Date: Mar 24, 2013 5:02 PM
Author: clicliclic@freenet.de
Subject: Re: Handling branch cuts in trig functions

"Nasser M. Abbasi" schrieb:> > I tried to simplify sqrt( sec(x)^2 ) but Mathematica will> only do this by assuming x is inside one branch, say> x>-Pi/2 && x<Pi/2  but Maple and maxima simplified it> but they gave the answer is terms of |sec(x)| to take> care of the sign which depends on the branch.> > Here is plot of sec(x)> > http://mathworld.wolfram.com/Secant.html> > -----------------------------> In[37]:= Assuming[x>-Pi/2&&x<Pi/2,Simplify[Sqrt[Sec[x]^2]]]> Out[37]= Sec[x]> > In[39]:= Assuming[x > Pi/2 && x < Pi, Simplify[Sqrt[Sec[x]^2]]]> Out[39]= -Sec[x]> ------------------------------> > If I just tell M that x>0, it will not simplify it.> > ------------------------------> In[38]:= Assuming[x>0,Simplify[Sqrt[Sec[x]^2]]]> Out[38]= Sqrt[Sec[x]^2]> -------------------------------> > but Maple did it only with the x>0 assumption:> > ----------------------> restart;> simplify(sqrt(sec(x)^2)) assuming  x::positive;> >                                 1>                              -------->                              |cos(x)|> restart;> simplify(sqrt(sec(x)^2));>                                /  1   \>                            csgn|------|>                                \cos(x)/>                            ------------>                               cos(x)> ---------------------------------> > On maxima 12.04.0> >    sqrt(sec(x)^2);>     |sec(x)|> > I think now that answer to sqrt(sec(x)^2) should be> |sec(x)| without need to give the branch. Since the only> different is the sign. Or is there something else here?> Mathematica and Maple by default work in the complex plane, and must therefore define ABS(z) = SQRT(RE(z)^2 + IM(z)^2), which agrees withyour SQRT(z^2) = SQRT(RE(z)^2 + 2*#i*RE(z)*IM(z) - IM(z)^2) only ifIM(z) = 0. So SQRT(SEC(z)^2) can be 'simplified' to ABS(SEC(z)) onlywhere SEC(z) = 1/COS(z) is real, that is for all real z only. Bothsystems should be able to simplify SQRT(SEC(z)^2) - ABS(SEC(z)) to zeroif z is restricted to real.Martin.