Simplification Problems

found by Rolf Sulanke

Date November 7, 2010
Mathematica version 7.0

Doing symbolical calculations in differential geometry I  had difficulties in simplifying certain large formulas.  The reason was that Mathematica could  not simplify some trigonometric  expressions hidden in these formulas. Sometimes  elementary limits  could not be calculated. Hoping that in future versions the simplification procedure can be improved I notice here the problematic expressions. 

1. Mathematica could not simplify

FullSimplify[(-Abs[Csch[w]])*
    Abs[Sinh[2*w]] + 2*Cosh[w], 
  {w != 0, Element[w, Reals]}]

in spite of the fact that Mathematica knows the correct result:

FullSimplify[
  (-Abs[Csch[w]])*Abs[Sinh[2*w]] + 
    2*Cosh[w] == 0, {w != 0, 
   Element[w, Reals]}]

2. Also the next cells show an unsatisfying behaviour of FullSimplify:

FullSimplify[Csch[w] - Abs[Csch[w]]*
     Abs[Csch[2*w]]*Sinh[2*w] == 0, 
  {Element[w, Reals], w != 0}]

FullSimplify[Csch[w] - Abs[Csch[w]]*
     Abs[Csch[2*w]]*Sinh[2*w] == 0, 
  {Element[w, Reals], w > 0}]

FullSimplify[Csch[w] - Abs[Csch[w]]*
     Abs[Csch[2*w]]*Sinh[2*w] == 0, 
  {Element[w, Reals], w < 0}]

3. The  next equation is true:

FullSimplify[-(Abs[Sinh[2*w]])*Sign[w] + Sinh[2*w] == 0, Element[w, Reals]]

only Mathematica does not recognize it, in spite of

FullSimplify[-(Abs[Sinh[2*w]])*Sign[w] + Sinh[2*w] == 0, w > 0]

FullSimplify[-(Abs[Sinh[2*w]])*Sign[w] + 
    Sinh[2*w] == 0, w < 0]

FullSimplify[-(Abs[Sinh[2*w]])*Sign[w] + 
    Sinh[2*w] == 0, w == 0]

4. We consider the function

FullSimplify[Sinh[w]*(Sqrt[-1 + Cosh[4*w]] + 
    Sqrt[2]*Sinh[2*w]), w < 0]

or its factor

FullSimplify[Sqrt[-1 + Cosh[4*w]] + Sqrt[2]*Sinh[2*w], w < 0]

They equal zero for w<0, only Mathematica does not recognize it.

Plot[Sinh[w]*(Sqrt[-1 + Cosh[4*w]] + 
     Sqrt[2]*Sinh[2*w]), {w, -2, 1}, 
  PlotStyle -> {Thick, Red}]

Plot[Sqrt[-1 + Cosh[4*w]] + 
     Sqrt[2]*Sinh[2*w], {w, -2, 1}, 
  PlotStyle -> {Thick, Red}]

FullSimplify[
  Sinh[w]*(Sqrt[-1 + Cosh[4*w]] +  Sqrt[2]*Sinh[2*w]) == 0, w < 0]

Answer from Katherine Bautista
Senior Academic Program Manager
Wolfram Research, Inc.

Date : November 18, 2010

Clear["Global`*"]

Example 1:

Sometimes, performig some manual conversions help. In this case, we convert the expression to exponential form to simplify it.

expr = (-Abs[Csch[w]])*Abs[Sinh[2*w]] + 2*Cosh[w] ; 
FullSimplify[TrigToExp[expr],{w≠0,Element[w,Reals]}]

Example 2:

expr = Csch[w]-Abs[Csch[w]]*Abs[Csch[2*w]]*Sinh[2*w]==0 ;

FullSimplify[TrigToExp@expr,{w≠0, Element[w,Reals]}]

Example 3:

FullSimplify[-(Abs[Sinh[2*w]])*Sign[w] + Sinh[2*w],Element[w, Reals]]

Adding the extra condition gives the desired result.5. Symbolically Mathematica does not find the limits
FullSimplify[Limit[((-1 + E^(a*t))*v)/a, t -> Infinity], a > 0]
Limit[(((-1+(e^(a t))) v)/a),t->∞]
Obviously, the limit is Infinity*Sign[v].
FullSimplify[Limit[((-1 + E^(a*t))*v)/a, t -> -Infinity], 
  a > 0]
Limit[(((-1+(e^(a t))) v)/a),t->-∞]
Obviously, the limit is -v/a.
FullSimplify[Limit[(-1 + E^(a*t))/a, t -> -Infinity], 
  a > 0]
Limit[((-1+(e^(a t)))/a),t->-∞]
FullSimplify[Limit[(-1 + E^(a*t))/a, t -> Infinity], 
  a > 0]
Limit[((-1+(e^(a t)))/a),t->∞]

expr = -(Abs[Sinh[2*w]])*Sign[w] + Sinh[2*w]; 
FullSimplify[TrigToExp[expr],Element[w, Reals]]

FullSimplify[TrigToExp[expr],{Element[w, Reals],Exp[w]<1}]

Example 4:

expr = Sinh[w]*(Sqrt[-1 + Cosh[4*w]] +Sqrt[2]*Sinh[2*w]) ; 
FullSimplify[TrigToExp[expr],w<0]

FullSimplify[TrigToExp[expr], {w < 0,Exp[w]<1}]

End of Kathy's  communication.

Further Example

Example 5

Rolf Sulanke ,     
Date : November 18, 2010

5. Symbolically Mathematica does not find the limits

FullSimplify[Limit[((-1 + E^(a*t))*v)/a, t -> Infinity], a > 0]

Obviously, the limit is Infinity*Sign[v].

FullSimplify[Limit[((-1 + E^(a*t))*v)/a, t -> -Infinity], 
  a > 0]

Obviously, the limit is -v/a.

FullSimplify[Limit[(-1 + E^(a*t))/a, t -> -Infinity], 
  a > 0]

FullSimplify[Limit[(-1 + E^(a*t))/a, t -> Infinity], 
  a > 0]