ABSTRACT
The values of many definite integrals have been compiled in the classical Table of Integrals, Series and Products by I. S. Gradshteyn and I. M. Ryzhik [40]. The table is organized like a phonebook: integrals that look similar are placed close together. For example, 4.229.4 gives
ln
( ln
x
)( ln
x
)µ−1 dx = ψ(µ)Γ(µ), (1.1.1)
for Reµ > 0, and 4.229.7 states that
ln ln tanx dx = π
2 ln
{ Γ ( 3 4
) Γ ( 1 4
) √2π } . (1.1.2)
In spite of a large amount of work in the development of this table, the latest version of [40] still contains some typos. For example, the exponent u in (1.1.1) should be µ. A list of errors and typos can be found in
https://www.mathtable.com/errata/gr6_errata.pdf
The fact that two integrals are close in the table is not a reflection of the difficulty involved in their evaluation. Indeed, the formula (1.1.1) can be established by the change of variables v = − lnx followed by differentiating the classical gamma function
Γ(µ) :=
tµ−1e−t dt, Reµ > 0, (1.1.3)
with respect to the parameter µ. The function ψ(µ) in (1.1.1) is simply the logarithmic derivative of Γ(µ) and the formula has been checked. The situation is quite different for (1.1.2). This formula is the subject of the lovely paper
Volume
Theory to check ingredients of the proof are quite formidable: the author shows that∫ π/2
ln ln tanx dx = d
ds Γ(s)L(s) at s = 1, (1.1.4)
where
L(s) = 1− 1 3s
+ 1
5s − 1
7s + · · · (1.1.5)
is the Dirichlet L-function. The computation of (1.1.4) is done in terms of the Hurwitz zeta function
ζ(q, s) =
(n+ q)s , (1.1.6)
defined for 0 < q < 1 and Re s > 1. The function ζ(q, s) can be analytically continued to the whole plane with only a simple pole at s = 1 using the integral representation
ζ(q, s) = 1
Γ(s)
e−qtts−1
1− e−t dt. (1.1.7)
The relation with the L-functions is provided by employing
L(s) = 2−2s ( ζ(s, 14 )− ζ(s, 34 )
) . (1.1.8)
The functional equation
L(1− s) = ( 2
π
)s sin
πs
2 Γ(s)L(s), (1.1.9)
and Lerch’s identity
ζ′(0, a) = log Γ(a)√ 2π
, (1.1.10)
complete the evaluation. More information about these functions can be found in [83].
