Chapte 8 Complex Numbes Motivatio: The ae used i a umbe of diffeet scietific aeas icludig: sigal aalsis, quatum mechaics, elativit, fluid damics, civil egieeig, ad chaos theo (factals. 8.1 Cocepts - Defiitio -- i 1 o i 1 -- all umbes ma be witte as: z = x i whee x = Re( z ad = Im( z - Powes of i -- the ol optios ae i, i,1, 1 -- examples with = 0, ± 1,... 1 i i = = = = = = = = i i 1 4 4 i = i i = 1 1= 1 i = 1 3 4 3 1 i i i 1 i i i i o i i = = = = 5 4 1 i i i i 1 1 i i i i 6 4 i = i i i = 1 1 1= 1 i = 1 8. Algeba of Complex Numbes Let two complex umbes be: z 1 = x 1 i 1 & z = x i - Equalit z = z if ad ol if x = x ad = 1 1 1 - Additio Ol the eal ad imagia compoets ma be added togethe o z1 z = Re( z1 z iiim ( z1 z = ( x1 x ii ( 1 - Multiplicatio As was the case i algeba all compoets will iteact with each othe z iz = ( x i ( x i 1 1 1 z iz = x ix x ii i ix i ii z z x x i i x x z iz = x ix i x x 1 1 1 1 1 = 1 1 1 ( ( 1 1 1 1 1 Re Im - The Complex Cojugate, z -- fomulated b evesig the sig of the imagia compoet of a umbe z1 = x1 i1 ad z = x i -- popeties z z = x i x i = x = i Re z --- additio: 1 1 1 1 1 1 1 ( --- subtactio: z1 z1 = x1 i1 x1 i1 = i1 = i Im( z --- multiplicatio: z ( ( z1 = x1 i1 x1 i1 = x1 i 1 = x1 1 Hece we ca use the multiplicatio popet to geeate a eal umbe
- Divisio z1 z1 Re Im z z ( x i ( x i z x i x i x i x x ix ix = = i = = z x i x i x i x x z1 x1x 1 x1 x1 = i z x x 1 1 1 1 1 1 1 1 1 1 1 8.3 Gaphical Repesetatio - complex plae is compised of the imagia -axis & the eal x-axis - as was the case i vecto math, the distace of a poit z fom the oigi (o vecto z o legth is detemied fom the modulus of z ( = x - Pola Repesetatio - usig figue 8.1 -- a pojectio of vecto uto the x-axis leads to x= cos θ -- a simila pojectio uto the -axis gives = siθ -- sice z = x i the z = ( cosθ isiθ - as is the case with the eal Catesia tasfomatio to pola coodiates, the value of z must emai uchaged fo multiples of π -- this is because pola coodiates ae based o a cicle which epeats itself fo π multiples -- we ca get θ: 1 si ta ( x if 0 θ x > taθ = = = θ x = cosθ 1 x ta ( x π if x < 0 - Aithmetic Opeatios agai thik vectos -- Additio & Subtactio
--- the vectos ca be added up ad subtacted as oe would do with vectos -- Multiplicatio & Divisio --- it is ofte easie to pefom multiplicatio & divisio opeatios whe the umbes ae expessed i tems of pola coodiates --- let z1 = 1( cosθ1 isi θ1 ad z = ( cosθ isiθ ---- fo multiplicatio: 1 = 1( cosθ1 isiθ1 i( cosθ isiθ 1 = 1 ( cosθ1cosθ cosθsiθ isiθ1cosθ isiθsiθ = ( cosθ cosθ siθ siθ cosθ isiθ isiθ cosθ Usig tig idetities: si ( x = si xcos cos xsi cos( x = cos xcos si xsi 1 = 1 cos( θ1 θ isi ( θ1 θ 1 1 1 1 1 1 ---- fo divisio: z1 1 = cos( θ θ i si ( θ θ z 1 1 ---- modulus: 1 = z1 z -- de Moive s Fomula skip this 8.4 Complex Fuctios - ou wavefuctios will ofte have a imagia compoet - i additio we will eed to use a fuctioal fom of the modulus to geeate a eal value fo the eeg of said wavefuctio - complex fuctio have a fom simila to complex umbes, let x be a eal vaiable s.t. f ( x = g( x ih( x -- like the cojugate of a complex umbe, f ( x = g( x ih( x -- also like a complex umbe, the modulus of a complex fuctio is
f ( x f ( x = f ( x = ( g( x ih( x ( g( x ih( x f ( x f ( x = g ( x g( x ih( x ih( x g( x h ( x f ( x f ( x = g ( x h ( x 8.5 Eule s Fomula - we ca epeset e z with the Talo seies 3 z z z z e = 1... 1!! 3! - suppose z is a imagia umbe o z = the 3 4 ( ( θ θ θ e i θ = 1... = 1...... 1!! 3!! 4! 3! -- it tus out that seies expasios fo si & cos ae: 3 5 4 x x x x si x = x... cos x = 1... 3! 5!! 4! e = θ i θ aka Eule s fomula = θ i θ -- hece: cos si - fo e cos si - ou ca veif that the modulus of both fuctios is 1 -- i fact, o matte what the value of θ both fuctios ae located o a uit cicle ceteed at the oigi as show below: - othe useful elatios cosθ = 1 = 1 ( e e siθ ( e e - fo pola coodiate sstem: z = e z e z = e 1 1 - de Moive s Fomula iθ e = cos θ isi θ iθ e = ( e ( ( e = cosθ isiθ = cos θ isi θ aka de Moive's fomula - Rotatio Opeatos i -- whe a complex umbe, z = e α i, is multiplied b e θ the esult is a otatio b θ i θ i α i( e e = e θ α -- i tems of Catesia coodiates: z = x i = cosα isiα e z = z = x i = cos( θ α isi ( θ α x = cos( θ α = ( cosθcosα siθsiα = xcosθ siθ = si ( θ α = ( siθcosα cosθsiα = xsiθ cosθ -- i othe wods, z has bee tasfomed to z though a otatio of θ o Rθ ( x, = ( x, 8.6 Peiodicit - sice complex umbes ae based o a oscillatig fuctio the ae peiodic i( θ π - mathematicall, e = e fo = 0, ± 1,... - gaphicall, a otatio b π leads back to the statig poit because a complete
evolutio is the esult - 3 examples: -- O a Cicle -- O a Lie -- Applicatio i Quatum ou will see this soo eough i lectue 8.7 Evaluatio of Itegals skip