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Unit Disk Integral

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The integral of 1/r over the unit disk U is given by

intint_(U)(dA)/r=intint_(U)(dxdy)/(sqrt(x^2+y^2))
(1)
=int_0^(2pi)int_0^1(rdrdtheta)/r
(2)
=2piint_0^1dr
(3)
=2pi.
(4)

In general,

intint_(U)r^ndA=2piint_0^1r^(n+1)dr=(2pi)/(2+n)
(5)

provided n>-2.

Additional integrals include

intint_(U)ln(x+iy)dxdy=-1/2pi
(6)
intint_(U)ln(x^2+y^2)dxdy=-pi
(7)
intint_(U)e^(x+iy)dxdy=pi.
(8)

See also

Double Integral, Unit Disk, Unit Square Integral

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Cite this as:

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

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