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Unit Square Integral


Integrals over the unit square arising in geometric probability are

 int_0^1int_0^1sqrt(x^2+y^2)dxdy=1/3[sqrt(2)+sinh^(-1)(1)] 
int_0^1int_0^1sqrt((x-1/2)^2+(y-1/2)^2)dxdy 
 =1/6[sqrt(2)+sinh^(-1)(1)],
(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

int_0^1int_0^1|x-y|^ndxdy=2/((n+1)(n+2))
(2)
int_0^1int_0^1|x+y|^ndxdy=(2(2^(n+1)-1))/((n+1)(n+2)),
(3)

for R[n]>-1 and R[n]>-2, respectively.

Another simple integral is given by

 int_0^1int_0^1(dxdy)/(sqrt(1+x^2+y^2))=4ln(2+sqrt(3))-2/3pi
(4)

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

int_0^1int_0^1(dxdy)/(x^2+y^2+1)=int_0^1(tan^(-1)(1/(sqrt(1+y^2))))/(sqrt(1+y^2))dy
(5)
=int_0^11/(sqrt(4x^2-1))[tan^(-1)(3/(sqrt(4x^2-1)))-tan^(-1)((1+2x^2)/(sqrt(4x^2-1)))]dx
(6)
=int_0^(pi/4)(tan^(-1)(costheta))/(costheta)dtheta
(7)
=1/2pisinh^(-1)1-K+1/6_3F_2(1/2,1,1;3/2,3/2;1/9)
(8)
=0.63951...
(9)

(OEIS A093754; M. Trott, pers. comm.), where K is Catalan's constant and _3F_2(a,b,c;d,e;z) is a generalized hypergeometric function. A related integral is given by

 int_0^1int_0^1(dxdy)/(x^2+y^2),
(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

Hint_0^1int_0^1(dxdy)/(x^2+y^2)=intint_(x^2+y^2>1; 0<x<1; 0<y<1)(dxdy)/(x^2+y^2)
(11)
=int_0^11/x[tan^(-1)(1/x)-tan^(-1)((sqrt(1-x^2))/x)]dx
(12)
=1/2piln2-K
(13)
=0.172827...
(14)

(OEIS A093753), where K 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

int_0^1int_0^1(x^(u-1)y^(v-1))/(1-xyz)[-ln(xy)]^sdxdy=Gamma(s+1)(Phi(z,s+1,v)-Phi(z,s+1,u))/(u-v)
(15)
int_0^1int_0^1((xy)^(u-1))/(1-xyz)[-ln(xy)]^sdxdy=Gamma(s+2)Phi(z,s+2,u),
(16)

for u,v>0, R[s]>-2 if z in C-[1,infty), and R[s]>-1 if z=1, where Gamma(s) is the gamma function and Phi(z,s,a) is the Lerch transcendent. In (15), to handle the case u=v, take the limit as v->u, which gives (16).

Another result is

 int_0^1int_0^1(1-x)/((1-xy)(-lnxy))(xy)^(u-1)dxdy=lnu-psi_0(u)
(17)

(Guillera and Sondow 2005), for u>0 and where psi_0(z) is the digamma function.

Guillera and Sondow (2005) also give

int_0^1int_0^1([-ln(xy)]^s)/(1-xy)dxdy=Gamma(s+2)zeta(s+2)
(18)
int_0^1int_0^1([-ln(xy)]^s)/(1+xy)dxdy=Gamma(s+2)eta(s+2)
(19)
int_0^1int_0^1([-ln(xy)]^s)/(1+x^2y^2)dxdy=Gamma(s+2)beta(s+2),
(20)

where the first holds for R[s]>-1, the second and third for R[s]>-2, zeta(s) is the Riemann zeta function, eta(s) is the Dirichlet eta function, and beta(s) is the Dirichlet beta function. (19) was found by Hadjicostas (2002) for s>=0 an integer. Formulas (18) and (19) are special cases of (16) obtained by setting u=1 then taking z=1 and z=-1, respectively.

The beautiful formulas

int_0^1int_0^1(dxdy)/(1-xy)=zeta(2)
(21)
int_0^1int_0^1(-ln(xy))/(1-xy)dxdy=2zeta(3)
(22)

were given by Beukers (1979). These integrals are special cases of (19) obtained by taking s=0 and 1, respectively. An analog involving Catalan's constant K is given by

 int_0^1int_0^1(dxdy)/((1-xy)sqrt(x(1-y)))=8K
(23)

(Zudilin 2003).

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

int_0^1int_0^1(x-1)/((1-xy)ln(xy))dxdy=gamma
(24)
int_0^1int_0^1(x-1)/((1+xy)ln(xy))dxdy=ln(4/pi)
(25)

