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Double Integral

A double integral is a two-fold multiple integral.

Examples of definite double integrals evaluating to simple constants include

int_0^1int_0^1(dxdy)/(1-x^2y^2)=1/8pi^2
(1)
int_0^1int_0^1(dxdy)/(1-xy)=1/6pi^2
(2)
int_(-1)^1int_(-1)^1(dxdy)/(sqrt(1+x^2+y^2))=4ln(2+sqrt(3))-2/3pi
(3)
int_0^1int_0^1(dxdy)/((x+y)sqrt((1-x)(1-y)))=4K,
(4)

where K is Catalan's constant (Borwein et al. 2004, pp. 48-49), and

int_0^1int_0^1(x-1)/((1-xy)ln(xy))dxdy=gamma,
(5)

where gamma is the Euler-Mascheroni constant (Sondow 2003, 2005; Borwein et al. 2004, pp. 48-49).

See also

Euler-Mascheroni Constant, Hadjicostas's Formula, Integral, Multiple Integral, Triple Integral, Unit Disk Integral, Unit Square Integral

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References

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.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.

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Double Integral

Cite this as:

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

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