Thursday, 27 November 2014

calculus - Obtain magnitude of square-rooted complex number



I would like to obtain the magnitude of a complex number of this form:



z=1α+iβ



By a simple test on WolframAlpha it should be




|z|=14α2+β2



The fact is that if I try to cancel the root in the denominator I still have a troublesome expression at the numerator:



z=α+iβα+iβ



And this alternative way seems unuseful too:



z=(α+iβ)12




If WolframAlpha gave the correct result, how to prove it?


Answer



If you convert the number to the complex exponential form, the solution is easy.



Let s=α+βi=reθi, then z=s12=r12eθ2i. The conjugate (written with an overbar) of a complex exponential reθi is just reθi, so calculating zˉz leads to the exponential terms cancelling and leaves zˉz=r1. Now r=α2+β2 and you need |z|=zˉz.


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