Re: Abs and derivative problems

*To*: mathgroup at smc.vnet.net*Subject*: [mg14651] Re: [mg14639] Abs and derivative problems*From*: Jurgen Tischer <jtischer at col2.telecom.com.co>*Date*: Sat, 7 Nov 1998 02:10:00 -0500*Organization*: Universidad del Valle*Sender*: owner-wri-mathgroup at wolfram.com

Hi Sylvan, playing around for five minutes I came up with the following. In[1]:= Sqrt[(a + I*b)/(c + I*d)*Conjugate[(a + I*b)/(c + I*d)]] Out[1]= Sqrt[((a + I*b)*Conjugate[a + I*b])/ ((c + I*d)*Conjugate[c + I*d])] Now if you apply ComplexExpand separately to both denominator and numerator you end up with the correct answer. This does not mean I can't understand your complain, but given the intents in this direction (package ReIm, a similar package in Mathsource), it looks like there is a serious problem behind. To give you an idea of what things can happen, have a look at the following example: In[1]:= FullSimplify[D[EllipticE[x, 1], x]] Out[1]= Sqrt[Cos[x]^2] In[2]:= Integrate[Sqrt[Cos[x]^2], x] Out[2]= Sqrt[Cos[x]^2]*Tan[x] I suggest you plot the two functions (EllipticE[x,1] and Sqrt[Cos[x]^2]*Tan[x]) and compare them closely. As to your second problem, there is an intuitive answer to it. You want to replace z and not z[t]. Maybe you remember your Calculus teacher saying (I admit not all say it) you have to distinguish between a function f and its value at x, f[x]? So use /. z->#^2& or /.z->Function[x,x^2]. By the way humans are typically not very consistent with respect to this rule, see what I wrote above, its just we are still a bit more (context) sensitive. Jurgen sylvan wrote: > > I could not calculate the modulus of a complex expression containing > imaginary parts in both denominator and numerator with Mathematica. An > Example: > > (a + I b) / (c + I d) > > a,b,c,d (real) symbolic variables. > > In pratice, this should be absolutely trivial. ComplexExpand is not > effective. > How do you "tell" mathematica that your variables are real ?? I > included an example below (cell format, you can cut and paste). > > Also, replacement rules like //. z[t_] -> t^2 do not work well on > expressions like z'[t] + b z[t]. the result is z'[t] + b t^2... I > could not force it to Evaluate z'[t] or D[z[t], t]. > > Could you help ?? I am sure there is a non-intuitive solution to that. > > Cell[OutputFormData["\<\ > Abs[(b0*\\[CapitalDelta]z* > (Es - I*\\[Eta]*\\[Omega]))/ > (-I*m*\\[Gamma]*\\[Omega] + > m*\\[Omega] + > b0*(Es - I*\\[Eta]*\\[Omega]) - > m*\\[Omega]^2)]\ > \>", "\<\ > Abs[(b0 \[CapitalDelta]z (Es - I \[Eta] \[Omega])) / > 2 > (-I m \[Gamma] \[Omega] + m \[Omega] + b0 (Es - I \[Eta] \[Omega]) > - > 2 > m \[Omega]0 )]\ > \>"], "Output", > CellLabel->"Out[110]//TextForm=", > LineSpacing->{1, 0}]