Are there two conventional definitions of “holomorphic”?

Question

In Walter Rudin's Real and Complex Analysis, second edition, on page 213, two definitions are stated. One of them says the derivative of $f$ at $z_0$ is

$$f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}.\tag 1$$

The other says $f$ is holomorphic in an open set if $f'(z_0)$ exists at every value of $z_0$ in that set.

By that definition, holomorphic doesn't just mean differentiable; it means differentiable at every point in some open set.

He never defines “holomorphic at” a point, but only “holomorphic in” an open set.

Before opening the book this afternoon, just remembering from years ago, I thought what it said was

• $f$ is differentiable at $z_0$ if $(1)$ holds; and
• $f$ is holomorphic at $z_0$ if the derivative exists at every point in some open neighborhood of $z_0$.

By that definition $f(z)=|z|^2$ would be differentiable at $0$ but not holomorphic at $0$.

Thus differentiability at an isolated point would be weaker than holomorphy at that point.

QUESTION: Do both conventions exist?

I take “$f$ is analytic at $z_0$” to mean $f$ has a convergent power series expansion at $z_0$, which actually converges to the right thing, the value of $f$, in the interior of its circle of convergence. Differentiability at every point in some open neighborhood of $z_0$ is enough to prove analyticity, but differentiability at $z_0$ is not. By one of the conventions above, one can say a function holomorphic at a point is analytic at that point; by the other one cannot.

Answer 1

There is almost never any reason to talk about functions that are (complex-) differentiable at isolated points (other than as exercises to show the students that the derivative may exist at some exceptional points, and to show that Cauchy-Riemann's equations don't quite tell the full story).

I can't think of any book or text that uses "holomorphic at a point", at least not meaning that $f'$ exists at that particular point. If I would see the term, I would interpret it as synonymous with "analytic at a point", i.e. holomorphic on some open neighbourhood of the point.

Answer 2

Yes, both conventions exist, and are, as far as I'm aware, standard. Differentiability is a pointwise property (in the sense that differentiablity at $x$ does not imply differentiability at any other point), and holomorphy (or analyticity) a local property (in the sense that holomorphy at $z_0$ implies holomorphy in a neighbourhood of $z_0$).

Fischer/Lieb, A Course in Complex Analysis, define

A complex-valued function $f$ defined on an open set $U \subset \mathbb{C}$ is (complex) differentiable at $z_0 \in \mathbb{C}$ if there exists a function $\Delta$ on $U$, continuous at $z_0$, such that $$f(z) = f(z_0) + \Delta(z)(z-z_0)\tag{1}$$ holds for all $z \in U$. If $f$ is complex differentiable at all points $z_0 \in U$, then we say that $f$ is holomorphic on $U$. We say that $f$ is holomorphic at $z_0$ if there exists an open neighbourhood of $z_0$ on which $f$ is holomorphic.

(The $z_0 \in \mathbb{C}$ is a typo, should of course read $z_0 \in U$.)

Ahlfors also differentiates between the derivative existing at a point and analyticity/holomorphy:

The class of analytic functions is formed by the complex functions of a complex variable which possess a derivative wherever the function is defined. The term holomorphic function is used with identical meaning.

Answer 3

According to Wikipedia, a function $f$ is said to be holomorphic at a point $z_0$ if it is holomorphic in some neighbourhood of $z_0$. It goes on to say that $f$ is defined to be holomorphic on a non-empty set $A$ if it is holomorphic on some open neighbourhood of $A$.

I don't know if any other reference uses this terminology though.

Answer 4

I am writing this like a answer, but only because for me to add a comment here I have t have 50 reputation (but I don't have still). So, Michael Hardy, I was with a similar doubt. Because in Churchill's Complex variables and its applications, there's a section where he define an analytical function just like a function which derivative exists in a point $z_{0}$ and in the points that are a neighborhood of the point $z_{0}$.

In the case of the function $f(z)={\vert}z{\vert}^{2}$, this function has a derivative defined in the point $z=0$, but in the $z=0$ neighborhood we cannot define the derivative (because the lateral limit does not exist, so the derivative does not exists too).

But there is another case: the function $g(z)={\frac{1}{z}}$. It's derivative is not defined for the point $z=0$, but it exists to the neighbor points of $z=0$. In this case, $z=0$ is a singular point of the function.

Note: I would like to write this like an comment (in fact, this is not the answer!), but my S.E. points are not enough to post a comment.