Verhalten undefinierter Funktionen

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JVR

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May 2, 2022, 1:51:57 AMMay 2
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Eine lustige und berühmte Formel ist diese:

Integral_0^infty [x^{s-1}(f(0) - x*f(1) + x^2*f(2) - + ...)] dx =
f(-s) * pi/sin(s*pi) .

oder dazu äquivalent:

Integral_0^infty [x^{s-1}(g(0) - (x/1!)*g(1) + (x^2/2!)*g(2) - + ...)] dx =
Gamma(s) * g(-s).

Wobei f(s) = g(s)/Gamma(1+s).
s = x + iy.

Es handelt sich also um eine Mellin-Transformation einer nicht
weiter beschriebenen Funktion. Die Formel ist offenbar eine Interpolation,
d.h. es werden neue Funktionswerte aus vorgegebenen Funktionswerten
berechnet; bzw. eine Funktion wird durch die Formel definiert. Jedenfalls
wird man einige Mühe haben für eine konkret vorgegebene Funktion f oder
g zu beweisen, dass die Formel gilt.
Man ist also sehr weit entfernt von der Vorstellung, dass Reihen und
Integrale nur Sinn haben, wenn Formeln vorgegeben sind.

Hardy kommentiert die Situation folgendermaßen:

"A mathematician may have stated a formula and advanced reasons for
its truth which are inadequate as they stand, in which case he cannot be
said to have 'proved' it. But it often happens that his method, when
restated and developed by a modern analyst, leads to a proof valid under
'natural' conditions, and in that case we may fairly say that he has
'really' proved the theorem. Thus Euler 'really' proved large parts of
the classical analysis, and there are a great many theorems which
Ramanujan had 'really' proved; but he had not 'really' proved any of the formulae
which I have quoted. It was impossible that he should have done so because
the 'natural' conditions involve ideas of which he knew nothing in 1914,
and which he had hardly absorbed before his death. He had also, as
Littlewood says, no clear-cut conception of proof: 'if a significant piece of
reasoning occurred somewhere, and the total mixture of evidence and
intuition gave him certainty, he looked no further'. In this case any 'real'
proof was inevitably beyond his grasp, and the 'significant pieces of
reasoning' which are indicated in the notebooks and reports, though we
shall find them curious and interesting, are quite inadequate for the
occasion."

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