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Can the equality

$$\dfrac{d^{n}}{ds^{n}}\Big[s^{n-1}\ln\Big(\pi^{-s/2}\Gamma\Big(1+\frac{s}{2}\Big)\Big)\Big]\Bigg|_{s=1} = \dfrac{d^{n}}{ds^{n}}\Big[s^{n-1}\ln\Big(s\pi^{-s/2}\Gamma\Big(\frac{s}{2}\Big)\Big)\Big]\Bigg|_{s=1}$$

hold for any positive real number $n$, where $\Gamma(x)$ is the usual gamma function in number theory and analysis ?

PS: This is a follow up to my earlier question: On the equality of derivatives of two functions.

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Q_p
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  • Can it? Can you even make sense out of it when $n=0.5$? Not even WolframAlpha (and probably Mathematica) can understand what $\frac{d^{1/2}}{ds^{1/2}}$ means. For example, is the following true? $$x{^?=^?}y\color{white}{\qquad \ddot\smile\qquad x=5,y=6}$$ if I don't tell you what $x$ and $y$ are? – Simply Beautiful Art Sep 29 '16 at 21:16
  • if $f(s)$ is analytic (or simply differentiable) at $s=1$ and $n \in \mathbb{N}$ then $\frac{d^n}{ds^n}[s^{n-1} f(s)]_{s=1} = f'(1)$ – reuns Sep 30 '16 at 01:51
  • @SimpleArt, what exactly is not clear to you ? – Q_p Sep 30 '16 at 07:23
  • @user43208, the MO post includes the Riemann zeta function and a minus sign, hence i don't see how the two posts are similar. – Q_p Sep 30 '16 at 07:25
  • @Isaac. My apologies. – user43208 Sep 30 '16 at 12:22
  • It holds for all positive integers $n$. Does your "for any" mean "there exists (a positive real number $n$ such that …)" or "for all (positive real numbers $n$ …)"? – Daniel Fischer Sep 30 '16 at 13:19
  • @Isaac. What is not clear to you is what I'm asking. For example, there are different definitions to a fractional derivative. Try telling my what any side of the equation means for $n=0.5$ before I can approach this problem. Also note that the constant of integration can come into play for positive non-integer derivatives, and discrepancies can indeed show up for the much much more simple case of $D^ne^x$ using different definitions. Here is a basic guide to fractional calculus you may need – Simply Beautiful Art Sep 30 '16 at 20:21

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