By Larry

2019-01-11 23:26:40 8 Comments

Through some calculation, I found that for all $r>0$ $$\int_{0}^{1}((1-x^r)^{1/r}-x)^2dx=\frac{1}{3}$$ $$\int_{0}^{1}((1-x^r)^{1/r}-x)^4dx=\frac{1}{5}$$ $$\int_{0}^{1}((1-x^r)^{1/r}-x)^6dx=\frac{1}{7}$$

It seems like for {$n=2k,~r>0~|~k∈ℕ$}


I want to prove this general form.

Someone suggested to make the substitution $$y=(1-x^r)^{1/r}$$ So I rewrote the integral into $$\int_{0}^{1}((1-x^r)^{1/r}-x)^ndx=\int_{0}^{1}(y-x)^ndx$$ and tried to use the binomial formula: $$(y-x)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}(-1)^{n-k}x^{n-k}y^{k}$$

The integral then becomes $$\begin{align} \int_{0}^{1}(y-x)^ndx&=\int_{0}^{1}\sum _{k=0}^{n}{\binom {n}{k}}(-1)^{n-k}x^{n-k}y^{k}dx\\ &=\sum _{k=0}^{n}{\binom {n}{k}}(-1)^{n-k}\int_{0}^{1}x^{n-k}y^{k}dx\\ &=\sum _{k=0}^{n}{\binom {n}{k}}(-1)^{n-k}\int_{0}^{1}x^{n-k}(1-x^r)^{k/r}dx\\ \end{align}$$ Now I think I need to use Beta function: $$B(x,y) = \frac{(x-1)!(y-1)!}{(x+y-1)!}= \int_{0}^{1}u^{x-1}(1-u)^{y-1}du=\sum_{n=0}^{\infty}\frac{{\binom{n-y}{n}}}{x+n}$$ Am I on the right track? Are here any easier ways to prove the general form?


@ablmf 2019-01-12 11:03:22

Here's a proof with Hypergeometirc function.

We have $$ \underset{j=1}{\overset{2 n+1}{\sum }} \left( \begin{array}{c} 2 n \\ j-1 \\ \end{array} \right) (-x)^{j-1} \left(\left(1-x^r\right)^{1/r}\right)^{-j+2 n+1} =\left(\left(1-x^r\right)^{1/r}-x\right)^{2 n} $$ by binomial expansion.

It is easy to verify that $$ \left( \begin{array}{c} 2 n \\ j-1 \\ \end{array} \right) (-x)^{j-1} \left(\left(1-x^r\right)^{1/r}\right)^{-j+2 n+1} = \frac{\mathrm d}{\mathrm d x}\left( \frac{1}{2 n+1} (-1)^{j+1} x^j \binom{2 n+1}{j} \, _2F_1\left(\frac{j}{r},-\frac{-j+2 n+1}{r};\frac{j}{r}+1;x^r\right) \right) $$ Therefore, we have $$ \int((1-x^r)^{1/r}-x)^{2 n} \mathrm dx = \sum _{j=1}^{2 n+1} \frac{1}{2 n+1} (-1)^{j+1} x^j \binom{2 n+1}{j} \, _2F_1\left(\frac{j}{r},-\frac{-j+2 n+1}{r};\frac{j}{r}+1;x^r\right). $$ When $j=2n+1$, the summand in the right hand equals $\frac{x^{2 n+1}}{2 n+1}$. This is the term which gives us $\frac 1 {2n+1}$.

@Sangchul Lee 2019-01-12 00:53:59

Let $f : [0, 1] \to \mathbb{R}$ be continuous. (This restriction is not a essential, and in fact, it can be relaxed to Lebesgue integrability.)

Define $\varphi : [0, 1] \to [0, 1]$ by $\varphi(x) = (1-x^r)^{1/r}$ for $r > 0$ and notice that $\varphi$ is an involution, i.e., $\varphi^{-1} = \varphi$. Now by the substitution $y = \varphi(x)$, or equivalently, $x = \varphi(y)$,

$$ I := \int_{0}^{1} f( |x - \varphi(x)| ) \, \mathrm{d}x = -\int_{0}^{1} f( |\varphi(y) - y| ) \varphi'(y) \, \mathrm{d}y. $$

Summing two integrals,

\begin{align*} 2I &= \int_{0}^{1} f( |x - \varphi(x)| ) (1 - \varphi'(x)) \, \mathrm{d}x \\ &= \int_{-1}^{1} f( |u| ) \, \mathrm{d}u, \tag{$u = x - \varphi(x)$} \\ &= 2\int_{0}^{1} f(u) \, \mathrm{d}u. \end{align*}


$$ I = \int_{0}^{1} f(u) \, \mathrm{d}u. $$

Now plug $f(x) = x^n$ for positive even $n$ and use the fact that $|x - \varphi(x)|^n = (\varphi(x) - x)^n$.

