I want to find a rule to link the variation of the square of the modulus of a complex number, $\delta \lvert z \rvert^2$, to the variation of the complex number, $\delta z$. Or are the following relations correct?
$$\delta \lvert z \rvert^2=2 \lvert z\rvert \delta \lvert z\rvert=2 \lvert z\rvert \delta \lvert z\rvert,$$
$$\delta\lvert z \rvert=\lvert \delta z \rvert.$$
Many thanks！
For all $z\in\Bbb C\setminus\{0\}$, $$\frac{\partial\lvert z\rvert}{\partial z}=\frac{\overline z}{2\lvert z\rvert}\\ \frac{\partial\lvert z\rvert}{\partial \overline z}=\frac{z}{2\lvert z\rvert}$$
The aforementioned are Wirtinger's derivatives. Wirtinger's derivatives of a function $f:\Bbb C\to\Bbb C$ at a point $a$ have this property: if $(z_1,z_2)$ is a pair of complex numbers such that $$\lim_{h\to 0}\frac1{\lvert h\rvert}\left(f(a+h)-f(a)-hz_1-\overline hz_2\right)=0,$$ then $z_1=\frac{\partial f}{\partial z}(a)$ and $z_2=\frac{\partial f}{\partial \overline z}(a)$.