Complex Representation of Harmonic Oscillations

1. Complex Representation of Harmonic Oscillations

2. 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

3. Two complex numbers are equal if their real and imaginary parts are equal:

4. 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.

5. Multiplication of complex numbers: z1 z2 = ( x1 + i y1 )(x2 + i y2 ) = ( x1x2 + i x1y2 + i x2y1 + i 2 y1y2 ) = (x1x2 - y1y2) +i (x1y2 + x2y1)

6. 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.

7. Magnitude (also called modulus) of a complex number: The magnitude of a complex number is its distance from the origin in the complex plane

8. 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) b) arg(4 + i5) = tan-1 (5 / 4) . 51.34E = 0.896 rad

9. In terms of magnitude and phase, we have

10. One of the most important relations in mathematics is Euler’s theorem: This can be proven by expanding both sides in a Taylor series and comparing the two sides term by term.

11. Euler’s theorem and the basic properties of exponents can be used to prove all trigonometric identities. For example

12. 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 So that

13. A complex number of magnitude unity is often called a pure 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

14. 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 The x and y coordinates represent points in simple harmonic motion:

15. Two points moving with the same angular speed but separated by an angle N can be represented by complex numbers The x coordinates represent points in simple harmonic motion with a phase difference N:

16. Oscillations are often expressed in the form of a complex amplitude function z (t) = Aeit where A is a complex number A= The real amplitude function (what would be observed in a measurement) is found by taking the real part: