The Feynman integral in ℝ1 ⊕ ℝm and complex expansion of 2F1. Issue 7 (2nd July 2016)

• Record Type: Journal Article
• Authors: Shpot, Mykola A.; Pogány, Tibor K.
• Is Part Of: Integral transforms and special functions. Volume 27:Issue 7(2016)
• Page Start: 533
• Page End: 547
• Publication Date: 2016-07-02
• Subjects: Feynman integral -- Gauss hypergeometric function -- generalized hypergeometric function -- Appell functions -- Horn functions -- hypergeometric transformation formulae; 33C05 -- 33C20 -- 33C65 -- 40C10 -- 40H05 -- 81Q30; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2016.1159560
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: Closed-form expressions are proposed for the Feynman integral I D, m ( p, q ) = ∫ d m y ( 2 π ) m ∫ d D x ( 2 π ) D 1 ( x − p / 2 ) 2 + ( y − q / 2 ) 4 × 1 ( x + p / 2 ) 2 + ( y + q / 2 ) 4 over d = D + m dimensional space with ( x, y ), ( p, q ) ∈ R D ⊕ R m, in the special case D =1. We show that I 1, m ( p, q ) can be expressed in different forms involving real and imaginary parts of the complex variable Gauss hypergeometric function 2 F 1, as well as generalized hypergeometric 2 F 2 and 3 F 2, Horn H 4 and Appell F 2 functions. Several interesting relations are derived between the real and imaginary parts of 2 F 1 and the function H 4 .