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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.

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    $\begingroup$ For what it's worth, I only saw holomorphic/analytic applied to functions in open (connected) sets. $\endgroup$ Sep 4, 2013 at 21:44
  • $\begingroup$ @PeterTamaroff : As I said, $z\mapsto|z|^2$ is differentiable only at one point, a fortiori isolated. Apply the definition and you'll see that. $\endgroup$ Sep 4, 2013 at 22:08
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    $\begingroup$ Doesn't "differentiable at an isolated point" convey that information? $\endgroup$ Sep 4, 2013 at 22:16
  • $\begingroup$ @MichaelHardy Maybe the first read felt weird. I'll clean up. $\endgroup$
    – Pedro
    Sep 4, 2013 at 22:26

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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.

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  • $\begingroup$ Indeed! In_practice, functions "holomorphic at a point" do not arise. This issue would only conceivably be contrived in a textbook to make a (rather pedantic) point. Thus, to avoid this pedantry, one might (semi-mysteriously) insist on differentiability on a non-empty open set in the "definition". In real life, typically one has differentiability on a (non-empty) open set, anyway, so this "extra requirement" is not a "burden", but simply a thing that naturally arises, and we "have". It is not good to overload language with delicate distinctions whose origins or purposes are suppressed. $\endgroup$ Sep 4, 2013 at 22:23
  • $\begingroup$ John Bush, an expert in fluid dynamics who has been affiliated with Harvard and MIT at various times, taught a course to prepare engineering graduate students with topics in differential equations and in complex variables that were to be used in fluid dynamics. He included an example of a fairly simply defined function that is differentiable everywhere on a straight line in the complex plane but nowhere else in the plane. It seems safe to predict they won't see that very often in fluid dynamics. $\endgroup$ May 15, 2020 at 21:59
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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.

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  • $\begingroup$ Whether a function $f$ is differentiable at a point $z_0$ depends on the behavior of $f$ in some neighborhood of $z_0$. So I'd have guessed you'd call it a "local" property. Can you tell me what definition of "pointwise" is consistent with this? $\endgroup$ Sep 4, 2013 at 22:14
  • $\begingroup$ Well, yes, of course the existence of the derivative at a point is local in the sense that it depends on the values in a neighbourhood of that point. What I meant, and don't know how to succinctly express, is that to say that "$f$ is differentiable at $z_0$" does not imply the existence of the derivative at any other point, while "$f$ is holomorphic at $z_0$" does imply the existence of the derivative in a full neighbourhood of $z_0$. If you (or anybody) can suggest a good wording, I'd be grateful. $\endgroup$ Sep 4, 2013 at 22:20
  • $\begingroup$ If there's no difference in meaning between "holomorphic" and "analytic", then what does "real-analytic" mean? I thought it meant being equal to its power-series expansion. $\endgroup$ Sep 4, 2013 at 23:32
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    $\begingroup$ "Real-analytic" means representable by a power series in the real coordinates (or, in the case of domains in $\mathbb{C}^n$, equivalently, power series in the $z_i$ and $\overline{z}_i$). "Analytic" sans "real" is - in complex analysis - reserved for holomorphic functions; historically, as far as I remember, people called functions analytic that were representable as power series (in $z$, resp the $z_i$, only), and holomorphic when they had a complex derivative/partial complex derivatives in an open set. It turned out, both conditions are equivalent, but that was not obvious from the start. $\endgroup$ Sep 4, 2013 at 23:38
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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.

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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.

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