Relation between complex analysis and harmonic function theory - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T13:44:43Z http://mathoverflow.net/feeds/question/52180 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52180/relation-between-complex-analysis-and-harmonic-function-theory Relation between complex analysis and harmonic function theory Jahnke 2011-01-15T19:02:20Z 2011-01-19T23:37:57Z <p>There are some theorems in harmonic function theory that resemble results in complex analysis, like:</p> <ul> <li>Holomorphic functions and complex functions are analytic;</li> <li>Cauchy's integral formula in complex analysis and the mean value theorem in harmonic function theory;</li> <li>The principle of maximum and minimum that works for harmonic and holomophic functions.</li> <li>The real and imaginary parts of a holomorphic function are harmonic;</li> </ul> <p>These results suggest that there are connections between these two areas and I would like to ask: how can each of these theories be used to develop the other?</p> <p>PS: I'm really sorry for my really bad English.</p> http://mathoverflow.net/questions/52180/relation-between-complex-analysis-and-harmonic-function-theory/52561#52561 Answer by Daniel Parry for Relation between complex analysis and harmonic function theory Daniel Parry 2011-01-19T23:37:57Z 2011-01-19T23:37:57Z <p>The Cauchy Reimann Equations imply that every holomorphic function satisfies Laplace's Equation and is therefore its real and imaginary components are harmonic.</p> <p>You can also take a harmonic function u and construct, up to a constant, its harmonic conjugate v so that u and v satisfy the Cauchy Reimann Equations. Thus u+iv is a holomorphic function.</p> <p>These proofs can be found in just about any Complex Analysis book. My favorite is Complex Analysis by Lars Ahlfors.</p> <p>So in a sense a harmonic function is just the real component of a holomorphic function. </p>