Analogue of a Fock-type integral arising from electromagnetism and its applications in number theory
- PDF / 606,124 Bytes
- 33 Pages / 595.276 x 790.866 pts Page_size
- 84 Downloads / 153 Views
RESEARCH
Analogue of a Fock-type integral arising from electromagnetism and its applications in number theory Atul Dixit1* and Arindam Roy2 * Correspondence:
[email protected] Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat 382355, India Full list of author information is available at the end of the article 1
Abstract Closed-form evaluations of certain integrals of J0 (ξ ), the Bessel function of the first kind, have been crucial in the studies on the electromagnetic field of alternating current in a circuit with two groundings, as can be seen from the works of Fock and Bursian, Schermann, etc. Koshliakov’s generalization of one such integral, which contains Js (ξ ) in the integrand, encompasses several important integrals in the literature including Sonine’s integral. Here, we derive an analogous integral identity where Js (ξ ) is replaced by a kernel consisting of a combination of Js (ξ ), Ks (ξ ) and Ys (ξ ). This kernel is important in number theory because of its role in the Voronoï summation formula for the sum-of-divisors function σs (n). Using this identity and the Voronoï summation formula, we derive a general transformation relating infinite series of products of Bessel functions Iλ (ξ ) and Kλ (ξ ) with those involving the Gaussian hypergeometric function. As applications of this transformation, several important results are derived, including what we believe to be a corrected version of the first identity found on page 336 of Ramanujan’s Lost Notebook. Keywords: Bessel functions, Generalized sum-of-divisors function, Voronoï summation formula, Analytic continuation Mathematics Subject Classification: Primary 11M06, 33E20; Secondary 33C10
1 Introduction and main results In his famous memoir on the propagation of waves in wireless telegraphy, Sommerfeld [36] developed a method of integral representation for calculating the electromagnetic field on a flat boundary where the solution sought is expressed in terms of a Fourier integral consisting of Bessel functions. The Sommerfeld integral, given by [33, p. 366]
∞ 0
√ √ 2 2 2 2 e−a t +ξ e−ξ a +ρ tJ0 (ρt) dt = , t2 + ξ 2 a2 + ρ 2
(1.1)
is valid for Re(a) > |Im(ρ)| and Re(ξ ) > 0 and is actually the special case s = 0, ν = 1/2 of [44, p. 416, Equation (2)] (see also [21, p. 693, Formula 6.596.7])
123
© Springer Nature Switzerland AG 2020.
0123456789().,–: volV
25
A. Dixit, A. Roy Res Math Sci (2020)7:25
Page 2 of 33
∞
t s+1 Js (ρt)
Kν a t 2 + ξ 2
0
(t 2 + ξ 2 )ν/2
ρs dt = ν a
ν−s−1 a2 + ρ 2 Kν−s−1 ξ a2 + ρ 2 , ξ (1.2)
where1 Re(a) > |Im(ρ)|, | arg ρ| < π, Re(s) > −1, ν ∈ C and Re(ξ ) > 0. Here, Js (ξ ), the Bessel function of the first kind of order s is defined by [44, p. 40]
Js (ξ ) :=
∞ (−1)m (ξ /2)2m+s m!(m + 1 + s)
(ξ , s ∈ C),
(1.3)
m=0
and Ks (ξ ) is the modified Bessel function of the second kind of order s defined by [44, p. 78, eq. (6)], Ks (ξ ) :=
π (I−s (ξ ) − Is (ξ )) , 2 sin πs
where Is (ξ ) is the modified Bessel function of the first kind of order
Data Loading...
