## quantum (2008)

### BibTeX

@MISC{Coffey08quantum,

author = {Mark W. Coffey},

title = {quantum},

year = {2008}

}

### OpenURL

### Abstract

evaluation of a ln tan integral arising in

### Citations

327 |
Polylogarithms and associated functions
- Lewin
- 1981
(Show Context)
Citation Context .... These other integrals permit us to explicitly write other conjectures directly in terms of linear combinations of specific Clausen function values. The Clausen function Cl2 can be defined by (e.g., =-=[14, 16]-=-) ∫ θ Cl2(θ) ≡ − 0 t ln ∣2 sin ∣dt = 2 ∫ 1 0 tan −1 ∫ 1 ln x = − sin θ 0 x2 dx = − 2x cosθ + 1 ( x sin θ ) dx 1 − x cosθ x ∞∑ n=1 (12) sin(nθ) n2 . (13) When θ is a rational multiple of π it is known ... |

47 |
Series Associated with the Zeta and Related Functions
- Srivastava, Choi
- 2001
(Show Context)
Citation Context ... 0, (17) n−1 ∑ j=1 Cl2 ( 2π n j ) = 0. (18) In (17), pairwise cancellation occurs, as Cl2(θ) = −Cl2(2π − θ). Further information on the special functions that we employ may readily be found elsewhere =-=[13, 14, 16, 9]-=-. In particular, with or Li2(z) = ∞∑ k=1 ∫ z Li2(z) = − 0 the dilogarithm function, we have the relation zk k2, |z| ≤ 1, (19) ln(1 − t) dt, (20) t Li2(e iθ ) = π2 6 − 1 4 θ(2π − θ) + iCl2(θ), 0 ≤ θ ≤ ... |

42 | Massive 3-loop Feynman diagrams reducible to SC ∗ primitives of algebras of the sixth root of unity, Eur
- Broadhurst
- 1999
(Show Context)
Citation Context ...s in a number of contexts and has received significant attention in the last several years [3, 4, 5, 6]. This and related integrals arise in hyperbolic geometry, knot theory, and quantum field theory =-=[6, 7, 8]-=-. Very recently [9] we obtained an explicit evaluation of (1) in terms of the specific Clausen function Cl2. However, much work remains. This is due to the conjectured relation between a Dirichlet L s... |

18 | Experimental mathematics: examples, methods and implications.” Notices of the American Mathematical Society 52.5 (2005):502-514. http://9003-sfx.calstate.edu.opac.sfsu.edu/sfsu?sid=AMS%3AMathSciNet&atitle=Experimental%20mathematics%3A%20examples%2C%20meth
- Bailey
- 2011
(Show Context)
Citation Context ...B30, 11M35, 11M06 1The particular integral I7 ≡ 24 7 √ ∫ π/2 7 π/3 tant + ln ∣ √ 7 tant − √ dt, (1) 7∣ occurs in a number of contexts and has received significant attention in the last several years =-=[3, 4, 5, 6]-=-. This and related integrals arise in hyperbolic geometry, knot theory, and quantum field theory [6, 7, 8]. Very recently [9] we obtained an explicit evaluation of (1) in terms of the specific Clausen... |

18 |
Structural Properties of Polylogarithms
- Lewin, Ed
- 1991
(Show Context)
Citation Context .... These other integrals permit us to explicitly write other conjectures directly in terms of linear combinations of specific Clausen function values. The Clausen function Cl2 can be defined by (e.g., =-=[14, 16]-=-) ∫ θ Cl2(θ) ≡ − 0 t ln ∣2 sin ∣dt = 2 ∫ 1 0 tan −1 ∫ 1 ln x = − sin θ 0 x2 dx = − 2x cosθ + 1 ( x sin θ ) dx 1 − x cosθ x ∞∑ n=1 (12) sin(nθ) n2 . (13) When θ is a rational multiple of π it is known ... |

15 |
The Dilogarithm function in Geometry and Number Theory. Number Theory and related topics
- Zagier
- 1988
(Show Context)
Citation Context ...r evaluation of I7, based upon a property of the Bloch-Wigner form of the dilogarithm function. We recall that the L series L−7(s) has occurred in several places before, including hyperbolic geometry =-=[19]-=- and Dedekind sums of analytic number theory [2]. Let ζ Q( √ −p) denote the Dedekind zeta function of an imaginary quadratic field Q( √ −p). 1 (7n + 6) 2 ] . (2) ( )] 6π , 7 (3) 2Then indeed we have ... |

11 | Asymptotic formulas and generalized Dedekind sums
- Almkvist
(Show Context)
Citation Context ...Bloch-Wigner form of the dilogarithm function. We recall that the L series L−7(s) has occurred in several places before, including hyperbolic geometry [19] and Dedekind sums of analytic number theory =-=[2]-=-. Let ζ Q( √ −p) denote the Dedekind zeta function of an imaginary quadratic field Q( √ −p). 1 (7n + 6) 2 ] . (2) ( )] 6π , 7 (3) 2Then indeed we have [2, 19, 20] ζ Q( √ −7)(s) = 1 2 ∑ m,n∈Z (m,n)̸=(... |

