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This question is from this following derivation from pages 204-205 of PDE Evans, 2nd edition. This is from the same proof of Example 9 on those pages of the textbook, and I asked a question about that as well (see Compact support).

Now $n=2k+1$ and $\alpha(n) = \frac{\pi^{n/2}}{\Gamma(\frac n2+1)} = \frac{\pi^{k+\frac 12}}{\Gamma(\frac n2+1)}$. Since $\Gamma\left(\frac 12\right)=\pi^{1/2}$ and $\Gamma(x+1)=x\Gamma(x)$ for $x > 0$, we can compute $$\frac{n \alpha(n)}{\pi^k 2^{k+1}}=\frac{n\pi^{1/2}}{2^{k+1}\Gamma(\frac n2+1)}\color{blue}{=\frac{1}{(n-2)(n-4)\cdots5\cdot3}=\frac{1}{\gamma_n}}$$

May I ask how were the last two equalities obtained (highlighted in $\color{blue}{\textbf{blue}}$)? All I know so far was that the first equality is due to plugging in the expression of $\alpha(n) = \frac{\pi^{k+\frac 12}}{\Gamma(\frac n2+1)}$.

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1 Answer 1

up vote 4 down vote accepted

The functional equation $\Gamma(x+1) = x\Gamma(x)$ yields

$$\begin{align} \Gamma\left(\tfrac{n}{2}+1\right) &= \Gamma\left(\tfrac{n}{2}\right)\tfrac{n}{2}\\ &= \Gamma\left(\tfrac{n}{2}-1\right)\tfrac{n}{2}\tfrac{n-2}{2}\\ &= \Gamma\left(\tfrac{n}{2}-r\right)\prod_{m=0}^r\tfrac{n-2m}{2} \\ &= \Gamma\left(\tfrac{1}{2}\right)\cdot\prod_{m=0}^{k}\left(\tfrac{n}{2}-m\right)\\ &= \pi^{1/2}\prod_{m=0}^k\frac{n-2m}{2}\\ &= \frac{n\pi^{1/2}}{2^{k+1}}\prod_{m=1}^k(n-2m). \end{align}$$

Plug that in, and you have the first of the equalities. For the second, the definition of $\gamma_n$ would probably be helpful.

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Yeah, that second inequality turns out to just be a definition by the textbook author. From page 79 (in section 2.4, Wave Equation - solution for odd $n$) of PDE Evans, 2nd edition, $$\gamma_n=1\cdot 3 \cdot 5 \cdots(n-2).$$ –  glace Jul 3 at 21:15
Just apply the functional equation repeatedly, $$\Gamma\left(\tfrac{n}{2}+1\right) = \tfrac{n}{2}\Gamma\left(\tfrac{n}{2}\right) = \tfrac{n}{2}\tfrac{n-2}{2}\Gamma\left(\tfrac{n}{2}-1\right) = \tfrac{n}{2}\tfrac{n-2}{2}\tfrac{n-4}{2}\Gamma\left(\tfrac{n}{2}-2\right)$$ etc. –  Daniel Fischer Jul 3 at 21:54
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