Unit Square Integral — from Wolfram MathWorld

Integral of sqrt(x) from 1 to 9
Integral of sqrt(x) from 1 to 9

Integrals over the unit square arising in geometric probability are

(1)

which give the average distances in square point picking from a point picked at random in a unit square to a corner and to the center, respectively.

Unit square integrals involving the absolute value are given by

(2)

(3)

for
and ,
respectively.

Another simple integral is given by

(4)

(Bailey et al. 2007, p. 67). Squaring the denominator gives

(5)

(6)

(7)

(8)

(9)

(OEIS A093754; M. Trott, pers. comm.), where
is Catalan’s constant and is a generalized
hypergeometric function. A related integral is given by

(10)

which diverges in the Riemannian sense, as can quickly seen by transforming to polar coordinates. However, taking instead Hadamard integral to discard the divergent portion inside the unit circle gives

(11)

(12)

(13)

(14)

(OEIS A093753), where is Catalan’s constant.

A collection of beautiful integrals over the unit square are given by Guillera and Sondow (2005) that follow from the general integrals

(15)

(16)

for ,

if ,
and
if ,
where
is the gamma function and is the Lerch transcendent.
In (15), to handle the case , take the limit as , which gives (16).

Another result is

(17)

(Guillera and Sondow 2005), for and where is the digamma function.

Guillera and Sondow (2005) also give

(18)

(19)

(20)

where the first holds for , the second and third for , is the Riemann zeta
function,
is the Dirichlet eta function, and is the Dirichlet
beta function. (19) was found by Hadjicostas (2002) for

an integer. Formulas (18) and (19) are
special cases of (16) obtained by setting then taking and , respectively.

The beautiful formulas

(21)

(22)

were given by Beukers (1979). These integrals are special cases of (19) obtained by taking
and 1, respectively. An analog involving Catalan’s
constant
is given by

(23)

(Zudilin 2003).

Other beautiful integrals related to Hadjicostas’s formula are given by

(24)

(25)

(Sondow 2003, 2005; Borwein et al. 2004, p. 49), where is the Euler-Mascheroni
constant.

A collection of other special cases (Guillera and Sondow 2005) includes

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

where
is the Riemann zeta function, is Apéry’s constant,

is the golden ratio, is Somos’s
quadratic recurrence constant, and is the Glaisher-Kinkelin
constant. Equation (57) appears in Sondow (2005), but is
a special case of the type considered by Guillera and Sondow (2005).

Corresponding single integrals over for most of these integrals can be found by making the
change of variables , . The Jacobian then gives , and the new limits of integration are , . Doing the integral with respect to then gives a 1-dimensional integral over . For details, see the first part of the proof of Guillera-Sondow’s
Theorem 3.1.

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