COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers...

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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

Transcript of COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers...

Page 1: COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers Complex conjugate, magnitude Operations: Addition, multiplication, reciprocal number

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

Page 2: COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers Complex conjugate, magnitude Operations: Addition, multiplication, reciprocal number

1. Definition of complex numbers

Page 3: COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers Complex conjugate, magnitude Operations: Addition, multiplication, reciprocal number

2. Representation of complex numbers in polar form

Page 4: COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers Complex conjugate, magnitude Operations: Addition, multiplication, reciprocal number
Page 5: COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers Complex conjugate, magnitude Operations: Addition, multiplication, reciprocal number

In short,

Anytime we write Ae j

we actually mean Acos() + j A Sin()

Ae j

is simply easier to manipulate

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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

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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)

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Page 17: COMPLEX NUMBERS - Portland State University...COMPLEX NUMBERS 1. Definition of complex numbers Complex conjugate, magnitude Operations: Addition, multiplication, reciprocal number
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