Simplification of algebraic expression

Question

I found an entry of a matrix from two different methods using mathematica. Which are given as;

g/(Sqrt[2] Sqrt[1 + g^2 - Sqrt[1 + g^2]])

and

1/Sqrt[2 + (2 - 2 Sqrt[1 + g^2])/g^2].

Logically these expression should be same, and they simplify to same result when i do it on paper. But when i plot these expressions for {g,-5,5}, it gives different plots on mathematica. What is happening?

@bbgodfrey, but these should reduce to single simplified form? — Usman, Jan 18 '15 at 17:07
@Usman All you need you could find here: Simplifying expressions with square roots — Artes, Jan 18 '15 at 17:15
@Usman, No. Sqrt[g^2] is a Mathematica function equal to g for g > 0 and to -g for g < 0. To go from your first expression to your second involves multiplying the first function by Sqrt[g^2]/g, which is not equal to one for negative g. — bbgodfrey, Jan 18 '15 at 17:18
The two plots are exactly the same for g>0 but not for g<0, as per bbgodfrey's comment. — Stelios, Jan 18 '15 at 17:21

score 2 · Answer 1 · answered Jan 19 '15 at 08:22

These are two simplifications of the same second expression with different ConplexityFunction used:

 Simplify[1/Sqrt[2 + (2 - 2 Sqrt[1 + g^2])/g^2], g > 0]

Simplify[1/Sqrt[2 + (2 - 2 Sqrt[1 + g^2])/g^2], g > 0, 
 ComplexityFunction -> (StringLength[ToString[#]] &)]

yielding the following:

(*  1/Sqrt[2 + (2 - 2 Sqrt[1 + g^2])/g^2]  *)
(*  g/(Sqrt[2] Sqrt[1 + g^2 - Sqrt[1 + g^2]])  *)

Have fun!

Simplification of algebraic expression

1 Answers1