       Re: Weird trigonometric integral and Simplification question

• To: mathgroup at smc.vnet.net
• Subject: [mg31789] Re: [mg31777] Weird trigonometric integral and Simplification question
• From: BobHanlon at aol.com
• Date: Sun, 2 Dec 2001 04:25:05 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```In a message dated 2001/12/1 4:50:18 AM, bta at attewode.com writes:

>Simplification question
>
>   If I set lambda=(1+Sqrt)/2, and then ask Mathematica to
>Expand[lambda^7] or ask for any other power of lambda, Mathematica
>gives a nice result that I can easily rewrite by hand in the form a + b
>lambda where a and b are integers. Is there any way to get Mathematica
>to
>write the result in this form? (I'm not interested in the answer involving
>the Fibonacci sequence, I'm interested in learning how to manipulate
>expressions in Mathematica.)

Once you give lambda a value you cannot express anything literally in
terms of lambda since Mathematica will substitute the assigned value.
However, the symbol GoldenRatio which is equivalent to your lambda
can be used

{GoldenRatio, FunctionExpand[GoldenRatio]}

{GoldenRatio, (1/2)*(1 + Sqrt)}

Clear[f];

f[n_Integer?NonNegative] :=
Module[{a,b, lambda = (1+Sqrt)/2, soln1, soln2},
Off[Solve::svars];
soln1 = FullSimplify[Solve[a+b*lambda == lambda^n, {a,b}][]];
soln2 =
Solve[Coefficient[FullSimplify[a /. soln1 /. Sqrt -> x], x] == 0,
b][];
On[Solve::svars];
Simplify[a+b*GoldenRatio /. Join[soln1 /. soln2, soln2]]];

f

8 + 13*GoldenRatio

Table[{GoldenRatio^n,"==", t=f[n],
FullSimplify[GoldenRatio^n - t]\[Equal]0}, {n, 0, 7}]//TableForm

>
>    Similarly if I ask Mathematica for Inverse[{{a,b},{c,d}}] it gives
>a
>nice result, but how do I get it to factor out a d-b c?
>

inv = Inverse[{{a,b},{c,d}}]

If you factor something out of a matrix, Mathematica will immediately
carry out the multiplication or division.  You can use an undefined
operator to delay the operations

CircleTimes[(t=Denominator[inv[[1,1]]]) * inv, 1/t]

inv == Times@@%

Bob Hanlon
Chantilly, VA  USA

```

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