Having a test soon and was looking at this Question without having any sure ideas on what to do. Thanks for your help.
Exist 2 Differential functions :
$ u(x,y)$ and $v(x,y) $
that also agree with
$$x=u^2-v^2 \\ y = 2uv $$ 2. (This is the tuff one) Proove that $u(x,y) \ v(x,y)$ agree with
$$u_{xx}+u_{yy}=0 \\v_{xx}+v_{yy}=0$$ Near $(x,y) = (1,1)$
On
Simply take a branch of the square root, i.e., $$ u(x,y)+iv(x,y)=\sqrt{x+iy}. $$ Or note that $$ x^2+y^2=\cdots= (u^2+v^2)^2 $$ and thus $u^2+v^2=(x^2+y^2)^{1/2}$. Hence $$ u^2=\frac{\sqrt{x^2+y^2}+x}{2}, \quad v^2=\frac{\sqrt{x^2+y^2}-x}{2}. $$ Choosing for example $$ u(x,y)=\sqrt{\frac{\sqrt{x^2+y^2}+x}{2}}, \quad v(x,y)=\sqrt{\frac{\sqrt{x^2+y^2}-x}{2}}, $$ works!
This is just quadratic equations. In fact, it is asking about the complex square root function.
Take the square of the second equation and remove $v$ with the help of the first equation. Then test the 4 solutions of the resulting biquadratic on the original equations.
Of course, you can also do what is demanded of you and compute the Jacobi matrix $\frac{\partial(x,y)}{\partial(u,v)}$ at $(u,v)=(1,1)$.