Test for pure real complex quotient

*To*: mathgroup at smc.vnet.net*Subject*: [mg49277] Test for pure real complex quotient*From*: mathma18 at hotmail.com (Narasimham G.L.)*Date*: Sun, 11 Jul 2004 02:16:12 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

We know that a quotiented complex number Z =(a + I b)/(c + I d) can be pure real if ratios of quotients (angle argument of each complex number,ArcTan[b/a]) are equal, i.e., (b/a=d/c). In general complex variable function theory,is there condition or a method to know if Im[Z1/Z2]=0 ? One way in Mathematica is to ParametricPlot3D [{Re[Z],Im[Z]},{t,,}] and verify a plane, while ignoring error messages ( not machine-size real number ) when result is contradictory. Is there a simple test for pure real (or for that matter, pure imaginary) numbers? This came last week while discussing area/volume relations of an oblate ellipsoid. http://mathforum.org/discuss/sci.math/a/m/614934/617098 y=Sqrt[1-x^2]; Plot [{Log[(1+y)/(1-y)],y,Log[(1+y)/(1-y)]/y},{x,0,2}]; It was not expected the third function would be real for x > 1 . TIA

**Follow-Ups**:**Re: Test for pure real complex quotient***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>