Download Complex Representation of Harmonic Oscillations

Complex Representation of
Harmonic Oscillations
The imaginary number i is defined by i 2 = -1.
Any complex number can be written as z = x + i y where x and y
are real numbers. x is called the real part of z ( symbolically, x = Re(z) )
and y is the imaginary part of z ( y = Im(z) ).
Complex numbers can be represented as points in the complex plane,
where the point (x,y) represents the complex number x + i y
Two complex numbers are equal if their real and
imaginary parts are equal:
z1  z2
if and only if
Re(z1 )  Re( z2 ) and Im( z1 )  Im( z2 )
Addition and subtraction of complex numbers:
z1 + z2 = ( x1 + i y1 ) + (x2 + i y2 ) = (x1 + x2) + i (y1 + y2)
I.e. Re (z1 + z2) = Re (z1) + Re(z2) and Im (z1 + z2) = Im (z1) +
Im(z2).
Similarly for subtraction: z1 - z2 = (x1 - x2) + i (y1 - y2)
Geometrically, addition of
complex numbers corresponds to
vector addition in the complex
plane.
Multiplication of complex numbers:
z 1 z2
= ( x1 + i y1 )(x2 + i y2 )
= ( x1x2 + i x1y2 + i x2 y1 + i 2 y1y2 )
= (x1x2 - y1y2) +i (x1y2 + x2 y1)
Complex conjugate: z*/ x - i y.
That is, Re(z*) = Re(z), Im(z*) = - Im(z)
example: ( 2 + i 3)* = 2 - i 3
Geometrically, the complex
conjugate represents a
reflection through the x-axis in
the complex plane
Properties of the conjugate:
A) (z*)* = z
B) zz* = x2 + y2 = a real number $0. Further, zz*=0 if and only
if Re(z) = 0 and Im(z) = 0.
Magnitude (also called modulus) of a complex number:
z  x  iy 
x2  y2 
z*z
The magnitude of a complex number is its distance from the origin in the
complex plane
It is often useful to use a polar
representation of complex numbers.
The angle between a radial line and the
positive x-axis makes an angle called
the argument of z or the phase of z.
In symbols, 2
= arg(z)
example: Find the magnitude and phase of 4 + i5
a)
4  i5  16  25  6.40
b) arg(4 + i5) = tan-1 (5 / 4) . 51.34E = 0.896 rad
In terms of magnitude and phase, we have
Re( z)  z cos , Im( z)  z sin 
therefore,
z = Re  z + i Im z  = z cos  i z sin   z cos  i sin  
One of the most important relations in mathematics is Euler’s
theorem:
e
i
 cos  i sin 
This can be proven by expanding both sides in a Taylor series and
comparing the two sides term by term.
examples: e
i0
 e

i 0
 1, cos(0)  i sin(0)  1  i 0  1
e i / 2  cos( / 2)  i sin( / 2)  0  i  i
e i  cos( )  i sin( )   1  i 0   1
Euler’s theorem and the basic properties of exponents can be used
to prove all trigonometric identities. For example
cos(2 )  i sin(2 )  e
i 2
 (e )  cos( )  i sin( )
i
2
2
 cos2 ( )  sin 2 ( )  i 2 cos( ) sin( )
Taking the real part of this equality gives cos(2 )  cos2 ( )  sin 2 ( )
and taking the imaginary part gives sin(2 )  2 cos( ) sin( )
We have seen that any complex number can be written in terms
of its magnitude and phase as
z = z  cos   i sin  
therefore we have
z  z ei   z ei arg( z )
This is called the polar form of a complex number. For example,
we have 4 + i 5 = 6.40 ei 0.896
Multiplication of complex numbers is easiest in polar form

z1 z2  z1 ei1
z
2

ei2  z1 z2 ei (1  2 )
So that
z1 z2  z1 z2 , arg( z1 z2 )  arg( z1 )  arg( z2 )
A complex number of magnitude unity is often called a pure
i
e
phase, and it can be written as
Multiplying a complex number by a pure phase rotates the
corresponding point in the complex plane counterclockwise by
an angle equal to the phase
Consider a point moving
clockwise on a circle of radius
A with angular speed T in
the complex plane. The
coordinates of the moving
point corresponds to the
complex number z  Ae  i  t
The x and y coordinates represent points in simple harmonic
motion:
Re( z )  A cos( t ), Im( z )   A sin( t )
Two points moving with the
same angular speed but
separated by an angle N can be
represented by complex
numbers
z1 (t )  Aeit and z2 (t )  Ae i (t  )
The x coordinates represent points in simple harmonic motion with
a phase difference N:
Re( z1 )  A cos( t ), Re( z2 )  A cos( t   )
We call the complex function Aei t   the "complex
form of the amplitude function". The quantity Aei
is called the complex amplitude.
Oscillations are often expressed in the form of a complex
amplitude function z (t) = AeiTt where A is a complex
i
number A= A e
The real amplitude function (what would be observed in
a measurement) is found by taking the real part:
x(t )  Re  z(t )  A cos  t   