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

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

int_0^1int_0^1(-xln(xy))/(1+x^2y^2)dxdy=K-1/(48)pi^2
(26)
int_0^1int_0^1(-xln(xy))/(1-x^2y^2)dxdy=1/(12)pi^2
(27)
int_0^1int_0^1(-ln(xy))/(1+x^2y^2)dxdy=1/(16)pi^3
(28)
int_0^1int_0^1(dxdy)/(1+x^2y^2)=K
(29)
int_0^1int_0^1(-dxdy)/((2-xy)ln(xy))=ln2
(30)
int_0^1int_0^1(dxdy)/(2-xy)=1/(12)pi^2-1/2ln^22
(31)
int_0^1int_0^1(-ln(xy))/(2-xy)dxdy=7/4zeta(3)-1/6pi^2ln2+1/3ln^32
(32)
int_0^1int_0^1(-x)/((2-xy)ln(xy))dxdy=lnsigma
(33)
int_0^1int_0^1(-dxdy)/((phi-xy)ln(xy))=2lnphi
(34)
int_0^1int_0^1(-dxdy)/((phi^2-xy)ln(xy))=lnphi
(35)
int_0^1int_0^1(dxdy)/(phi-xy)=1/(10)pi^2-ln^2phi
(36)
int_0^1int_0^1(dxdy)/(phi^2-xy)=1/(15)pi^2-ln^2phi
(37)
int_0^1int_0^1(-ln(xy))/(phi^2-xy)dxdy=8/5zeta(3)-4/(15)pi^2lnphi+4/3ln^3phi
(38)
int_0^1int_0^1(-x)/((1+xy)ln(xy))dxdy=ln(1/2pi)
(39)
int_0^1int_0^1(-x)/((1+x^2y^2)ln(xy))dxdy=ln[(sqrt(2pi))/(Gamma^2(3/4))]
(40)
int_0^1int_0^1(-1)/((1+x^2y^2)ln(xy))dxdy=1/4pi
(41)
int_0^1int_0^1x/(1-x^3y^3)dxdy=pi/(3sqrt(3))
(42)
int_0^1int_0^1(dxdy)/(1-x^2y^2)=1/8pi^2
(43)
int_0^1int_0^1(ln(2-x))/(1-xy)dxdy=1/4pi^2ln2-zeta(3)
(44)
int_0^1int_0^1(ln(2-xy))/(1-xy)dxdy=5/8zeta(3)
(45)
int_0^1int_0^1(ln(1+x))/(1-xy)dxdy=5/8zeta(3)
(46)
int_0^1int_0^1(ln(1+xy))/(1-xy)dxdy=1/4pi^2ln2-zeta(3)
(47)
int_0^1int_0^1(-ln(1-x))/(1-xy)dxdy=2zeta(3)
(48)
int_0^1int_0^1(-ln(1-xy))/(1-xy)dxdy=zeta(3)
(49)
int_0^1int_0^1(xdxdy)/([-ln(xy)]^(3/2))=2(sqrt(2)-1)sqrt(pi)
(50)
int_0^1int_0^1(dxdy)/([-ln(xy)]^(3/2))=sqrt(pi)
(51)
int_0^1int_0^1(dxdy)/([-ln(xy)]^(5/4))=Gamma(3/4)
(52)
int_0^1int_0^1(xln^2(xy))/((1+x^2y^2)^2)dxdy=K-1/(48)pi^2+1/(32)pi^3
(53)
int_0^1int_0^1(ln^4(xy))/((1+xy)^2)dxdy=(225)/2zeta(5)
(54)
int_0^1int_0^1(-dxdy)/((1+x^2y^2)^2ln(xy))=1/8(pi+2)
(55)
int_0^1int_0^1(-xdxdy)/((1+xy)^2ln(xy))=ln((A^6)/(2^(1/6)sqrt(pie)))
(56)
int_0^1int_0^1(1-x)/((1+xy)ln^2(xy))dxdy=ln((pi^(1/2)A^6)/(2^(7/6)e))
(57)
int_0^1int_0^1(1-x^2)/((1+x^2y^2)ln^2(xy))dxdy=ln[(Gamma(1/4))/(2Gamma(3/4))]+(2K)/pi-1/2
(58)
int_0^1int_0^1(1-x)/([-ln(xy)]^(5/2))dxdy=1/3sqrt(pi)(8sqrt(2)-10),
(59)

where zeta(n) is the Riemann zeta function, zeta(3) is Apéry's constant, phi is the golden ratio, sigma is Somos's quadratic recurrence constant, and A 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 [0,1] for most of these integrals can be found by making the change of variables x=X/Y, y=Y. The Jacobian then gives dxdy=Y^(-1)dXdY, and the new limits of integration are {X,0,1}, {Y,X,1}. Doing the integral with respect to Y then gives a 1-dimensional integral over [0,1]. For details, see the first part of the proof of Guillera-Sondow's Theorem 3.1.


See also

Double Integral, Hadjicostas's Formula, Hypercube Line Picking, Square Point Picking, Unit Disk Integral, Unit Square

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References

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Beukers, F. "A Note on the Irrationality of zeta(2) and zeta(3)." Bull. London Math. Soc. 11, 268-272, 1979.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005. http://arxiv.org/abs/math.NT/0506319.Hadjicostas, P. "Some Generalizations of Beukers' Integrals." Kyungpook Math. J. 42, 399-416, 2002.Sloane, N. J. A. Sequences A093753 and A093754 in "The On-Line Encyclopedia of Integer Sequences."Sondow, J. "Criteria for Irrationality of Euler's Constant." Proc. Amer. Math. Soc. 131, 3335-3344, 2003. http://arxiv.org/abs/math.NT/0209070.Sondow, J. "Double Integrals for Euler's Constant and ln(4/pi) and an Analog of Hadjicostas's Formula." Amer. Math. Monthly 112, 61-65, 2005.Zudilin, W. "An Apéry-Like Difference Equation for Catalan's Constant." Electronic J. Combinatorics 10, No. 1, R14, 1-10, 2003. http://www.combinatorics.org/Volume_10/Abstracts/v10i1r14.html.

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Unit Square Integral

Cite this as:

Weisstein, Eric W. "Unit Square Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UnitSquareIntegral.html

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