I am trying to track down at what point mathematicians started to use the terminology of expanding a function "around a point" or in the "neighbourhood of a point". Neither Taylor nor Maclaurin would appear to use such terminology, and while I would have guessed that Weierstrass would be using it, I have yet to see a publication of his where that is the case.
Does anyone happen to know?
The terminology "neighbourhood of a point" (in German, "Umgebung einer bestimmten Stelle") with its current meaning, dates back at least to 1841 when Weierstrass wrote his Zur Theorie der Potenzreihen ("On the theory of power series"), published in Mathematische Werke, Band I, S. 67-74. The definition of neighbourhood of a point in $\mathbb{R}^n$ is in the footnote at page 70 and is used throughout the paper:
Unter Umgebung einer bestimmten Stelle $(a_1,a_2,...,a_\rho)$ im Gebiete von $\rho$ Veränderlichen $x_1,x_2,...,x_\rho$ ist ein Bereich von folgender Beschaffenheit zu verstehen: Wenn $(x'_1,x'_2,...,x'_\rho)$ irgend eine Stelle des Bereiches ist, so gilt dasselbe von jeder Stelle $(x''_1,x''_2,...,x''_\rho)$, die den Bedingungen $|x''_1-a_1|\leq |x'_1-a_1|$, $|x''_2-a_2|\leq |x'_2-a_2|$, ..., $|x''_\rho-a_\rho|\leq |x'_\rho-a_\rho|$ entspricht. Ein solcher Bereich ist z. B. der Convergenzbezirk eder gewöhnlichen Potenzreihe von $x_1,x_2,...,x_\rho$.
i.e.
The neighbourhood of a particular point $(a_1,a_2,...,a_\rho)$ in the domain of $\rho$ variables $x_1,x_2,...,x_\rho$ is to be understood as a domain of the following nature: if $(x'_1,x'_2,...,x'_\rho)$ is a point of the domain, then the same is true of any point $(x''_1,x''_2,... ,x''_\rho)$ such that $|x''_1-a_1|\leq |x'_1-a_1|$, $|x''_2-a_2|\leq |x'_2-a_2|$, ..., $|x''_\rho-a_\rho|\leq |x'_\rho-a_\rho|$. Such a region is, for example, the convergence region of the ordinary power series of $x_1,x_2,...,x_\rho$.
Harnack was well-aware of the work of Weierstrass, and in fact at page 130 he cites in footnote Zur Theorie der eindeutigen analytischen Functionen (1876) where one can read
Ich will von einer eindeutigen analytischen Function $f(x)$ der complexen Veränderlichen $x$ sagen, sie verhalte sich regulär in der Umgebung einer hestimmten Stelle ($x = a$), wenn sie innerhalb eines gewissen Bezirks[*], dessen Mittelpunkt $a$ ist, überall einen endlichen und mit $x$ stetig sich ändernden Werth hat.
i.e.
I say that an analytic function $f(x)$ of the complex variable $x$ is regular in the neighbourhood of a point ($x = a$) if it has a finite value everywhere in a circle whose centre is $a$ and which varies continuously with $x$.
Probably Harnack read the French translation, which he cites at page 148:
In the French Translation by E. Picard of the above Memoir of Weierstrass, which appeared under the revision of the Author in the Annales de l'Ecole Normale, 2e Serie, T. VIII, 1879, entitled: "Memoire sur les fonctions analytiques uniformes", the opening statement reads as follows:
Among unique (uniformes) functions of a single variable, rational functions form a distinct class which we proceed to define by their characteristic property.
We shall say that a unique function $f(z)$ of the complex variable $z$ is regular in the neighbourhood of a point $a$, when for all values of $z$ comprised within a circle having its centre at $a$ and a radius sufficiently small, the function can be developed in a series of the form [...]. In case the point a were at infinity we should replace $z-\infty$ by $\frac{1}{z}$.
Every point $a$ in whose neighbourhood the function $f(x)$ is not regular will be called a singular point [...].
Back to Weierstrass: he used the terminology "Umgebung einer bestimmten Stelle" throughout his work, and the term Umgebung appears at lest 59 times in the first volume of his Werke and 106 in the second one. Moreover, the terminology was in use in the German mathematical community, for example Über die Integration der linearen Differentialgleichungen durch Reihen (1873) start with the sentence
Wenn alle Integrale einer homogenen linearen Differentialg $\lambda^{\text{ter}}$ Ordnung in der Umgebung einer bestimmten Stelle,[...]
i.e.
If all integrals of a homogeneous linear differential of order $\lambda^{\text{ter}}$ in the neighbourhood of a certain point, [...]
Anyway, the idea (but not the definition) of neighbourhood predates Weierstrass, and as TonioElGringo says, it is present also in Cauchy. For example in the Cours d'analyse de l'École royale polytechnique (1821) one can find (page 35)
On dit encore que la fonction $f(y)$ est, dans le voisinage d'une valeur[**] particulière, attribuée à la variable $x$, fonction continue de cette variables, toutes les fois quelle est continue entre deux limites de $x$, même très-rapprochées, qui renferment la valeur dont il s'agit.
i.e.
We also say that the function $f(y)$ is, in the vicinity of a particular value attributed to the variable $x$, a continuous function of this variable, whenever it is continuous between two limits of $x$, even very close together, which contain the value in question.
In my opinion, there is a fundamental difference with respect to Weierstrass' use of it: in Cauchy the idea of neighbourhood is intuitive and has a vaguely geometric meaning (a sort of unspecified proximity), Weierstrass on the other hand gives a formal definition of it (again based on a geometric idea, of course) and which in some respects prefigures the topological definition.
[*]: Weierstrass uses the word "Bezirk", which nowadays means normally "district" (I believe), but actually comes from the Latin circulus, "circle", while for the geometric figure one normally uses "Kreis" (from Proto-Germanic), even it is also used to denote a particular administrative division.
[**]: Also notice that Cauchy always (not only in the Cours d'Analyse) says "voisinage d'une valeur" (neighbourhood of a value) and never of a point.
According to Miller's Earliest Known Uses, "neighborhood of a point" first appears in Cayley's 1877 paper (in a different context), and then in Harnack's calculus text (1891), where it is used extensively throughout in connection with continuity, partial derivatives, regularity (holomorphy), series convergence, etc. When it comes to the Taylor series in chapter 2 of book IV, we read passages like:
§187 "If we know that a function loses the property of a unique analytic function at certain points $c_1,c_2\dots$ of the plane, and we desire to expand this function in a series of powers in the neighbourhood of a point $a$, the circle of convergence may only be so large as to pass through one or more of the points $c$, but must include none of them. For, the series of powers is a unique analytic function in the entire circle, while by hypothesis the function loses this property in each point $c$."
§189 "Having established that analytic functions can be expressed by series of ascending positive integer powers, it is still necessary that we should discuss their singular points, in order that we may ascertain how analytic functions admit of expansion in the neighbourhood of any singular point according to its kind."