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Polar Form of Complex Numbers I. Exponential Form The Cartesian form, z=x+iy, can be plotted on the x and iy coordinates. Then as a vector, z=1+i*Sqrt(3) has magnitude 2 at angle 60 degrees and we could write z =2*cos(60) +i*2*sin(60); this is abbreviated as z = 2cis( where cis stands for cos(x) plus i times sin(x). Euler’s formula writes this with exponential notation, z=2*exp(i*Pi/3). In general, exp(i*x)=cos(x)+i*sin(x). Here are some examples: z1 2 i 2 3 4cis (60) 4e i / 3 z 2 5 5i 5 2cis (135) 5 2e i 3 / 4 z1 * z 2 20 2cis (195) 20 2e i13 /12 z 2 / z1 5 2 i 5 /12 e 4 Multiplication of z by exp(i*t) rotates z by t. If z=1, the vector z is rotated by exp(i*Pi) to -1: e i 1 Multiplication of z1=3cis(15) by z2=2cis(45) results in a vector with length 6 at angle 60. 3 * cis (15) * 2 * cis (45) 6cis (60) 3 i3 3 . Here is a full example, check with your calculator: z1 cis (30), z 2 cis (45), z1 * z 2 cis (75) 0.2588 i 0.9659 To add we must use the cartesian form : 3 i 1 i , z2 , z1 * z 2 0.2588 i 0.9659 2 2 2 3 3i z1 z 2 1 2 2 z1 If we are given z=-1+i*sqrt(2) we must determine the angle, t, using the arctan(-sqrt(2)). However, the arctan returns only values between -90 and 90. If z is in the II or III quadrant we need to add 180 to the arctan. Here is an example with z in the II quadrant. z 1 i 2 , z 3, t arctan( 2 ) 54.736 180 125.264 z 3cis (125.264) May 4, 2011 Dr. G. Boyd Swartz HeroesGifted.org Polar Form of Complex Numbers Problems 1: Here are some problems, find both Cartesian and polar forms in decimal and degrees: z1 1 i 3 , z 2 1 i 2 z1 * z 2, z 2 / z1, z1 z 2 Repeat the operations for : z1 I. 2 (1 i ), z 2 2cis (225) 2 Powers of z: De Moivre’s Theorem is: cis(t)^n=cis(n*t) since exp(it)^n=exp(itn). We can use this to compute powers: z1=3cis(15), z^3=27cis(45) and z2=(1+i)^3=2*sqrt(2)cis(135). Let’s check by expanding z2: (1 i)3 2(1 i) 2 2 (cos(135) i sin( 135)) 2 2i We can also use the theorem to derive trig identities: e it e ir e i (t r ) (cos(t ) i sin( t ))(cos( r ) i sin( r )) cos(t r ) i sin( t r ) cos(t ) cos( r ) sin( t ) sin( r ) i (sin( t ) cos( r ) sin( r ) cos(t )) cos(t r ) i sin( t r ) cos(t ) cos( r ) sin( t ) sin( r ) cos(t r ) sin( t ) cos( r ) sin( r ) cos(t ) sin( t r ) Problems 2: Use De Moivre to find an identity for sin3x and cos3x. II. Roots of a complex number Functions of complex numbers can be multivalued, thus we have sqrt(4) = -2,2 and the Sqrt(i) ={ exp(i*(Pi/4 + Pi*i*k))}^(1/2) for k=0, 1= +/-(sqrt(2)/2)*(1 +i ). The polar form z = exp(i*z + 2*Pi*i*k) for k = 0, 1, 2, …. Has the same value for each k The complex number w = z ^ (m/n) will have multiple values as k varies. This is central to the fundamental theorem of algebra that z^n has n roots. Guass invented the theory of complex numbers in order to prove it. Problems 3: Find the four roots of unity. May 4, 2011 Dr. G. Boyd Swartz HeroesGifted.org Polar Form of Complex Numbers May 4, 2011 Dr. G. Boyd Swartz HeroesGifted.org