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Complex Representation of Harmonic Oscillations

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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

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

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Addition and subtraction of complex numbers: z 1 + z 2 = ( x 1 + i y 1 ) + (x 2 + i y 2 ) = (x 1 + x 2 ) + i (y 1 + y 2 ) I.e. Re (z 1 + z 2 ) = Re (z 1 ) + Re(z 2 ) and Im (z 1 + z 2 ) = Im (z 1 ) + Im(z 2 ). Similarly for subtraction: z 1 - z 2 = (x 1 - x 2 ) + i (y 1 - y 2 ) Geometrically, addition of complex numbers corresponds to vector addition in the complex plane.

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Multiplication of complex numbers: z 1 z 2 = ( x 1 + i y 1 )(x 2 + i y 2 ) = ( x 1 x 2 + i x 1 y 2 + i x 2 y 1 + i 2 y 1 y 2 ) = (x 1 x 2 - y 1 y 2 ) +i (x 1 y 2 + x 2 y 1 )

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Geometrically, the complex conjugate represents a reflection through the x-axis in the complex plane Properties of the conjugate: A) (z*)* = z B) zz* = x 2 + y 2 = a real number $ 0. Further, zz*=0 if and only if Re(z) = 0 and Im(z) = 0. Complex conjugate: z* / x - i y. That is, Re(z*) = Re(z), Im(z*) = - Im(z) example: ( 2 + i 3)* = 2 - i 3

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

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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). 51.34 E = 0.896 rad

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In terms of magnitude and phase, we have

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

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Eulers theorem and the basic properties of exponents can be used to prove all trigonometric identities. For example

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This is called the polar form of a complex number. For example, we have 4 + i 5 = 6.40 e i 0.896 Multiplication of complex numbers is easiest in polar form So that

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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

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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:

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

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Oscillations are often expressed in the form of a complex amplitude function z (t) = Ae i 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:

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4.6 – Perform Operations with Complex Numbers Not all quadratic equations have real-number solutions. For example, x 2 = -1 has no real number solutions.

4.6 – Perform Operations with Complex Numbers Not all quadratic equations have real-number solutions. For example, x 2 = -1 has no real number solutions.

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