A. La Rosa
Lecture Notes
PSU-Physics
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COMPLEX NUMBERS
1. Definition of complex numbers
Complex conjugate, magnitude
Operations: Addition, multiplication, reciprocal
number
2. Representation of complex numbers in polar form
The Euler’s representation z = a + ib = Aeiθ
3. Solving differential
variable
equations
using
complex
1. Definition of complex numbers
2. Representation of complex numbers in polar
form
In short,
 Anytime we write Ae
j
we actually mean Acos() + j A Sin()
 Ae
j
is simply easier to manipulate
3. Solving differential equations using complex
variables

Consider the following equation, where all the quantities are
real numbers,
m
d 2x
dt 2
b
dx
 kx  FoCos(t )
d
(1)
This is the Eq. that governs the dynamic response of an
oscillator under the influence of a harmonic external force
FoCos(t ) .
We are looking for a solution x = x(t)

We can always consider a parallel Eq.
m
d2y
dt 2
b
dy
 ky  Fo Sin(t )
d
Notice the force is now Fo Sin(t )
(Different force, different solution; hence the use of y instead
of x.)
Judiciously, and since the Eq. is linear, we multiply the Eq. by
the complex number j; thus
m

d 2 jy
dt 2
b
djy
 kjy  Fo jSin(t )
d
(2)
Adding (1) and (2)
d 2 [ x  jy]
d [ x  jy]
m

b
 k[ x  jy]  Fo [Cos(t )  jSin(t )]
2
d
dt
By defining
z  x  jy
(3)
The above Eq. takes the form
d 2z
dz
m 2 b
 kz 
d
dt
Foe jt
(4)
Compare Eq. (4) with Eq. (1)

Thus, if we managed to find the complex function z(t) that
satisfies (4), then the solution of Eq (1) can be obtained using,
x= Real (z)
In section 2.2.c we will show how to solve Eq.4.
(5)