The name is absent

As we have already noted before in [16], there seem to be some misunder-
standings as to the meaning and extent of the feature of uniqueness with which
the Weierstrass continuation is endowed. The fact is that this uniqueness is
much less “absolute” than sometimes vaguely taken for granted, and we
are going to explain what we mean by this. Our present explanation will be
somewhat different from and more detailed than the one given in [76].

Let D₁ be a bounded simply connected domain in the z-plane, and D₂
such a domain in the w-plane ; let

(8)                                   w = <∕>(z)

be a one-one holomorphic mapping from D₁ to D₂; let ∕₂(w) be a holo-
morphic function on D₂, and let∕₁(z) ≡∕₂(≠(^z)) be its preimage on D_l. If
we form the two ensembles

(9)                      {D_l,f_l}, {D₂,f₂}

then (8) is also a holomorphic transformation of the first ensemble into the
second, in an obvious sense. We now form the Weierstrass continuation
of each of the functions f₂,f₂. This gives rise to two “larger” ensembles

(10)                      {D_i,f_i}, {D₂,f₂},

and, contrary to what one might vaguely expect, these two larger ensembles
need no longer be holomorphic images of each other, either by the mapping
(8) itself, or by any other mapping. One can easily construct counter-
examples, and we now choose the following.

Let D₁ be the disc

(11)                      Z>₁t[z∣<l.

Let D₂ be the interior of a Jordan curve B₂ no arc of which is real-analytic,
and let

∕₂(w) ≡ w.

Then

∕₁(z) ≡ ≠(z).

By known properties of conformal mapping, the function φ(z) has no
analytic continuation at all, so that

(12)                      {D_l,f_l} ≡ {D_l,f₁}.

But f₂(w) = w can be continued analytically from D₂ into all of C^l(>v),
so that

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