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)