### D.5 2D elliptical transformation

To bring two-dimensional elliptical equations in the two-dimensional canonical form, you need to solve, say, the ordinary differential equation

Note now that even if you take to be real, will be complex. And that means that you need to know what happens to the coefficients , , and when you depart the real -plane into the complex domain. That may be fine if you know the coefficients analytically, but otherwise it is a problem.

Assuming that you can solve the system, call the integration constant . Assuming that it is a differentiable function of and , it will satisfy

Now set

If you plug that in the equation above and multiply out, you get

Now within the square brackets above, you find the generic expressions for the coefficients , and , respectively, of the transformed partial diffential equation. For a complex number to be zero, both its real and its imaginary part must be zero. It follows that and that .