Simplification Rules



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Simplification Rules

 

Derive provides the menu items Simplify, Expand, and Factor to simplify expressions in which special functions like trigonometric functions, logarithm, and exponential occur. Derive is very precise in checking whether transformations of formulae are valid or not on the domain of computation. If necessary, you have to issue the Declare Variable command and restrict the domain of a variable. For functions with simplification rules in two directions such as the rule you also have to specify the direction via the Manage menu item. Examples show best what can be achieved.

Let us consider the transformation in both directions. It is only valid when is real or complex and is nonnegative (or with the roles of and interchanged). Derive refuses to do the transformation unless the necessary condition is satisfied. So, let us first issue the Declare Variable command and specify x as a nonnegative number. Next, we issue the Manage Logarithm command to control the direction of simplification and choose the Expand option.

1:   x :epsilon Real [0,inf)

2:   Logarithm := Expand
The expression
3:   LN(x y)
is now Simplified to
4:   LN(x) + LN(y)
The Collect option in the Manage Logarithm menu must be chosen to Simplify in the opposite direction and combine sum of logarithms into the logarithm of a product.
5:   Logarithm := Collect

6:   LN(x y)
In order to illustrate that Derive only does those simplifications about which it is sure that the transformations are valid, we enter under the above circumstances the following sum of logarithms.
7:   LN(x) + LN(y) + LN(z)
Simplification only partly combines logarithms.
8:   LN(x y) + LN(z)
You can get the simplification
9:   LN(x z) + LN(y)
by choosing in the Manage Ordering command the , , order of variables.

Trigonometric simplification is another highlight of Derive. Use the Manage Trigonometry menu and choose the direction of simplification and the preference of simplification of powers of sines or cosines. The Auto option can be chosen when you leave it up to Derive to make a heuristic choice. For example, in the Auto mode, Derive simplifies

           2           2 
10:  COS(x)  + 2 SIN(x)
into
           2    
11:  SIN(x)  + 1
But if you prefer cosines over sines, you select Manage Trigonometry Toward: Cosines and get
12:  Trigpower := Cosines

               2
13:  2 - COS(x)
When you enter
14:  SIN(x + y)
and select Manage Trigonometry Direction: Expand,
15:  Trigonometry := Expand
then simplification yields
16:  COS(x) SIN(y) + SIN(x) COS(y)
On its turn, choosing the Collect direction, gives back the original expression upon simplification.



next up previous contents
Next: Derive Packages Up: A Tour of Previous: Recursion



Andre Heck
Thu Mar 23 17:40:24 MET 1995