COMPLEX NUMBERS - Portland State University...A. La Rosa Lecture Notes PSU-Physics _____ COMPLEX...
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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 = Ae iθ 3. Expressing the equation for the “forced harmonic oscillator” in complex variable 4. More on complex function formalism Time averaging of sinusoidal products
Transcript of COMPLEX NUMBERS - Portland State University...A. La Rosa Lecture Notes PSU-Physics _____ COMPLEX...
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
4. More on complex function formalism Time averaging of sinusoidal products
1. Definition of complex numbers
2. Representation of complex numbers in polar form
z
zZ
z
zZ
z
Euler’s
formula
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,
)( tCosFkxdt
dxb
dt
xdm o2
2
(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.
)( tSinFkydt
dyb
dt
ydm o2
2
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
)( tjSinFkjydt
djyb
dt
jydm o2
2
(2)
Adding (1) and (2)
)]()([][][][
ωtjSinωtCosFjyxkdt
jyxdb
dt
jyxdm o2
2
By defining
jyxz (3)
The above Eq. takes the form
tjeFkz
dt
dzb
dt
zdm o2
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)
In Section 2.2C “Studying atomic electronic excitations using a mechanically forced harmonic oscillator model,” shows how to solve Eq. (4).
4. More on complex function formalism
Ref: Amnon Yariv, Introduction to Optical Electronics
In problems that involve sinusoidally varying time functions we can save
a great deal of manipulation and space by using the complex function
formalism. As an example consider the function
The exceptions are cases that involve the product (or powers) of sinusoidal functions. In these cases we must use the real form of the function 1.1-3.
Example where the distinction between the real and complex form is not
necessary: Consider the problem of taking the derivative of a(t) dt
ta d )(~
Real
Complex ã(t)
dt
ta d )(~
Example in which we have to use the real form of the function: Consider
the product of two sinusoidal functions a(t) and b(t)
Using the real functions we get
Were we to evaluate the product a(t) b(t) using the complex form of the functions, we would get
Time averaging of sinusoidal products Another problem often encountered is that of finding the time average of the product of two sinusoidal functions of the same frequency.
)(~
)(~ tbta
.
