simplify doesn't work with symbolic powers ???

Posted on 2008-07-30 15:44 By summertan (7)

Categories:

How do I ...? with Maple T.A.

How to make Maple simplify expression like

sqrt( x^(2*a+2) )

to x^(a+1), where x is assumed positive?

The Maple's simplify works only if the constant 'a' is already assigned to a number, otherwise it refuses to simplify it, even if I tell it assume(x>0).

more assumptions

Comment on 2008-07-30 16:12 from djc (

64)

Try this:

simplify(sqrt( x^(2*a+2))) assuming x::positive,a::real;

Or without assumpiton with the symbolic keyword:

simplify(sqrt( x^(2*a+2) ),symbolic);

example

Comment on 2008-07-30 16:18 from acer ( Maple Rating 4

1377)

What happens if 0<x<1 and a is nonreal?

For example, when x=0.1 and a=I ?

acer

Complex powers

Comment on 2008-07-31 00:27 from Robert Israel ( Maple Rating 4

1460)

By definition, x^p = exp(p ln(x)), where Maple uses the principal branch of ln.

So (x^(2*b))^(1/2) = exp(1/2*ln(x^(2*b))) = exp(1/2*ln(exp(2*b*ln(x))))

Now ln(exp(z)) = z + 2*Pi*I*n where n is an integer such that

-Pi < Im(z) + 2*Pi*n <= Pi

i.e. n = floor((1 - Im(z/Pi))/2)

resulting in exp(1/2*ln(exp(z))) = z if n is even, -z if n is odd.

In your case,, with z = 2*b*ln(x) = 2*(a+1)*ln(x), 0 < x < 1,, we get n=0 iff
-Pi/(2*ln(x)) > Im(a) >= Pi/(2*ln(x)). For example, Pi/(2*ln(0.1)) is approximately -.6821881772.

> plot(map(Re, [sqrt(0.1^(2+2*I*t)), 0.1^(1+I*t)]), t=-3..3,
 colour=[red,blue]);

nice

Comment on 2008-07-31 00:37 from acer ( Maple Rating 4

1377)

A nice clear analysis, thank you. I kept it short just to illustrate that the expected simplification originally posted need not always hold. (Ie, evaluate both, at the named point.) But giving a result like -Pi/(2*ln(x)) > Im(a) >= Pi/(2*ln(x)) is better.

acer

Thanks, both suggestions

Comment on 2008-07-30 19:24 from summertan (7)

Thanks, both suggestions work.

Real vs. Complex

Comment on 2008-07-31 00:48 from alec ( Maple Rating 4

1497)

It seems as if a lot of problems with understanding Maple results (especially for Calculus students) are caused by initial assumption that all variables are complex. A better design (at least in Student versions) probably, would be assuming initially that all the variables are real, and use assumptions if they are complex.

Alec