quadrature domains from circles? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:12:03Z http://mathoverflow.net/feeds/question/86987 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86987/quadrature-domains-from-circles quadrature domains from circles? John Mangual 2012-01-29T21:13:49Z 2012-08-31T23:48:13Z <p>If $h(z)$ is analytic on the disk centered at 0 of radius r, by the Cauchy Residue formula $\int \int_D h(z)\, dx dy = \pi r^2 h(0)$ The disk is the simplest example of a <a href="http://www.math.kth.se/~gbjorn/schottky.pdf" rel="nofollow">quadrature domain</a> since the integral of a holomorphic function over the domain is determined by the value at a single point. </p> <p>How about the next simplest cases? What are <em>connected</em> quadrature domains whose integrals only depend on a few points (e.g. 2 or 3)? </p> <p>$\int \int_D h(z)\, dx dy = c_1 h(z_1) + c_2 h(z_2) + c_3 h(z_3)$</p> <p>Probably these will all be close to the union of a few circles (with jumps in the coefficients as the radius changes).</p> <p><img src="http://s16.postimage.org/vzkdfcqnn/domain.gif" alt="alt text"></p> <p>It looks exact quadrature domains can be constructed <a href="http://www2.imperial.ac.uk/~dgcrowdy/PubFiles/Review.pdf" rel="nofollow">using meromorphic functions on Riemann surfaces</a> and is related to uniformation of surfaces. There are relations to <a href="http://www.unige.ch/~hongler/ascona/slides/wiegmann.pdf" rel="nofollow">Laplacian growth</a>, <a href="http://temple.birs.ca/~07w5008/Teodorescu.pdf" rel="nofollow">Random Matrix Theory</a> &amp; <a href="http://www.physics.ubc.ca/pitp/archives/theory/2004talks/wiegmann.pdf" rel="nofollow">Integrable Hierarchies</a>.</p> http://mathoverflow.net/questions/86987/quadrature-domains-from-circles/106079#106079 Answer by Alexandre Eremenko for quadrature domains from circles? Alexandre Eremenko 2012-08-31T23:48:13Z 2012-08-31T23:48:13Z <p>A bounded simply connected domain with analytic boundary is a quadrature domain if and only if the inverse of the Riemann mapping function (mapping the disc onto the domain) is rational.</p> <p>See, for example, P. J. Davis, The Schwarz function and its applications, The Mathematical Association of America, Bualo, N. Y., 1974. The Carus Mathematical Monographs, No. 17. or A. Varchenko and P. Etingof, Why the bundary of a round drop becomes a curve of order 4, AMS 1991. </p>