It is well-known that one of the form of the general solution of $u_{xx}+u_{yy}=0$ is $u(x,y)=C_1(x+iy)+C_2(x-iy)$ , where $C_1$ and $C_2$ are arbitrary functions.
However it is not quite convenient to have complex functions. Can we find another form of the general solution of $u_{xx}+u_{yy}=0$ : $u(x,y)=C_1(f(x,y))+C_2(g(x,y))$ , where $f(x,y)$ and $g(x,y)$ are real functions of $x$ and $y$ ?
Looking back to the second order linear ODE with constant coefficients $ay''+by'+cy=0$ , when $b^2-4ac<0$ , besides it has the form of the general solution $y=C_1e^{\frac{-b+\sqrt{b^2-4ac}}{2a}x}+C_2e^{\frac{-b-\sqrt{b^2-4ac}}{2a}x}$ , it also has the better form of the general solution $y=C_1e^{\frac{-bx}{2a}}\sin\dfrac{\sqrt{4ac-b^2}}{2a}x+C_2e^{\frac{-bx}{2a}}\cos\dfrac{\sqrt{4ac-b^2}}{2a}x$ so that we can avoid the complex function. Does the form of the general solution of $u_{xx}+u_{yy}=0$ have the similar trick?
Note that $f(x,y)$ and $g(x,y)$ I required can be any type of real functions and are not restricted to real polynomial functions only. In fact $u(x,y)=C_1(x+iy)+C_2(x-iy)$ can rewrite to $u(x,y)=C_1(e^{x+iy})+C_2(e^{x-iy})$ . So can $u(x,y)=C_1(e^x\sin y)+C_2(e^x\cos y)$ be the one of the form of the general solution of $u_{xx}+u_{yy}=0$ ?