Summary
G. H. Hardy proved 1916 for the first time, that the function R(x) of Riemann in the title has no finite differential quotient for all x, except two classes of rational numbers, noted in the text by 3) and 4). We prove in the following by Laplace-Transformation: R(x) is also not differentiable for the class 4) and it is differentiable for the class 3) with R′(x) always equal to −1/2.
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Literatur
Festschrift sur Gedächtnisfeier für Karl Weierstraβ 1815–1965, herausgegeben von H. Behnke und K. Kopfermann, Westdeutscher Verlag Köln und Opladen, 1966, Seite 89, Zeile 11 von oben.
G. H. Hardy,Weierstraβ's Nondifferentiable Function, Trans. Amer. Math. Soc.,17 (1916), S. 301–325 und speziell für die Riemannsche Funktion, S. 322 ff.
G. Doetsch,Handbuch der Laplace-Transformation, Band I:Theorie der Laplace-Transformation, Verlag Birkhäuser, Basel, 1950.
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Mohr, E. Wo ist die riemannsche funktion\(\sum\limits_{n = 1}^\infty {\frac{{\sin n^2 x}}{{n^2 }}}\) nichtdifferenzierbar?. Annali di Matematica pura ed applicata 123, 93–104 (1980). https://doi.org/10.1007/BF01796541
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DOI: https://doi.org/10.1007/BF01796541