Friday, March 3, 2017

Properties of the principal square root of a complex number



I am studying the principal square root function of complex numbers. On Wikipedia they present a complex number z using polar coordinates as



z=reiφ,r0, π<φπ.



Further, they define the principal square root of z as




z=reiφ/2.



Continuing, it is mentioned that




The principal square root function is thus defined using the
nonpositive real axis as a branch cut. The principal square root

function is holomorphic everywhere except on the set of non-positive
real numbers (on strictly negative reals it isn't even continuous).




I do not understand these two statements. My questions are




  1. Why is the principal square root function defined using the nonpositive real axis as a branch cut? It seems to me that for z=eiπ, we obtain by equation (1) the principal square root z=1eiπ/2=i.

  2. Why is the principal square root function not continuous on the negative reals?



Answer



That is a convention that for principal value in general we must use π<argz<π for other values you can change the branch cut.


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