Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$.

The problem is finding or approximating the following integral (The actual problem is more complex than this !) $$I(k) = \int\limits_{\mathbb{S}^7}\lvert(z_1z_4 - z_2z_3)^k\rvert,$$ where $\mathbb{S}^7$ is the $7$-sphere, and the integral is over its surface measure. The ultimate aim is to find or get an estimate of the ratios of the type $\displaystyle \frac{I(k)}{I(k+1)}$.

I have never done an integration over such a surface, the hypersphere and don't know where to begin. Any hint or any pointers would really help me to get started.

(I have been suggested to make use of the fact that the points $x_i$'s and $y_i$'s lie on a sphere; hence we can use it to reduce one variable and make use of Fubini's Theorem with proper limits for integration to proceed. But I can't see how to make use of this.)

What have I done till now?

As the problem say, it would be acceptable if we can approximate the ratios or the integral, so $$\lvert z_1z_4 - z_2z_3\rvert \, \leq \, \lvert z_1z_4\rvert + \lvert z_2z_3\rvert \, = \, \lvert z_1\rvert\lvert z_4\rvert + \lvert z_2\lvert\rvert z_3\rvert.$$ This converts everything into real variables, and then I have used Mathematica to calculate this. But the problem is Mathematica gives the solution for small values of $k$, but as I have to find it for the general case, it is not of much use.

Also, I have tried using the spherical coordinates to help with the integral, but they make it more complicated.

Any pointers on this front are also welcome.

Here are the values of $I(k)$ for $k=1,\,\cdots,\,10$.

$z_1 = x_1 + ix_2, \ \ z_2 = x_3+ix_4, \ \ z_3 = x_5+ix_6, \ \ z_4 = x_7+ix_8$ then

$$\mid z_1z_4 - z_2z_3| \ \ = \ \ \sqrt{\left(x_4 x_5+x_3 x_6-x_2 x_7-x_1 x_8\right){}^2+\left(x_3 x_5-x_4 x_6-x_1 x_7+x_2 x_8\right)^2} $$

To integrate over $\mathbb{S}^7$, I have used two methods, one is using `NIntegrate`

and the second using `integrateSphere`

function from HFT Software package.

(**Note** : HFT package doesn't work for odd values of $k$, but for even values of $k$, the results match perfectly with that of `INtegrate`

)

$$\begin{align} I(1) \ \ &\approx \ \ 9.54903\\ I(2) \ \ &= \ \ \frac{\pi^4}{3\times 10} \ \ \approx \ \ 3.24697\\\\ I(3) \ \ &\approx \ \ 1.19691\\ I(4) \ \ &= \ \ \frac{\pi^4}{3\times70} \ \ \approx \ \ 0.463853\\\\ I(5) \ \ &\approx \ \ 0.188085\\ I(6) \ \ &= \ \ \frac{\pi^4}{3\times420} \ \ \approx \ \ 0.0773088\\\\ I(7) \ \ &\approx \ \ 0.0329181\\ I(8) \ \ &= \ \ \frac{\pi^4}{3\times2310} \ \ \approx \ \ 0.0140561\\\\ I(9) \ \ &\approx \ \ 0.00620447\\ I(10) \ \ &= \ \ \frac{\pi^4}{3\times12012} \ \ \approx \ \ 0.0027031\\\\ \end{align}$$

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