I'm having trouble with this showing question.
Any help would be much appreciated. thanks!
Setting $\displaystyle z= x + i\cdot y$ and $\displaystyle f(z)= u(x,y) + i\cdot v(x,y)$, $\displaystyle f(*)$ is analytic in a region $\displaystyle A$ of the complex plane if and only if in that region is...
$\displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$
$\displaystyle \frac{\partial u}{\partial y}=- \frac{\partial v}{\partial x}$ (1)
But $\displaystyle f(*)$ is a real function in $\displaystyle A$ so that is $\displaystyle v(z)=0$ and therefore for the (1) is $\displaystyle \frac{\partial u}{\partial x} = \frac{\partial u}{\partial y}=0$...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$