I would like to obtain the magnitude of a complex number of this form:
z=1√α+iβ
By a simple test on WolframAlpha it should be
|z|=14√α2+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=s−12=r−12e−θ2i. The conjugate (written with an overbar) of a complex exponential reθi is just re−θi, so calculating zˉz leads to the exponential terms cancelling and leaves zˉz=r−1. Now r=√α2+β2 and you need |z|=√zˉz.
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