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 $z=x+iy$, $x\in \mathbb{R}$, $|y|\le r$.

Then, what partial differential equation with respect to the real variables $t>0$ and $x\in \mathbb{R}$ satisfies $v(t, x)$ ?

(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 ?)

Thank you very much in advance.

  • $\begingroup$ What is $v(t,x)$? Are you fixing a specific value for $y$? $\endgroup$ – YangMills Jul 26 '12 at 1:13
  • $\begingroup$ $v(t, x)=v(t, x+i 0)$, that is $y=0$. $\endgroup$ – galsorin Jul 26 '12 at 11:09
  • $\begingroup$ Explicitly: Expand $\partial_z = \frac12 \partial_x - i\frac12 \partial_y$. Use the Cauchy-Riemann equations to change $y$ derivatives into $x$ derivatives. $\endgroup$ – Willie Wong Jul 26 '12 at 11:21

$v(t,x)$ 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.

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  • 2
    $\begingroup$ But that's only because $v$ is a holomorphic function of $z$, right? $\endgroup$ – Deane Yang Jul 26 '12 at 3:01

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