7 | Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links,” [arXiv:hep-th/9811173
- Borwein, Broadhurst
- 1998
(Show Context)
Citation Context ...B30, 11M35, 11M06 1The particular integral I7 ≡ 24 7 √ ∫ π/2 7 π/3 tant + ln ∣ √ 7 tant − √ dt, (1) 7∣ occurs in a number of contexts and has received significant attention in the last several years =-=[3, 4, 5, 6]-=-. This and related integrals arise in hyperbolic geometry, knot theory, and quantum field theory [6, 7, 8]. Very recently [9] we obtained an explicit evaluation of (1) in terms of the specific Clausen... |

7 | Alternative evaluation of a ln tan integral arising in quantum field theory
- Coffey
(Show Context)
Citation Context ... and has received significant attention in the last several years [3, 4, 5, 6]. This and related integrals arise in hyperbolic geometry, knot theory, and quantum field theory [6, 7, 8]. Very recently =-=[9]-=- we obtained an explicit evaluation of (1) in terms of the specific Clausen function Cl2. However, much work remains. This is due to the conjectured relation between a Dirichlet L series and I7 [6], I... |

7 |
Robertson Some Properties of Dirichlet L-series
- Zucker, M
- 1976
(Show Context)
Citation Context ... and Dedekind sums of analytic number theory [2]. Let ζ Q( √ −p) denote the Dedekind zeta function of an imaginary quadratic field Q( √ −p). 1 (7n + 6) 2 ] . (2) ( )] 6π , 7 (3) 2Then indeed we have =-=[2, 19, 20]-=- ζ Q( √ −7)(s) = 1 2 ∑ m,n∈Z (m,n)̸=(0,0) 1 (m 2 + mn + 2n 2 ) s (4) where ( ) ν 7 = ζ(s)L−7(s) = ζ(s)7 −s 6∑ ν=1 ( ) ( ν ζ 7 s, ν 7 ) , (5) is a Legendre symbol, ζ(s, a) is the Hurwitz zeta function,... |

7 |
Solving differential equations for 3-loop diagrams: Relation to hyperbolic geometry and knot theory,” arXiv:hep-th/9806174
- Broadhurst
(Show Context)
Citation Context ...s in a number of contexts and has received significant attention in the last several years [2, 3, 4, 5]. This and related integrals arise in hyperbolic geometry, knot theory, and quantum field theory =-=[5, 6, 7]-=-. Very recently [8] we obtained an explicit evaluation of (1) in terms of the specific Clausen function Cl2. However, much work remains. This is due to the conjectured relation between a Dirichlet L s... |

6 |
Computer-assisted discovery and proof
- Bailey, Borwein
- 2008
(Show Context)
Citation Context ...B30, 11M35, 11M06 1The particular integral I7 ≡ 24 7 √ ∫ π/2 7 π/3 tant + ln ∣ √ 7 tant − √ dt, (1) 7∣ occurs in a number of contexts and has received significant attention in the last several years =-=[3, 4, 5, 6]-=-. This and related integrals arise in hyperbolic geometry, knot theory, and quantum field theory [6, 7, 8]. Very recently [9] we obtained an explicit evaluation of (1) in terms of the specific Clausen... |

5 |
Formulae concerning the computation of the Clausen integral
- Grosjean
- 1984
(Show Context)
Citation Context ...0 x2 dx = − 2x cosθ + 1 ( x sin θ ) dx 1 − x cosθ x ∞∑ n=1 (12) sin(nθ) n2 . (13) When θ is a rational multiple of π it is known that Cl2(θ) may be written in terms of the trigamma and sine functions =-=[11, 13]-=-. This Clausen function is odd and periodic, Cl2(θ) = −Cl2(−θ), and Cl2(θ) = Cl2(θ + 2π). It also satisfies the duplication triplication and quadriplication 1 2 Cl2(2θ) = Cl2(θ) − Cl2(π − θ), (14) 1 3... |

2 |
et al., Experimental Mathematics in Action
- Bailey
- 2007
(Show Context)
Citation Context |

2 | On a three-dimensional symmetric Ising tetrahedron, and contributions to the theory of the dilogarithm and Clausen functions
- Coffey
- 2008
(Show Context)
Citation Context ...Γ(1 − s)Γ(s) = π/ sin(πs), this functional equation may also be written in the form L−k(1 − s) = 2(2π) −s k s−1/2 sin ( ) πs Γ(s)L−k(s). (7) 2 Integral representations are known for these L-functions =-=[20, 10]-=-. From the functional equation (6) we find ∂ ∂s L−k(s) ∣ s=−1 = k3/2 4π L−k(2). (8) In turn, we have where we used ζ(−1) = −1/12 and L−k(−1) = 0. ζ ′ Q( √ −k) (−1) = −k3/2 48π L−k(2), (9) 3We have Pr... |

2 |
de Doelder, On the Clausen integral Cl2(θ) and a related integral
- J
- 1984
(Show Context)
Citation Context ...0 x2 dx = − 2x cosθ + 1 ( x sin θ ) dx 1 − x cosθ x ∞∑ n=1 (12) sin(nθ) n2 . (13) When θ is a rational multiple of π it is known that Cl2(θ) may be written in terms of the trigamma and sine functions =-=[11, 13]-=-. This Clausen function is odd and periodic, Cl2(θ) = −Cl2(−θ), and Cl2(θ) = Cl2(θ + 2π). It also satisfies the duplication triplication and quadriplication 1 2 Cl2(2θ) = Cl2(θ) − Cl2(π − θ), (14) 1 3... |

1 | Apparently in (11.22) in [16], the lower limit of the integral for D(z) is intended to be 0, consistent with our - Srivastava, Choi |