Mathematica does not recognize pattern when changing by numerical values

Question

The following returns False:

MatchQ[(-1 + (1 + E^-x)^0.1 )/(1 + E^-x)^0.2, ((1 + Exp[-x_])^v_. - 1)/(1 + Exp[-x_])^u_.]

False

And the following returns True:

    MatchQ[(-1 + (1 + E^-x)^0.1 )/(1 + E^-x)^h, ((1 + Exp[-x_])^v_. - 1)/(1 + Exp[-x_])^u_.]

True

It is strange because it looks pretty similar but Mathematica says that the pattern does not match.

Answer 1

The expression 1/(E^B)^C Has full form

Power[Power[E,B],Times[-1,C]]

But 1/(E^B)^0.2 has full form

Power[Power[E,B],-0.2`]

Notice the difference. Since you have literal -0.2 and not a symbol, Mathematica changed it from Times[-1,C] to -0.2 instead of Times[-1,0.2] so same pattern will not work on both. To match both of the above you need pattern which will capture both cases. Literal numbers and symbols. For this small example, this will do (need to use Alternatives)

pat   =  Power[Power[E,any1_],Alternatives[Times[-1,any2_],any3_]];
expr  =  1/(E^B)^C;
MatchQ[expr,pat]
 (* True *)
expr  = 1/(E^B)^0.2;
MatchQ[expr,pat]
 (* True *)

This explains why in your case (1 + E^-x)^0.2 and (1 + E^-x)^h did not give same result using the same pattern.

I have to run now. But will try later on to adjust your pattern to handle both case as in the small example I showed earlier.

Always use FullForm to find the actual pattern for expression.