This is a HW problem of 2 parts. I solved the first part, but just need a hint of how solve the second part. I am not asking for solution, just what is the idea.
Here is the problem. Part (a). Prove that the complex change of variables $x=x,t=i y$, maps the Laplace equation $u_{xx}+u_{yy}=0$ to the wave equation $u_{tt}=u_{xx}$.
Part (b) Explain why the type of a partial differential equation is not necessarily preserved under a complex change of variables.
I do not know even where to start from part (b). I think there should be something invariant when doing coordinates transformations and using complex variables breaks this? But not really sure.
Any hint where/how to start or the concept behind this part I need to look into more?
The leading symbol of a PDE is a quadratic form. Quadratic forms (non-degenerated) can only be distinguished over the real vector space by their signatures. Over complex vector space, all non-degenerated quadratic forms are equivalent to a standard one, so by picking a real subspace you can make it whatever you want. Intuitively (using low dimension picture) this is because all conic sections are obtained by cutting a plane through a conic.
EDIT: leading symbol for a 2nd order PDE is are the 2nd order term treated as a polynomial by replacing derivatives with variables.
I'm not sure if there is an intuitive way to explain quadratic form if you haven't know about it, it's from linear algebra. Think about a homogeneous quadratic polynomial in 2 variables. To make the symbol, take all the 2nd order terms and replace them with variable, so $u_{xx}$ turn into $X^{2}$ or $u_{xy}$ into $XY$. Then the level surfaces of this polynomial are either ellipses or hyperbola (hence the name). Change of variable produce linear change of variable at the derivative level, so what they can only do is to stretch and rotate those level curve, and it's impossible to change one into the other. But when you allow the domain to take on complex values, then all of ellipse and hyperbola become a conic surface so now you have a conic as your level curve, and the real part of the domain is just a plane cutting through this conic. When you allow yourself to do complex change of variables, you are stretching and rotating this conic and see what come out when this real plane cut it, which is equivalent to just rotating the plane. But since conic sections come from cutting planes through conic, both ellipse and hyperbola can be produced.