For a real number b>0 and a real number x, define b^x, and show

Show that every complex number is the square of a complex number.

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- October 24th 2008, 12:39 AMterr13Real Analysis Problem
For a real number b>0 and a real number x, define b^x, and show

Show that every complex number is the square of a complex number. - October 24th 2008, 03:42 AMHallsofIvy
It's hard to believe you are serious- or that whoever gave you those problems is serious. Both of these are very deep proofs. For the first one it is probably best to start by proving it for x and y positive integers and work your way up through the various number systems (integers, rational numbers, real numbers).

The second is a special case of the "fundamental theorem of algebra" but perhaps not as hard as I thought at first. Given any complex number a+ bi, write (x+ yi)^2= a+ bi. Multiply it out, equate real part to real part, and solve the two real number equations for x and y.