EDIT. Here is an anternative solution. Write $p = 1/r$. Then using the substitution $x = u^p$,

\begin{align*} \int_{0}^{1} \left( (1 - x^r)^{1/r} - x \right)^n \, \mathrm{d}x &= \int_{0}^{1} \left( (1 - u)^{p} - u^p \right)^{n} pu^{p-1} \, \mathrm{d}u \\ &= \sum_{k=0}^{n} (-1)^k \binom{n}{k} p \int_{0}^{1} (1-u)^{p(n-k)} u^{p(k+1)-1} \, \mathrm{d}u \\ &= \sum_{k=0}^{n} (-1)^k \binom{n}{k} p \cdot \frac{(p(n-k))!(p(k+1)-1)!}{(p(n+1))!} \end{align*}

Here, $s! = \Gamma(s+1)$. Now define $a_k = (pk)!/k!$. Then the above sum simplifies to

\begin{align*} \int_{0}^{1} \left( (1 - x^r)^{1/r} - x \right)^n \, \mathrm{d}x &= \frac{1}{(n+1)a_{n+1}} \sum_{k=0}^{n} (-1)^k a_{n-k}a_{k+1} \\ &= \frac{1}{(n+1)a_{n+1}} \left( a_0 a_{n+1} + \sum_{k=0}^{n-1} (-1)^k a_{n-k}a_{k+1} \right). \end{align*}

So it suffices to show that $\sum_{k=0}^{n-1} (-1)^k a_{n-k}a_{k+1} = 0$. But by the substitution $l = n-1-k$, we have

$$ \sum_{k=0}^{n-1} (-1)^k a_{n-k}a_{k+1} = - \sum_{l=0}^{n-1} (-1)^l a_{l+1}a_{n-l}. $$

(Here the parity of $n$ is used.) So the sum equals its negation, hence is zero as required.

@Zacky 2019-01-12 01:02:48

Beautiful! This looks like Glasser's Master theorem little brother :D

@user 2019-01-12 01:04:12

Was it necessary to start with absolute value in the argument of function?

@Sangchul Lee 2019-01-12 01:06:31

@user, It is kind of necessary, in the sense that $$ \int_{0}^{1} g\left(x-(1-x^r)^{1/r}\right) \, \mathrm{d}x = \int_{0}^{1} g(u) \, \mathrm{d}u $$ may fail if $g$ is not an even function on $[-1, 1]$.

@Sangchul Lee 2019-01-12 01:12:11

@Zacky, Both Glasser's master theorem and my answer deals with specific examples of measure-preserving transformations, hence the conclusion should look similar. Of course, the beauty of Glasser's result is that its proof is very elementary. (The result itself was known much prior to his paper.)

@clathratus 2019-01-12 00:59:24

Here's your Beta integral $$S=\int_0^1x^{n-k}(1-x^r)^{k/r}dx$$ Setting $w=x^r$, we see that $$S=\frac1r\int_0^1w^{\frac{n+1-k-r}r}(1-w)^{k/r}dw$$ $$S=\frac1r\int_0^1w^{\frac{n+1-k}r-1}(1-w)^{\frac{k+r}r-1}dw$$ $$S=\frac1r\mathrm{B}\bigg(\frac{n+1-k}r,\frac{k+r}r\bigg)$$ So $$I(r,n)=\int_0^1[(1-x^r)^{1/r}-x]^ndx$$ $$I(r,n)=\frac1r\sum_{k=0}^{n}(-1)^{n-k}{n\choose k}\frac{\Gamma(\frac{n+1-k}r)\Gamma(\frac{k+r}r)}{\Gamma(1+\frac{n+1}r)}$$ $$I(r,n)=\frac1{r}\frac{\Gamma(n+1)}{\Gamma(1+\frac{n+1}r)}\sum_{k=0}^{n}(-1)^{n-k}\frac{\Gamma(\frac{n+1-k}r)\Gamma(\frac{k+r}r)}{\Gamma(k+1)\Gamma(n-k+1)}$$

Which is a closed form

@user 2019-01-12 01:45:11

Try to find an error in your derivation as $I(r,n)=\frac{1}{n+1}$ for even $n$ and any $r$.

@clathratus 2019-01-12 01:57:23

@user Ah yes I forgot $n$ was even

@DavidG 2019-01-12 05:31:40

Nice approach. Seems almost obvious after reading it, but I was stuck when I first saw it (+1)

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