Airy Zeta Function
المؤلف:
Borwein, J.; Bailey, D.; and Girgensohn, R
المصدر:
Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters
الجزء والصفحة:
...
13-8-2018
1862
Airy Zeta Function
Define the Airy zeta function for 
, 3, ... by
 |
(1)
|
where the sum is over the real (negative) zeros 
of the Airy function 
. This has the closed-form representation
 |
(2)
|
where 
is the gamma function,
![T_n(z)=C^((n))(z)A+(d^(n-1))/(dz^(n-1))[Ai(z)Bi(z)]
-sum_(j=1)^n(n; j)C^((n-j))(z)(d^(j-1))/(dz^(j-1))[Ai(z)]^2,](http://mathworld.wolfram.com/images/equations/AiryZetaFunction/NumberedEquation3.gif) |
(3)
|
where
and
 |
(6)
|
(Crandall 1996; Borwein et al. 2004, p. 61).
Surprisingly, defining
gives 
as a polynomial in 
(Borwein et al. 2004, pp. 61-62). The first few such polynomials are
(OEIS A096631 and A096632). The corresponding numerical values are approximately 0.531457, 
, 0.0394431,
, and 0.00638927, ....
REFERENCES:
Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 61-62, 2004.
Crandall, R. E. "On the Quantum Zeta Function." J. Phys. A: Math. General 29, 6795-6816, 1996.
Sloane, N. J. A. Sequences A096631 and A096632 in "The On-Line Encyclopedia of Integer Sequences."
الاكثر قراءة في التفاضل و التكامل
اخر الاخبار
اخبار العتبة العباسية المقدسة
