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February 27th, 2018, 02:12 PM  #11  
Senior Member Joined: Aug 2012 Posts: 2,411 Thanks: 754  Quote:
In complex variable theory, the square root function $w = \sqrt{z}$ has two distinct outputs. Unlike in the case of a real variable, there's no algebraic way to distinguish them (for example by choosing the positive one by convention. There are no positive or negative complex numbers). There are two ways to address this problem. One is by "choosing a branch." When you specific a multivalued complex function, you also need to say which output value you're choosing. You can always do this in a consistent manner. [I'm leaving out a lot of technicalities with that statement]. The other way is to consider all the outputs at once, as a single geometric structure. The Wiki article contains some nice pictures of how this works.  
February 27th, 2018, 09:21 PM  #12 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 464 Thanks: 29 Math Focus: Number theory 
I recall that z^(1/2) could be represented on the unit complex circle by half its perimeter between z and real value one. Of course, the circle has countless rotations corresponding to the arc from z^(1/2) to the above mentioned complex origin, thus one complex number would project to innumerable others. Would this establish a mapping from a finite set (that of the complex number z) to a set of limitless corresponding rotations? Or is a circle disqualified for not being a function, although the results are unique? Did Riemann consider this approach, if tenable? In principle, can there exist a finite set of functions which map onto a set of unique, countless solutions? 

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finite, infinite, mapped, set 
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