Unit Square Integral

Unit square integrals are double integrals over the unit square. Examples arising in geometric probability are

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

for and , respectively. At , the second right-hand side is interpreted by its limit, which is .

Another simple integral is given by

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

(OEIS A093754; M. Trott, pers. comm.), where is Catalan's constant and is a generalized hypergeometric function. In the second representation, the principal branches of the square root and inverse tangent are used, with the integrand extended continuously at to the value . A related integral is given by

which diverges as an improper integral, as can quickly be seen by transforming to polar coordinates. Subtracting from the integral over and taking the limit as gives the radial Hadamard finite part

(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

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). For and , the right-hand side of (15) is interpreted by its continuous limit.

Another result is

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

Guillera and Sondow (2005) also give

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. (18) 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

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

(Zudilin 2003).

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

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

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

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.