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Everybody knows that $\Gamma(z) = (z-1)!$. Seeing as factorials can be analytically continued to all complex numbers (with a few exceptions), can a similar thing be said for product operators such as $$\prod_{n=1}^x e^{\frac{1}{n}}$$ Because I'm trying to convert that function to a continuous form such that I can take its derivative with respect to x.

Alexandra
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  • $\prod_{n=1}^x e^{\frac{1}{n}} = \exp \left( \sum_{n=1}^x \frac 1n \right)$, so your question is equivalent to this: https://math.stackexchange.com/q/3058500/42969 – Martin R Jan 03 '24 at 13:40
  • @MartinR Your answer works, but are there other ways? I cannot explicitly use the Euler-Mascheroni constant in my work, which means no digamma functions – Alexandra Jan 03 '24 at 14:10
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    There are various representations in that Q&A, for example $H_{z} = \int_0^1 \frac{1-x^{z}}{1-x},dx $. See also https://en.wikipedia.org/wiki/Harmonic_number#Harmonic_numbers_for_real_and_complex_values – Martin R Jan 03 '24 at 14:14

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