Sectios 13.1 & 13.2 Comple umbers ad comple plae Comple cojugate Modulus of a comple umber 1. Comple umbers Comple umbers are of the form z = + iy,, y R, i 2 = 1. I the above defiitio, is the real part of z ad y is the imagiary part of z. y The comple umber z = + iy may be represetedithe comple plae as the poit with cartesia coordiates (, y). 1 0 1 z=3+2i
Comple umbers ad comple plae Comple cojugate Modulus of a comple umber Comple cojugate The comple cojugate of z = + iy is defied as z = iy. As a cosequece of the above defiitio, we have Re(z) = z + z 2, Im(z) =z z 2i If z 1 ad are two comple umbers, the, z z = 2 + y 2. (1) z 1 + = z 1 +, z 1 = z 1. (2) Comple umbers ad comple plae Comple cojugate Modulus of a comple umber Modulus of a comple umber The absolute value or modulus of z = + iy is It is a positive umber. z = z z = 2 + y 2. Eamples: Evaluate the followig i 2 3i
2. Youshouldusethe same rules of algebra as for real umbers, but remember that i 2 = 1. Eamples: # 13.1.1: Fid powers of i ad 1/i. Assume z 1 =2+3i ad = 1 7i. Calculatez 1 ad (z 1 + ) 2. Get used to writig a comple umber i the form z =(real part)+i (imagiary part), o matter how complicated this epressio might be. (cotiued) Remember that multiplyig a comple umber by its comple cojugate gives a real umber. Eamples: Assume z 1 =2+3i ad = 1 7i. Fid z 1. Fid z 1. ( Fid Im 1 z 1 3 ). # 13.2.27: Solve (8 5i)z +40 20i =0.
3. I polar coordiates, y = r cos(θ), y = r si(θ), where r = 2 + y 2 = z. y 0 θ z=+iy The agle θ is called the argumet of z. Itisdefiedforall z 0, ad is give by arcta ( y ) if 0 arg(z) =θ = arcta ( y ) + π if < 0 ad y 0 arcta ( y ) ± 2π. π if < 0 ad y < 0 Pricipal value Arg(z) Because arg(z) is multi-valued, it is coveiet to agree o a particular choice of arg(z), i order to have a sigle-valued fuctio. The pricipal value of arg(z), Arg(z), is such that ta (Arg(z)) = y, with π<arg(z) π. y Note that Arg(z) Arg(z)=π jumps by 2π whe 1 oe crosses the 0 1 egative real ais Arg(z) > - π from above.
Pricipal value Arg(z) (cotiued) Eamples: Fid the pricipal value of the argumet of z =1 i. Fid the pricipal value of the argumet of z = 10. y 1 0 1 Polar ad cartesia forms of a comple umber Youeedtobeabletogobackadforthbetweethepolar ad cartesia represetatios of a comple umber. z = + iy = z cos(θ)+i z si(θ). I particular, you eed to kow the values of the sie ad cosie of multiples of π/6adπ/4. ( π ) ( π ) Covert cos + i si to cartesia coordiates. 6 6 What is the cartesia form of the comple umber such that z =3adArg(z) =π/4?
reads ep(iθ) =cos(θ)+i si(θ), θ R. As a cosequece, every comple umber z 0cabe writte as z = z (cos(θ)+i si(θ)) = z ep(iθ). This formula is etremely useful for calculatig powers ad roots of comple umbers, or for multiplyig ad dividig comple umbers i polar form. To fid the -th power of a comple umber z 0, proceed as follows 1 Write z i epoetial form, z = z ep (iθ). 2 The take the -th power of each side of the above equatio z = z ep (iθ) = z (cos(θ)+i si(θ)). 3 I particular, if z is o the uit circle ( z = 1), we have (cos(θ)+i si(θ)) = cos(θ)+i si(θ). This is De Moivre s formula.
(cotiued) Eamples of applicatio: Trigoometric formulas cos(2θ) =cos 2 (θ) si 2 (θ), si(2θ) = 2 si(θ) cos(θ). (3) Fid cos(3θ) ad si(3θ) i terms of cos(θ) ad si(θ). Product of two comple umbers The product of z 1 = r 1 ep (iθ 1 )ad = r 2 ep (iθ 2 )is z 1 = (r 1 ep (iθ 1 )) (r 2 ep (iθ 2 )) = r 1 r 2 ep (i (θ 1 + θ 2 )). (4) As a cosequece, arg(z 1 )=arg(z 1 )+arg( ), z 1 = z 1. We ca use Equatio (4) to show that cos (θ 1 + θ 2 )=cos(θ 1 ) cos (θ 2 ) si (θ 1 ) si (θ 2 ), si (θ 1 + θ 2 )=si(θ 1 ) cos (θ 2 )+cos(θ 1 ) si (θ 2 ). (5)
Ratio of two comple umbers Similarly, the ratio z 1 is give by z 1 = r 1 ep (iθ 1 ) r 2 ep (iθ 2 ) = r 1 r 2 ep (i (θ 1 θ 2 )). As a cosequece, ( ) z1 arg =arg(z 1 ) arg( ), z 1 = z 1. Eample: Assume z 1 =2+3i ad = 1 7i. Fid z 1. To fid the -th roots of a comple umber z 0, proceed as follows 1 Write z i epoetial form, with r = z ad p Z. z = r ep (i(θ +2pπ)), 2 The take the -th root (or the 1/-th power) ( z = z 1/ = r 1/ ep i θ +2pπ ) = r ep ( i θ +2pπ ). 3 There are thus roots of z, give by ( ( ) ( )) z k = θ +2kπ θ +2kπ r cos + i si, k =0,, 1.
(cotiued) The pricipal value of z is the -th root of z obtaied by takig θ = Arg(z) adk =0. The -th roots of uity are give by ( ) ( ) 2kπ 2kπ 1=cos + i si = ω k, k =0,, 1 where ω = cos(2π/)+i si(2π/). I particular, if w 1 is ay -th root of z 0,thethe-th roots of z are give by w 1, w 1 ω, w 1 ω 2,, w 1 ω 1. (cotiued) Eamples: Fid the three cubic roots of 1. Fid the four values of 4 i. Give a represetatio i the comple plae of the pricipal value of the eighth root of z = 3+4i.
If z 1 ad are two comple umbers, the z 1 + z 1 +. This is called the triagle iequality. Geometrically, it says that the legth of ay side of a triagle caot be larger tha the sum of the legths of the other two sides. More geerally, if z 1,,..., z are comple umbers, the z 1 + + + z z 1 + + + z.