Integrals of inverse trigonometric and polylogarithmic functions
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Integrals of inverse trigonometric and polylogarithmic functions Anthony Sofo1 Received: 11 February 2019 / Accepted: 8 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we study the representation of integrals whose integrand involves the product of a polylogarithm and an inverse or inverse hyperbolic trigonometric function. We further demonstrate many connections between these integrals and Euler sums. We develop recurrence relations and give some examples of these integrals in terms of Riemann zeta values, Dirichlet values and other special functions. Keywords Polylogarithm function · Recurrence relations · Euler sums · Zeta functions · Dirichlet functions Mathematics Subject Classification 11M06 · 11M32 · 33B15
1 Introduction In the paper [11], Mezo investigated the sequences of polylogarithmic integrals 1 In =
1 Lin (sin π x) dx and Jn =
0
Lin (cos π x) dx. 0
In the spirit of the Mezo study, we investigate families of integrals of the form, 1 J (a, δ, p, m) =
y m−1 f (y) Li p δ y 2am dy,
(1.1)
0
B 1
Anthony Sofo [email protected] Victoria University, P. O. Box 14428, Melbourne, VIC 8001, Australia
123
A. Sofo
where f (y) is the inverse of tangent or hyperbolic tangent evaluated at y m . The parameters satisfy the conditions a ∈ R+ , δ = ±1, p ∈ N, m ∈ R+ . It will be shown that these integrals are connected to sums of harmonic numbers, commonly known as Euler sums. Current symbolic software is unable to evaluate the integrals presented here. This work extends the results given in [8], where the author examined integrals with positive arguments of the polylogarithm. Devoto and Duke [4] list many identities of lower order polylogarithmic integrals and their relations to Euler sums. Classical sources on polylogarithm functions include [9,10]. We shall encounter a number of special functions throughout this paper and some useful definitions are listed below. For a more detailed list of special functions we refer the reader to the Digital Library of Mathematical Functions [12]. The Lerch transcendent (z, t, a) =
∞ m =0
zm (m + a)t
is defined for |z| < 1 and (a) > 0 and satisfies the recurrence (z, t, a) = z (z, t, a + 1) + a −t . The Lerch transcendent generalizes the Hurwitz zeta function at z = 1, (1, t, a) =
∞ m =0
1 (m + a)t
and the polylogarithm, or de-Jonquière’s function, when a = 1, L it (z) :=
∞ zm , t ∈ C when |z| < 1; (t) > 1 when |z| = 1. mt
m =1
Let Hn =
n 1 r =1
r
1
= 0
∞
1 − tn n ddt = γ + ψ (n + 1) = , 1−t j ( j + n)
H0 := 0
j=1
(m)
be the nth harmonic number, where γ denotes the Euler–Mascheroni constant, Hn = n 1 r =1 r m is the mth order harmonic number and ψ(z) is the digamma (or psi) function defined by ψ(z) :=
123
(z) 1 d {log (z)} = and ψ(1 + z) = ψ(z) + . dz (z) z
Integrals of inverse trigonometric and polylogarithmic functions
Moreover, ψ(z) = −γ +
∞ n=0
1 1 − n+1 n+z
.
The polygamma function ψ (k) (z) =
∞
1 dk {ψ(z)} = (−1)k+1 k! k dz (r +
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