Let $z = x + i y$ be a complex number and consider the modulus squared function: $$ f(z) = |z|^2 = z z^* = x^2 + y^2 = u(x, y) $$ where the asterix denotes complex conjugation and $u(x, y)$ is the real part of $f(z)$ (the imaginary part is clearly zero $v(x,y) = 0$). For $f(z)$ to be analytic the Cauchy Riemann-condition condition requires: $$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $$ Clearly, these conditions are not satisfied unless $z=0$: $$ 2x \neq 0 \,\, \forall x \neq 0 \quad \text{and} \quad 2y \neq 0 \,\, \forall y \neq 0. $$
Therefore $|z|^2$ is non-analytic away from $z=0$. Nevertheless, as a physicist I often see the derivative of $f(z)$ calculated by simply treating $z^*$ as an independent variable: $$ \frac{\partial f(z)}{\partial z} = z^* $$
I have just learned, that the derivative above really means the Wirtinger derivative defined by: $$ \frac{\partial}{\partial z} = \frac{1}{2} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) $$ Clearly this partial differential operator is well-defined even though $f(z)$ is not analytic.
In the case where a function is analytic one can use the Cauchy Riemann conditions to show that the Wirtinger derivative is the same as the complex derivative with respect to $z$ as is e.g. done on the Wiki page: https://en.wikipedia.org/wiki/Wirtinger_derivatives.
Here are my questions:
- I have just established that the function $f(z)$ is not analytic at any $z_0 \neq 0$, therefore the limit of the difference quotient for $z \to z_0$ is not well-defined. What then represents the Wirtinger derivative?
- Is there a term for non-analytic functions which still have a well-defined Wirtinger derivative?
- More broad question: In physics I often see differentiation done by treating $z$ and $z^*$ as independent variables (e.g. when we optimize a function with respect to $z$). I assume it is then the Wirtinger derivatives we really calculate. What is the reason that it is the Wirtinger derivatives which is commonly used in physics text books? And why is it, that (if I'm not mistaken) a lot of mathematical litterature on complex analysis concerns analytic functions and less attention is given to the Wirtinger derivatives?