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I've just read about geometric integrals and noticed a possible connection between geometric integrals and geometric means. I also know that integrals can be expressed as arithmetic means. But is there a type of integration that may be expressed as a harmonic mean? I tried googling it but the only thing I found was some convoluted old paper that seems to review an older paper by Hodge.

My question is two-fold. First, what is the definition of the Harmonic integral and what functions may it operate on. Second, is it expressible in terms of the standard arithmetic integral?

Edit: The geometric integral I'm thinking of is defined as: $$\prod^a_bf(x)^{dx}=lim_{\Delta x\rightarrow0}\prod f(x_i)^{\Delta x}$$ Wikipedia lists it as a type 1 product integral.

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  • $\begingroup$ @ThomasAndrews sorry, will edit. $\endgroup$
    – tox123
    Aug 27, 2017 at 21:28

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Let's attempt to construct a normal integral from it's arithmetic mean counterpart, and then follow the same process for constructing a harmonic integral.

An arithmetic mean is defined as follows. The arithmetic mean of a sequence $a_1,a_2,...,a_n$ is defined by $$\frac{1}{n}\sum_{i=1}^n a_i$$ The arithmetic mean of a function is then $$\int_0^1 f(x)dx = \lim_{n \rightarrow \infty} \frac{1}{n}\sum_{i=1}^nf\Big(\frac{i}{n}\Big)$$

Now for the harmonic integral we have the harmonic mean of a sequence $a_1,a_2,...,a_n$ is defined as $$\bigg(\frac{1}{n}\sum_{i=1}^{n}a_i^{-1}\bigg)^{-1}$$ So we could define a sort of harmonic integral as follows $$\mathfrak{H}_0^1 f(x) = \bigg(\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{i=1}^nf\Big(\frac{i}{n}\Big)^{-1}\bigg)^{-1}$$ Now we can see a remnant of the normal integral here, so we can define this in terms of a normal integral. $$\mathfrak{H} f(x)=\bigg(\int f(x)^{-1}dx\bigg)^{-1}$$

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  • $\begingroup$ Thanks! Do you know of any papers that discuss this integral, or are you just extrapolating? $\endgroup$
    – tox123
    Aug 28, 2017 at 0:14
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    $\begingroup$ I am just extrapolating. I know of no papers discussing this, nor could I find any on the internet albeit it is sometimes difficult to search mathematics on the internet. $\endgroup$ Aug 28, 2017 at 0:19

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