The name is absent

On one occasion Felix Klein gave an evaluation of Riemann’s originality
which mixed much admiration with a dose of incomprehension (25, pp.
118-119]. We translate it thus:

Riemann invariably investigates functions thus: at the head of the investigation
he places their defining properties, and from these properties he deduced every-
thing else, especially the formulas holding for the functions.

This procedure OfRiemann appears difficult only as long as it does not lean on any
concrete knowledge. Assoon as the latter takes place [that is, the leaning], it appears
most peculiarly simple and transparent.

Wecould also express this in the following way: Riemann's procedure is scienti-
fically excellent, but pedagogically unusable [unbrauchbar}. One must not begin
with it, but bring it at the end. A first example of this Riemannian treatment is his
theory of abelian integrals; he defines them as functions which by a closed circum-
ambulation on the Riemann surface vary only by an additive constant. A second
topic which Ricmann treats in this way is linear differential equations ....

Accordingto RiemannJn the theory of linear differentia! equations, one considers
simultaneously n functions ʃi,... ,ʃ,,, which, on the Riemann surface, experience
linear transformations after encircling certain “points of ramification” as also
closed circumambulations.

Klein’s criticism of Riemann₅ even if limited to “pedagogy,” was not
prophetically inspired because in the second half of the 20th century, a
hundred years after Riemann’s death; his “procedure” is flourishing with a
vengeance. There is an expanding mathematical “industry” which for any
compact complex manifold — algebraic or not, in one or several variables —
conceives and examines all sorts of classes of objects, scalar or tensorial,
tangential or fibral, holomorphic or meromorphic, “periodic” or auto-
morphic; and generally each of the classes has some kind affinité basis,
additive, multiplicative, or algebraic. Also, from time to time the name of
Riemann injects itself into the context.

Analytic Continuation. Riemann showed little interest in “arbitrary”
analytic functions, that is, in functions not “generated” in some algebraic
manner from the complex variable z. He knew, even for arbitrary functions,
that their analytic continuation is unique, but he did not make much of
the fact. Thus, Riemann never formulated the statement, of which he was
undoubtedly aware, that a holomorphic (or meromorphic) function in a
domain D of the complex plane — for instance, in a disc — gives rise by
analytic continuation along paths in the complex plane to a unique maximal
Riemann (covering) surface S over the complex plane into which f(z) can be
analytically continued (and similarly for the Gaussian sphere instead of
the complex plane). We will call this analytic continuation of/(z) from D
into D its "Weierstrass continuation” and denote it on D by /(z).

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