Relation between different spatial derivatives of a random field (related to complex integral and/or bessel function) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:59:00Z http://mathoverflow.net/feeds/question/46352 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46352/relation-between-different-spatial-derivatives-of-a-random-field-related-to-comp Relation between different spatial derivatives of a random field (related to complex integral and/or bessel function) Straybird 2010-11-17T14:05:13Z 2010-11-17T14:05:13Z <p>2 random fields $b$ and $c$ are derived from random field $a$ by </p> <p>$b=\nabla^2a\equiv(\partial_{xx}+\partial_{yy})a$ </p> <p>and</p> <p>$c \equiv c_1+i c_2 = (\partial_{xx}-\partial_{yy}+2i \partial_{xy}) a$.</p> <p>(all fields are 2-dimensional, $a$ and $c$ are real while $b$ is complex)</p> <p>The 2-point correlation function of $b$: $\langle bb\rangle(x)$ is known. <strong>What is the 2-point correlation function of $c$?</strong></p> <p>What about higher order correlation functions?</p> <p>This problem originates from astronomy. There exists a result at 2-point level. But the derivation process involves either integral of product of two Bessel functions or surface integral on a complex plane with poles. We cannot understand the mathematics involved rigorously, either can we generalize the result to higher order statistics. </p> <p>Is there any established mathematical theory related to this? Thanks a lot!</p>