Vardi's Integral
Vardi's integral is the beautiful definite integral
| 👁 int_(pi/4)^(pi/2)lnlntanxdx | 👁 = | 👁 pi/2ln[sqrt(2pi)(Gamma(3/4))/(Gamma(1/4))] |
(1)
|
| 👁 Image | 👁 = | 👁 pi/4ln[(4pi^3)/(Gamma^4(1/4))] |
(2)
|
| 👁 Image | 👁 = | 👁 -0.2604428063009... |
(3)
|
![👁 pi/2ln[sqrt(2pi)(Gamma(3/4))/(Gamma(1/4))]](https://mathworld.wolfram.com/images/equations/VardisIntegral/Inline3.svg){kind=link}
![👁 pi/4ln[(4pi^3)/(Gamma^4(1/4))]](https://mathworld.wolfram.com/images/equations/VardisIntegral/Inline6.svg){kind=link}
(OEIS A115252; Gradshteyn and Ryzhik 1980, p. 532; Bailey et al. 2007, p. 160; Moll 2006), where 👁 Gamma(z)
is the gamma function.
Other examples similar to these include
| 👁 int_0^1lnln(1/x)(dx)/(1+x+x^2) | 👁 = | 👁 pi/(sqrt(3))ln[(2pi^(5/6))/(Gamma(1/6))] |
(4)
|
| 👁 int_0^1lnln(1/x)(dx)/(1-x+x^2) | 👁 = | 👁 pi/(sqrt(3))ln[(32pi^5)/(Gamma^6(1/6))] |
(5)
|
![👁 pi/(sqrt(3))ln[(2pi^(5/6))/(Gamma(1/6))]](https://mathworld.wolfram.com/images/equations/VardisIntegral/Inline13.svg){kind=link}
![👁 pi/(sqrt(3))ln[(32pi^5)/(Gamma^6(1/6))]](https://mathworld.wolfram.com/images/equations/VardisIntegral/Inline16.svg){kind=link}
(Vardi 1988; Gradshteyn and Ryzhik 1980, pp. 571-572).
See also
Definite IntegralExplore with Wolfram|Alpha
More things to try:
References
Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Bierens de Haan, D. Nouvelles Tables d'intégrales définies. Amsterdam, 1867.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 4th ed. San Diego, CA: Academic Press, 1980.Moll, V. H. "Some Questions in the Evaluation of Definite Integrals." MAA Short Course, San Antonio, TX. Jan. 2006. http://crd.lbl.gov/~dhbailey/expmath/maa-course/Moll-MAA.pdf.Sloane, N. J. A. Sequence A115252 in "The On-Line Encyclopedia of Integer Sequences."Vardi, I. "Integrals, an Introduction to Analytic Number Theory." Amer. Math. Monthly 95, 308-315, 1988.Referenced on Wolfram|Alpha
Vardi's IntegralCite this as:
Weisstein, Eric W. "Vardi's Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VardisIntegral.html