Difference between revisions of "Airy zeta function"
From specialfunctionswiki
(→References) |
(→References) |
||
| Line 4: | Line 4: | ||
$$\zeta_{\mathrm{Ai}}(2) = \dfrac{3^{\frac{5}{3}}\Gamma^4(\frac{2}{3})}{4\pi^2}.$$ | $$\zeta_{\mathrm{Ai}}(2) = \dfrac{3^{\frac{5}{3}}\Gamma^4(\frac{2}{3})}{4\pi^2}.$$ | ||
=References= | =References= | ||
| − | [http://en.wikipedia.org/wiki/Airy_zeta_function Airy zeta function (Wikipedia)] | + | [http://en.wikipedia.org/wiki/Airy_zeta_function Airy zeta function (Wikipedia)]<br /> |
[http://mathworld.wolfram.com/AiryZetaFunction.html Airy zeta function (Mathworld)] | [http://mathworld.wolfram.com/AiryZetaFunction.html Airy zeta function (Mathworld)] | ||
Revision as of 02:06, 21 October 2014
The Airy function $\mathrm{Ai}$ is oscillatory for negative values of $x$. This yields a sequence of zeros $\{a_i\}_{i=1}^{\infty}$. We define the Airy zeta function using these zeros in the following way: $$\zeta_{\mathrm{Ai}}(z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{|a_k|^z}.$$ It can be shown that $$\zeta_{\mathrm{Ai}}(2) = \dfrac{3^{\frac{5}{3}}\Gamma^4(\frac{2}{3})}{4\pi^2}.$$
References
Airy zeta function (Wikipedia)
Airy zeta function (Mathworld)
