How do you make Mma assume a parameter is real and positive?

*To*: mathgroup at smc.vnet.net*Subject*: [mg7343] How do you make Mma assume a parameter is real and positive?*From*: "C. Woll" <carlw at u.washington.edu>*Date*: Tue, 27 May 1997 22:27:26 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Hi all, If I give mma the input Sqrt[3^2] I get back 3, but when I give it Sqrt[a^2] I don't get back a. This is exactly the behavior I expect. However, suppose that I want to tell mma that a is a real positive number, so that I want Sqrt[a^2] to return a. How can I do this? Let me give you the background for this question. As a simple example, suppose I am trying to find the series expansion of 2 b -------------- 2 2 a - Sqrt[a -b ] So, I try b^2/(a-Sqrt[a^2-b^2])+O[b]^4 and mma returns 2 b ---------- + O[b]^4 2 a-Sqrt[a ] which is certainly the wrong answer. The correct answer is 2 b 2 a - --- + O[b]^4 2 a So, I need to tell mma that Sqrt[a^2] is a while it is doing its series expansion algorithm. Another manifestation of this problem occurs when taking limits, since mma gives 2 b Limit[-------------, b->0] ---> 0 2 2 a-Sqrt[a -b ] which is also incorrect (it should be 2 a). The usual solution of using ComplexExpand doesn't help here. My solution for this particular example was to add new rules for Power: Unprotect[Power]; Power[Power[a,n_],m_]:=Power[a,n m]; Protect[Power]; I would have preferred to add an upvalue to a, but a would be too deeply buried if I were to try a /: Power[Power[a,n_],m_]:=Power[a,n m] I am not crazy about the above method, since it slows down the application of Power everywhere, so I am curious if anybody has a better method. Thanks for any suggestions. Carl