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I'm wondering if there is a way to represent a fractional power of an integral as a function of the integrand in the following sense: The binomial theorem for complex/fractional exponents represents a power of a simple sum as an infinite sum of powers of the elements. Is there a similar procedure for powers of integrals? Obviously, for integer powers of an integral you have something like \begin{equation} \left(\int \mathrm{d}x f(x)\right)^{n}=\prod_{i=1}^{n} \int \mathrm{d}x_{i} \,f(x_{i})\end{equation} which is a more or less analogous process. Is there an extension to complex powers? I suspect it may take the form of an iterated integral of sorts (or multiplicative product integrals), but I can't think of a good way to show this. I'm coming from a physics background, so I'm sorry this description isn't as clear and rigorous as it could probably be, and I would appreciate it if any responses could be written in terms a mere physics student could understand.

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