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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 ei1 z 2 ei2 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 ) Aeit 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 Aei 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