Simple Nature - Light and Matter

factored as (x − i)(x + i). In general, the fundamental theorem ofalgebra states that in the complex number system, any nth-orderpolynomial can be factored completely into n linear factors, and wecan also say that it has n complex roots, with the understanding thatsome of the roots may be the same. For instance, the fourth-orderpolynomial x 4 +x 2 can be factored as (x−i)(x+i)(x−0)(x−0), andwe say that it has four roots, i, −i, 0, and 0, two of which happento be the same. This is a sensible way to think about it, becausein real life, numbers are always approximations anyway, and if wemake tiny, random changes to the coefficients of this polynomial, itwill have four distinct roots, of which two just happen to be veryclose to zero.Discussion QuestionsA Find arg i, arg(−i), and arg 37, where arg z denotes the argument ofthe complex number z.B Visualize the following multiplications in the complex plane usingthe interpretation of multiplication in terms of multiplying magnitudes andadding arguments: (i)(i) = −1, (i)(−i) = 1, (−i)(−i) = −1.C If we visualize z as a point in the complex plane, how should wevisualize −z? What does this mean in terms of arguments? Give similarinterpretations for z 2 and √ z.D Find four different complex numbers z such that z 4 = 1.E Compute the following. Use the magnitude and argument, not thereal and imaginary parts.|1 + i| , arg(1 + i) ,( )11∣1 + i ∣ , arg1 + iBased on the results above, compute the real and imaginary parts of1/(1 + i).10.5.6 Euler’s formulaHaving expanded our horizons to include the complex numbers,it’s natural to want to extend functions we knew and loved fromthe world of real numbers so that they can also operate on complexnumbers. The only really natural way to do this in general is touse Taylor series. A particularly beautiful thing happens with thefunctions e x , sin x, and cos x:e x = 1 + 1 2! x2 + 1 3! x3 + . . .cos x = 1 − 1 2! x2 + 1 4! x4 − . . .sin x = x − 1 3! x3 + 1 5! x5 − . . .If x = iφ is an imaginary number, we havee iφ = cos φ + i sin φ ,,Section 10.5 LRC Circuits 605