Problem with simplification of $\sqrt{1+\frac{y^2}{x^2}}$

Question

When I evaluate my Mathematica code I get out a term of the following form: $\sqrt{1+\frac{y^2}{x^2}}$. However, I also have $\sqrt{x^2+y^2}$ terms that I would like it to cancel with.

For example, I would like the following to simplify:

u[x_, y_, z_, t_] := (-2 Sqrt[x^2 + y^2] - ((1 - E^(-x^2 + y^2)) y)/x)/Sqrt[1 + y^2/x^2]

I have tried using FullSimplify, but that has got me nowhere. I also tried using Replace but can't get it to work.

Can anybody help me out? Or has anybody had a similar problem?

Answer 1

Just give (Full)Simplify its assumptions in the second argument, for example:

Simplify[u[x,y,z,t], {x > 0, y > 0}]

(* -2 x + ((-E^x^2 + E^y^2)*y)/(E^x^2*Sqrt[x^2 + y^2]) *)

Answer 2

Try this:

expr1 = (-2 Sqrt[x^2 + y^2] - ((1 - E^(-x^2 + y^2)) y)/x)/
  Sqrt[1 + y^2/x^2];

then

expr2 = MapAt[HoldForm, expr1, {2, 1, 2, 2}] // 
   Simplify[#, {x > 0, y > 0}] & // ReleaseHold

(*  (-y + E^(-x^2 + y^2) y - 2 x Sqrt[x^2 + y^2])/Sqrt[x^2 + y^2]  *)

Take care that the assumptions x>0 and x<0 give different results.

Have fun!