Re: math -- int(f^r - g^r, x = -1..1) = 0

On Sun, 20 Apr 2008 23:03:09 GMT, Gerry Myerson
<gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article <480B30B3.2040307@xxxxxx>,
Jannick Asmus <jannick.news@xxxxxx> wrote:

On 20.04.2008 13:36, quasi wrote:
This might be easy, I'm not sure ...

Prove or disprove:

If f,g : [-1,1] --> (0,infinity) are continuous functions of x such
that

int(f^r - g^r, x = -1..1) = 0

holds for infinitely many r > 0, then it holds for all r > 0.

Yes, if the infinitely many r's have an accumulation point: Consider the
analytic function z -> int(exp(z.ln(f)) - exp(z.ln(g), x = -1...1) on C.

On the other hand,
if f is the constant 17,
and g is the constant minus 17,
then the integral is zero for all even integers r,
but not for all r > 0.

Although the problem requires f,g to be positive, for all x.

So the problem is not resolved for the case where the set of known
values of r doesn't have an accumulation point.

quasi
.

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