COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers...
Transcript of COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers...
A. La Rosa Lecture Notes PSU-Physics ________________________________________________________________________
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. Expressing the equation for the “forced harmonic oscillator” in complex variable
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. Expressing differential equations in complex variable
Consider the following equation, where all the quantities are real numbers,
)(2
2
tCosFkxd
dxb
dt
xdm o
(1)
This is the Eq. that governs the dynamic response of an oscillator under the influence of a harmonic external force
)( tCosFo .
We are looking for a solution x = x(t)
We can always consider a parallel Eq.
)(2
2
tSinFkyd
dyb
dt
ydm o
Notice the force is now )( tSinFo
(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
)(2
2
tjSinFkjyd
djyb
dt
jydm o
(2)
Adding (1) and (2)
)]()([][][][
2
2
tjSintCosFjyxkd
jyxdb
dt
jyxdm o
By defining
jyxz (3)
The above Eq. takes the form
tjeFokzd
dzb
dt
zdm
2
2
(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) (5)