p
p
|z| = x2 + y 2
y
sin θ = p
2
x + y2
2
|z|2 = reiθ = reiθ re−iθ = r2 e(iθ−iθ) = r2
x
cos θ = p
.
2
x + y2
(NOT r2 e2iθ ).
Application to problems in interference and diffraction
When there are only a small number of sources, the total intensity of the waves
produced by all the sources can be calculated by use of simple trigonometric formula.
However, when the number of sources becomes large, this can be an exceedingly
complicated procedure. However, complex number methods can offer considerable
simplification.
We can note that the kind of waves encountered can be expressed in the form
y = a sin(ωt − kx + φ)
(10)
and that, typically, we have to combine or superimpose two or more waves, that is,
add them together e.g. for two sources
y = a1 sin(ωt − kx1 + φ1 ) + a2 sin(ωt − kx2 + φ2 )
(11)
and so on – later we will be combining many such terms.
The value of complex nmbers comes from the need, as we shall see, of carrying out
a sum of the form
S = sin b + sin(b − δ) + sin(b − 2δ) + sin(b − 3δ) + . . . + sin b − (N − 1)δ (12)
i.e. a sum of N rather similar trignometric terms.
As it stands, this is quite a tricky sum to carry out, but we can turn it into something
much simpler by writing
sin(b − nδ) = Im ei(b−nδ)
(13)
so that
h
i
S =Im eib + ei(b−δ) + ei(b−2δ) + . . . + eb−i((N −1)δ)
h n
oi
=Im eib 1 + e−iδ + e−2iδ + . . . + e−i(N −1)δ .
Here we have used the fact that Im z1 + z2 = Im[z1 ] + Im[z2 ].
2
(14)
If we put r = e−iδ , we recognize the series between the curly brackets {. . .} as a
geometric series with a common ratio r:
h n
oi
ib
2
3
N −1
S =Im e 1 + r + r + r + . . . r
#
"
N
1
−
r
=Im eib
1−r
"
#
−iN δ
1
−
e
=Im eib
.
(15)
1 − e−iδ
The next step is to try to make use of the formulae Eq. (8) and Eq. (9) given above
for sin and cos. To do this we ‘extract’ a factor exp(−iN δ/2) from the numerator
and exp(−iδ/2) from the denominator of the fraction appearing in Eq. (15) which
produces the result:
#
"
−iN δ/2 iN δ/2
−iN δ/2
e
e
−
e
S =Im eib −iδ/2
e
eiδ/2 − e−iδ/2
"
#
sin
N
δ/2
=Im eib e−i(N −1)δ/2
sin δ/2
sin N δ/2 h i(b−(N −1))δ/2) i
=
Im e
(16)
sin δ/2
from which follows
S=
sin N δ/2
sin b − (N − 1)δ/2 .
sin δ/2
(17)
Thus we have shown that
sin b + sin(b − δ) + sin(b − 2δ) + sin(b − 3δ) + . . . + sin b − (N − 1)δ
sin N δ/2
=
sin b − (N − 1)δ/2 (18)
sin δ/2
a very useful result that will be applied to the case of large number of identical
sources, and later to the case of diffraction through a single slit.
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