Heat equation of spatial complex variable - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:29:05Z http://mathoverflow.net/feeds/question/103135 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103135/heat-equation-of-spatial-complex-variable Heat equation of spatial complex variable galsorin 2012-07-25T23:57:46Z 2012-08-09T02:22:00Z <p>Suppose that $v(t, z)$ is analytic with respect to the complex variable $z$ and differentiable with respect to the real variable $t$ and satisfies the partial differential equation $$\frac{\partial v}{\partial t}(t, z)+\frac{\partial^{2} v}{\partial z^{2}}(t, z)=0,$$ for all $t>0$ and $z=x+iy$, $x\in \mathbb{R}$, $|y|\le r$.</p> <p>Then, what partial differential equation with respect to the real variables $t>0$ and $x\in \mathbb{R}$ satisfies $v(t, x)$ ?</p> <p>(It is obtained simply by replacing the complex $z$ by the real $x$ in the original equation, and if this is the answer, how could be proved that ?)</p> <p>Thank you very much in advance.</p> http://mathoverflow.net/questions/103135/heat-equation-of-spatial-complex-variable/103144#103144 Answer by Robert Israel for Heat equation of spatial complex variable Robert Israel 2012-07-26T01:30:38Z 2012-07-26T01:30:38Z <p>$v(t,x)$ for $x \in {\mathbb R}$ is the restriction of $v(t,z)$ to ${\mathbb R}_+ \times {\mathbb R}$, and that satisfies the ordinary heat equation because the real derivative is a special case of the complex derivative. </p>