For $u_{tt}=u_{xx}+u_{yy}$ , the complex variable $z=x+iy$ and $u(x,y,t)$ are given.Then can you formulate new Cauchy Riemann Eq for $u(x,y,t)$
My firend ask about this to me and we think about it long, but we didn't get answer. Help us to solve this problem.(this must be exercise of engineering mathematics,I'm sorry for ask exercise but we want to solve Intensely.)
We think that what about $u(x(t),y(t))$ but it seems difficult than first one....
The function $u(x,y,t)$ can be rewritten as:
$$u(x,y,t) = f(x,y,t) + ig(x,y,t).$$ If it is homomorphic, then the Cauchy-Riemann equations are:
$$\begin{cases} f_x = g_y\\ f_y = -g_x \end{cases}.$$
In second partial derivatives are continuous, then also:
$$\begin{cases} f_{xx} + f_{yy} = 0\\ g_{xx} + g_{yy} = 0 \end{cases}.$$
Since $u_{xx} = f_{xx}+ig_{xx}$ and $u_{yy} = f_{yy}+ig_{yy}$, then:
$$u_{tt} = u_{xx}+u_{yy} = 0.$$
That is:
$$u_{tt} = 0.$$