The derivative of an analytic function of complex variable at infinity Let $f(z)$ be an analytic function of complex variable.
Please explain me how $f'(\infty)$ is defined. I cannot find it in any textbook.
What is the strict definition of the derivative of an analytic function of complex variable at infinite point?
 A: It depends. Definitions are made by people according to their convenience. Some may find convenient to define $f'(\infty)=\lim_{z\to\infty}f'(z)$, some prefer another definition, others let it be undefined. If you use this notation in your own writing, be sure to explain it. If you see it elsewhere, search for clues in the context.
Here is the definition that I see most often. First one needs to define $f(\infty)$, for which the only natural candidate is $\lim_{z\to\infty}f(z)$ (if this limit doesn't exist, we're not going to have $f'(\infty)$ either). 
Having taken care of that, the most common definition I've seen is 
$$f'(\infty) = \lim_{z\to\infty} z(f(z)-f(\infty))\tag{1}$$
So, for example, $f(z) = 3/z$ has $f'(\infty)=3$, while $f(z)=3/z^2$ has $f'(\infty)=0$, and for $f(z)=3z$ we have $f'(\infty)$ undefined. 
Equivalently, $f'(\infty)$ is the coefficient of $z^{-1}$ in the Laurent series of $f$ in a neighborhood of $\infty$ (assuming no positive powers of $z$ are present in the series). 
This matches   the fact that at a finite point, $f'(a)$ is the coefficient of $(z-a)$ in the Taylor series of $f$ in a neighborhood of $a$.
