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Question: simplification challenge. Exponentials in expression.

Question:simplification challenge. Exponentials in expression.

nm 10239 Maple 2024
simplify simplification
June 06 2024
1

These two expressions are the same

e1:=-sqrt(-(exp(-2 + 2*x) - 2)*exp(-2 + 2*x))/(exp(-2 + 2*x) - 2);
e2:=1/sqrt(2*exp(-2*x)*exp(2) - 1);

Is there an automated way to simplify e1 to e2? Below are my attempts. The closest I got is 

simplify(e1) assuming real;

But that still does not give same as e2. I can do it by "hand" as shown. But I like to find automated way since this is done in code without looking at expression. So I can't use the "hand" method there.

We can assume everything in real domain.


 

15244

> 

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

> 

e1:=-sqrt(-(exp(-2 + 2*x) - 2)*exp(-2 + 2*x))/(exp(-2 + 2*x) - 2);
e2:=1/sqrt(2*exp(-2*x)*exp(2) - 1);
plot([e1,e2],x=-3..3)

-(-(exp(-2+2*x)-2)*exp(-2+2*x))^(1/2)/(exp(-2+2*x)-2)

1/(2*exp(-2*x)*exp(2)-1)^(1/2)

> 

simplify(e1,size);
simplify(e1,symbolic);
simplify(e1) assuming real; #closest but still no cigar
 

-(-(exp(-2+2*x)-2)*exp(-2+2*x))^(1/2)/(exp(-2+2*x)-2)

-I*exp(-1+x)/(exp(-2+2*x)-2)^(1/2)

exp(-1+x)/(-exp(-2+2*x)+2)^(1/2)

> 

#can do it "by hand" by dividing upstairs and downstrais by numerator
A:=exp(-1 + x);
B:=-exp(-2 + 2*x) + 2;
e3:=1/sqrt( simplify(expand(B/A^2)))

exp(-1+x)

-exp(-2+2*x)+2

1/(-1+2*exp(2-2*x))^(1/2)

> 

#verify
plot([e3,e2],x=-3..3)

 

 


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