Transcription of Some Useful Properties of Complex Numbers
1 some Useful Properties of Complex NumbersComplex Numbers take the general formz=x+iywherei= 1 and wherexandyare both real Numbers . There are a few rules associated with the manipulation ofcomplex Numbers which are worthwhile being thoroughly familiar with. They aresummarized below. Real and imaginary partsThe real and imaginary parts of the complexnumberz=x+iyare given byReal part Rez=xImaginary part Imz=y.(1)withRe(az1+bz2) =aRe(z1) +bRe(z2) and Im(az1+bz2) =aIm(z1) +bIm(z2) (2)whereaandbare both real Numbers . Complex conjugateThe Complex conjugate of a Complex numberz, writtenz (or sometimes, in mathematical texts, z) is obtained by the replacementi i, so thatz =x iy. The modulus of a Complex numberThe product of a Complex numberwith its Complex conjugate is a real, positive number :zz = (x+iy)(x iy) =x2+y2(3)and is often writtenzz =|z|2=x2+y2(4)where|z|= x2+y2(5)is known as themodulusofz.
2 Euler s theoremThe Complex numbereixcan be writteneix= cosx+isinx(6)from which follows:(a) cosx= Re[eix]sinx= Im[eix](b) The Complex conjugate ofeixise ixso thate ix= cosx isinx.(7)(c) which leads us to the following important results, the first by addingEq. (6) and Eq. (7), the second by finding their difference:cosx=eix+e ix2(8)sinx=eix e ix2i.(9)The last two results are well worth trying to commit to Polar formA Complex numberzcan be written in the form:z=rei wherer= |z|= x2+y2sin =y x2+y2cos =x x2+ |z|2= rei 2=rei re i =r2e(i i )=r2(NOTr2e2i ).Application to problems in interference and diffractionWhen there are only a small number of sources, the total intensity of the wavesproduced by all the sources can be calculated by use of simple trigonometric , when the number of sources becomes large, this can be an exceedinglycomplicated procedure.
3 However, Complex number methods can offer can note that the kind of waves encountered can be expressed in the formy=asin( t kx+ )(10)and that, typically, we have to combine or superimpose two or more waves, that is,add them together for two sourcesy=a1sin( t kx1+ 1) +a2sin( t kx2+ 2)(11)and so on later we will be combining many such value of Complex nmbers comes from the need, as we shall see, of carrying outa sum of the formS= sinb+ sin(b ) + sin(b 2 ) + sin(b 3 ) +..+ sin[b (N 1) ](12) a sum ofNrather similar trignometric it stands, this is quite a tricky sum to carry out, but we can turn it into somethingmuch simpler by writingsin(b n ) = Im[ei(b n )](13)so thatS=Im[eib+ei(b )+ei(b 2 )+.]
4 +eb i((N 1) )]=Im[eib{1 +e i +e 2i +..+e i(N 1) }].(14)Here we have used the fact that Im[z1+z2]= Im[z1] + Im[z2].2If we putr=e i , we recognize the series between the curly brackets{..}as ageometric series with a common ratior:S=Im[eib{1 +r+r2+r3+..rN 1}]=Im[eib1 rN1 r]=Im[eib1 e iN 1 e i ].(15)The next step is to try to make use of the formulae Eq. (8) and Eq. (9) given abovefor sin and cos. To do this we extract a factor exp( iN /2) from the numeratorand exp( i /2) from the denominator of the fraction appearing in Eq. (15) whichproduces the result:S=Im[eibe iN /2e i /2eiN /2 e iN /2ei /2 e i /2]=Im[eibe i(N 1) /2sinN /2sin /2]=sinN /2sin /2Im[ei(b (N 1)) /2)](16)from which followsS=sinN /2sin /2sin[b (N 1) /2].
5 (17)Thus we have shown thatsinb+ sin(b ) + sin(b 2 ) + sin(b 3 ) +..+ sin[b (N 1) ]=sinN /2sin /2sin[b (N 1) /2](18)a very Useful result that will be applied to the case of large number of identicalsources, and later to the case of diffraction through a single
