Necessary condition for conformal mapping I want to understand the reason behind the condition of conformal mapping ie f'(z) is not equal to zero. Why we use this condition as a necessary condition I mean what is the effect of this case in conformal mapping.
Thank you.
 A: A conformal mapping $f : U \to \Bbb C$, $U \subseteq \Bbb C$ should preserve angles at all point. For holomorphic functions $f$, angles are preserved at every point $z$ where $f'(z) \neq 0$: This follows from the fact that, the Jacobian of $f$, viewed as a real map under the identification of $\Bbb C$ with $\Bbb R^2$, is
$$\pmatrix{\textrm{Re} \,f'(z) & -\textrm{Im} \,f'(z)\\ \textrm{Im} \,f'(z)& \textrm{Re} \,f'(z)},$$ which we can factor as a dilation by $|f'(z)|$ and a rotation by the angle $\arg f'(z)$, both of which preserve angle (in fact, oriented angle).
However, at a point $z_0$ where $f'(z_0) = 0$, we can see that the Jacobian vanishes and so what happens to directions through the point depends on some higher derivative of $f$ at $z_0$, and we are no longer guaranteed the angle-preservation property.
For example, the function $f(z) = z^2$, which satisfies $f'(0) = 0$, sends the positive real direction to itself and the positive imaginary direction to the negative real direction. Thus, it maps the angle of $\frac{\pi}{2}$ at which the positive real and imaginary half-axes meet to an angle of $\pi$; hence $f$ is not conformal at $0$.
