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It is widely known that $0^0$ is usually defined to be 1. I wonder why we cannot employ a similar technique to ascribe values to functions having poles in a point.

Now take the Gamma function. The real part $\Re(\Gamma(i x))$ is equal to $\gamma$ (Euler-Masceroni constant). We thus can use the formulas:

$$\Gamma(-n)=\lim_{h\to0} \Re (\Gamma(-n+ih))$$

or, alternatively,

$$\Gamma(-n)=\lim_{h\to0} \frac{\Gamma(-n+h)+\Gamma(-n-h)}2$$

Thus we can ascribe natural values to Gamma function at negative integers: $$\Gamma(0)=-\gamma$$ $$\Gamma(-1)=\gamma-1$$ $$\Gamma(-2)=\frac{3}{4}-\frac{\gamma }{2}$$ $$\Gamma(-3)=\frac{\gamma }{6}-\frac{11}{36}$$ $$\Gamma(-4)=\frac{25}{288}-\frac{\gamma }{24}$$

etc.

I wonder why these values are not given in tables and not used in computer algebra systems, this could symplify things a lot.

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  • $\begingroup$ Your suggestions are not consistent with $\Gamma(t+1)=t\Gamma(t)$, which is a key property of the Gamma function $\endgroup$ – Henry Oct 31 '14 at 8:07
  • $\begingroup$ @Henry good point indeed... But on the other hand they are consistent with Fourier differintegral: $$(t^n)^{(s)}|_{t=1}=\frac{1}{2\pi}\int_{-\infty}^{+\infty} \frac{e^{- i \omega }}{(-i\omega)^s} \int_{-\infty}^{+\infty}t^n e^{i\omega t}dt \, d\omega=\Gamma(s+1)(-1)^s t^{n-s}$$ $\endgroup$ – Anixx Oct 31 '14 at 8:27
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    $\begingroup$ Thanks ! This is very interesting. Cheers. $\endgroup$ – Claude Leibovici Oct 31 '14 at 8:57
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    $\begingroup$ I really appreciate ! Very nice question and solution ! Any idea about the pattern ? Cheers :-) $\endgroup$ – Claude Leibovici Oct 31 '14 at 9:58
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    $\begingroup$ this could simplify things a lot - What things ? Applications of the $\Gamma$ and beta functions $($which are generalizations of factorials and binomial coefficients$)$ require those values to be infinite. Replacing them with something else will simply yield false results. Which, of course, is not to say that at other times, one might not need to compute their normalized values. Furthermore, there are alternative ways of ascribing finite values to the factorials of negative integers. $\endgroup$ – Lucian Oct 31 '14 at 16:40

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