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The manuscript of the note (then entitled ‘On two theorems of Mr. S. Wigert’) was sent to Prof.Mittag-Leffler in 1917. I was at that time unaware of the existence of Mr.Carlson's dissertation (‘Sur une classe de séries de Taylor’, Uppsala, 1914). This dissertation was given to me by Prof.Mittag-Leffler in September 1919; and I found at once that Mr.Carlson had anticipated not only Mr.Wigert's theorem of 1916, referred to in § 2, but my own generalisation of this theorem and indeed the substance of most that I had to say. The note, however, contains something in substance, and a good deal in presentation, that is new; and I have therefore agreed to Prof.Mittag-Leffler's suggestion that it should still appear. Except as regards §§ 1–2, I have left it substantially in its original form.
Wigert (Sur un théorème concernant les fonctions entières,Arkiv för Matematik, vol. 11, 1916, no. 22, pp. 1–5) proves a theorem which is less general in that (I) the angle is supposed to cover the whole plane and (2)f(z) is supposed to vanish for all positive and negative integral values ofz. Carlson (l. c., p. 58) proves a theorem which contains the present theorem as a particular case (but is in fact substantially equivalent to it). His method of proof is similar to that of the first two proofs given here.Wigert (l. c.) refers to previous and only partially succesful attempts to prove his theorem, and gives a proof based on a theorem ofPhragmén (‘Sur une extension d'un théorème classique de la théorie des fonctions’,Acta Mathematica, vol 28, 1904, pp. 351–369). He deduces as a corollary a result relating to the case in whichf(z) vanishes only for positive integral values ofz; in this the number π is replaced by the less favourable number 1/2π. I may add that a similar result, in which 1/2π is replaced by the still less favourable number I, was found independently byPólya (‘Über ganzwertige ganze Funktionen’,Rendiconti del Circolo Matematico di Palermo, vol. 40, 1915, pp. 1–16). It should be added that this result ofPólya appears only incidentally as a corollary of theorems of a somewhat different character and of the highest interest.
E. Phragmén andE. Lindelöf, ‘Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogènes dans le voisinage d'un point singulier’,Acta Mathematica, vol. 31, 1908, pp. 381–406.
This follows from the argument of pp. 404–405 ofPhragmén andLindelöf's memoir. This argument presupposes that the value ofh(θ) is not always −8, a possibility excluded by the theorem of p. 385.
Phragmén and Lindelöf,l. c., p. 400.
Ibid., p. 399.
Ibid., p. 399.
Ibid., p. 403.
This artifice is due toMellin, from whose work the ideas of the proof are borrowed. SeeHj. Mellin, ‘Über die fundamentale Wichtigkeit des Satzes von Cauchy für die Theorien der Gamma und der Hypergeometrischen Funktionen;Acta Societatis Fennicae, vol. 21, no. 1. 1896, pp. 1–111 (pp. 37et seq.).
A. Pringsheim, ‘Elementare Theorie der ganzen transcendenten Funktionen von endlicher Ordnung’,Mathematische Annalen, vol. 58, 1904, pp. 257–342.
S. Wigert, ‘Sur les fonctions entières’,Öfversikt auf K. Vet.-Ak. Förhandlingar, Årg. 57, 1900, pp. 1001–1011.
E. le Roy, ‘Sur les séries divergentes et les fonctions données par un développement de Taylor’,Annales de la Faculté des Sciences de Toulouse, ser. 2, vol. 2, 1900, pp. 317–430 (pp 350–353).
E. Lindelöf,Le calcul des résidus, Paris, 1905, p. 127. See also ‘Quelques applications d'une formule sommatoire générale’,Acta Societatis Fennicae, vol. 31, no. 3, 1902, pp. 1–46.
G. Faber, ‘Über die Fortsetzbarkeit gewisser Taylorscher Reihen’,Mathematische Annalen, vol. 57, 1903, pp. 369–388.
A. Pringsheim, ‘Über einige funktionentheoretische Anwendungen der Eulerschen Reihen-Transformation’,Münchner Sitzungsberichte, 1912, pp. 11–92 (pp. 40–45).
See,e. g.,Pringsheim,l. c., pp. 38.
J. Hadamard, ‘Essai sur l'étude des fonctions données par leur développement de Taylor’,Journal de mathématiques, ser 4, vol. 8, 1892, pp. 101–186.
CompareG. H. Hardy, ‘A method for determining the behaviour of a function represented by a power series near a singular point on the circle of convergence’,Proc. London Math. Soc., ser. 2, vol. 3, 1905, pp. 381–389.
The argument fails forn=0 unlessf(∞)=0. CompareWigert's paper.
See pp. 337–342 ofPringsheim's paper in theMathematische Annalen quoted above., pp. 337–342.
The substance of these results is contained in the work ofle Roy andLindelöf. Cf.le Roy,l. c.. andLindelör,Le calcul des résidus, p. 135-136. A less complete result is given byPringsheim: see p. 46 of his paper in theMünchener Sitzungsberichte already referred to.A. Pringsheim, ‘Über einige funktionentheoretische Anwendungen der Eulerschen Reihen-Transformation’,Münchner Sitzungsberichte, 1912, pp. 46.
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Hardy, G.H. On two theorems of F. Carlson and S. Wigert. Acta Math. 42, 327–339 (1920). https://doi.org/10.1007/BF02404414
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DOI: https://doi.org/10.1007/BF02404414