Wikipedia:Reference desk/Archives/Mathematics/2013 January 30

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January 30

Generalization of integral 2

It is known $\int _{0}^{1}x^{a}(1-x)^{b}dx={\frac {\Gamma (a+1)\Gamma (b+1)}{\Gamma (a+b+2)}}$.

Consider the following generalization

$\int _{0}^{1}x^{a}(1-x)^{b}f(x)dx$

where $f$ satisfies $\int _{0}^{1}f(x)dx=1$ and $f(x)\geq 0$ for all x.

What is the result then? --AnalysisAlgebra (talk) 11:04, 30 January 2013 (UTC)

You might find Cauchy–Schwarz inequality useful, otherwise I don't think there is much one can say. Dmcq (talk) 23:26, 30 January 2013 (UTC)

OK. I want to find this limit: $\lim _{n\rightarrow \infty }{\frac {n^{-1/2}(1/2)^{n}}{\int _{0}^{1}x^{n/2}(1-x)^{n/2}f(x)dx}}$ --AnalysisAlgebra (talk) 02:54, 31 January 2013 (UTC)

Hint: what is the maximum value of x(1-x) in that interval, and what value of x does it occur at? Looie496 (talk) 03:38, 31 January 2013 (UTC)

1/4 and 1/2, respectively. Are you trying to bound the expression or something? But that tells you nothing about the maximum value of $x^{n/2}(1-x)^{n/2}f(x)$ with the f in there.--AnalysisAlgebra (talk) 05:28, 31 January 2013 (UTC)

As n tends to infinity with f remaining constant, $x(1-x)$ becomes a more significant factor in determining $x^{n/2}(1-x)^{n/2}f(x)$. -- Meni Rosenfeld (talk) 05:46, 31 January 2013 (UTC)

Is that also true for any $\alpha$ in $x^{\alpha n}(1-x)^{(1-\alpha )n}f(x)$--AnalysisAlgebra (talk) 05:57, 31 January 2013 (UTC)

Then the dominant factor is $x^{\alpha }(1-x)^{1-\alpha }$. -- Meni Rosenfeld (talk) 14:33, 31 January 2013 (UTC)

Are you saying the limit is independent of $f$? The limit is ${\sqrt {2/\pi }}$ if $f(x)=1$ but that's not the case if for example $f(x)=3x^{2}$?--AnalysisAlgebra (talk) 09:33, 31 January 2013 (UTC)

It depends on f, but only on its behavior in a specific area. -- Meni Rosenfeld (talk) 14:33, 31 January 2013 (UTC)

Or the integral can be done exactly using the above identity by just substituting f for its taylor series evaluated at either x=0 or x=1. As a note, naively treating the distribution as a nascent delta could lead you to erroneously conclude that if f(1/2)=0 the integral is zero. — Preceding unsigned comment added by 123.136.64.14 (talk) 06:13, 31 January 2013 (UTC)

Only if you know $f$, and also only if $f$ is analytic. --AnalysisAlgebra (talk) 09:33, 31 January 2013 (UTC)

@User:AnalysisAlgebra: Please do not remove other's comments at Help Desk/Math as you did here: [1] This is considered bad form. See here for the guideline. El duderino ^(abides) 11:36, 31 January 2013 (UTC)

If the function is of the form $x^{\alpha }(1-x)^{\beta }$ then the integral is a beta function. So write your function as a linear combination of such functions, $f(x)=\sum _{i}k_{i}x^{\alpha _{i}}(1-x)^{\beta _{i}}$ . Bo Jacoby (talk) 12:41, 31 January 2013 (UTC).

What if $f$ can't be written as a linear combo of those functions? Even if you allow an infinite sum, I'm pretty sure there are functions satisfying the conditions above which are not linear combinations of such functions.--AnalysisAlgebra (talk) 12:47, 31 January 2013 (UTC)

I doubt you'll find any reasonable closed form for the general case. The limit however should be quite easy. -- Meni Rosenfeld (talk) 14:33, 31 January 2013 (UTC)

If the function f is continuous, then the Stone–Weierstrass theorem tells you that it can be approximated by polynomials. So don't be pretty sure that it can't be done. Bo Jacoby (talk) 08:27, 1 February 2013 (UTC).

So, I was right, then. --AnalysisAlgebra (talk) 09:27, 1 February 2013 (UTC)

No, you were wrong. Bo Jacoby (talk) 13:01, 1 February 2013 (UTC).

If f is sufficiently differentiable at 1/2, the asymptotics of $\int _{0}^{1}x^{n/2}(1-x)^{n/2}f(x)dx$ should only depend on the even derivatives of f at 1/2, and probably only the first nonzero (even) derivative. — Arthur Rubin (talk) 17:32, 3 February 2013 (UTC)

When put that way, the result is obvious. If f is differentiable at 1/2, and integrable over (0,1), then:

$\lim _{n->\infty }{\frac {\int _{0}^{1}x^{n/2}(1-x)^{n/2}f(x)dx}{\int _{0}^{1}x^{n/2}(1-x)^{n/2}dx}}=f\left({\tfrac {1}{2}}\right)$

— Arthur Rubin (talk) 17:48, 3 February 2013 (UTC)