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Calculus Level 5

$\large \dfrac{\pi^3}{\displaystyle \int_0^1\int_0^1\int_0^1\dfrac{dx\,dy\,dz}{1 + x^2\,y^2\,z^2}\;}=?$

The answer is 32.

1 solution

First Last
Feb 14, 2018

$\displaystyle\int_0^1\int_0^1\int_0^1\frac{dxdydz}{1+x^2y^2z^2}=\int_0^1\int_0^1\int_0^1\sum_{n=0}^\infty (-1)^n(xyz)^{2n}dxdydz=$

$\displaystyle\sum_{n=0}^\infty (-1)^n\int_0^1\int_0^1\int_0^1(xyz)^{2n}dxdydz=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^3}$

Now this is the Dirichlet Beta Function: $\displaystyle\beta (s)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}\quad\boxed{\beta (3)=\frac{\pi^3}{32}}$

Could you also explain how $\beta(3)$ is calculated?

Digvijay Singh - 3 years, 3 months ago

I had only looked it up. Can you direct me to a proof?

First Last - 3 years, 3 months ago

I couldn't find a proof either.

Digvijay Singh - 3 years, 3 months ago

@Digvijay Singh – You could use the trilogarithm function. The summation is the immaginary part of (Li(i)-Li(-i))/2. (Here Li denotes the trilogarithm function) And we have the functional equation Li(-z)-Li(-1/z)= (-1/6)(ln(z))^3 -1/6(pi^2)(ln(z)) Now substitute z= -i to get the value. And make sure to evaluate only the principal value of ln(i) otherwise it leads to infinite solutions.

Arghyadeep Chatterjee - 7 months ago

∫ \int ∫ um

The answer is 32.

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