Complex Numbers: A Brief Review•Any complex numberzcan be written in theCartesian formz=x+iy,wherexandyare real andiis one of the square roots of-1 (the other being-i). Here,x≡Rezisthereal partofzandy≡Imzis itsimaginary part.Rezand Imzcan be interpretedas thexandycomponents of a point representingzon thecomplex plane.•By comparing Taylor series expansions, one can verify Euler’s formula:eiθ=cosθ+isinθfor any realθ(expressed in radians). This leads to an alternativepolar formz=reiθfor any complex number in terms of two reals:r≡|z|≥0isthemagnitudeormodulusofz,andθ≡argzis itsargumentorphase.•We can convert between Cartesian and polar forms forcomplex numberzusingr=px2+y2x=rcosθθ=atan(y/x)y=rsinθArgumentsθandθ+2πnfor integerndescribe the samepoint on the complex plane.Theprincipal argumentArgzis deﬁned to satisfy-π<Argz≤π.rxyzzReImzθ•The real and imaginary parts of complex numbers are added/subtracted separately:z1±z2=(x1+iy1)±(x2+iy2)=(x1±x2)+i(y1±y2).•Complex numbers are multiplied/divided by multiplying/dividing their magnitudesand adding/subtracting their arguments:
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This note was uploaded on 12/15/2011 for the course PHY 4604 taught by Professor Field during the Fall '07 term at University of Florida.