On the Represenation of a Function By a Trigonometric Series
On the Represenation of a Function By a Trigonometric Series
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, 1880, p. 104). Since this is rather condensed, I have expanded it a little in a few places. In particular, I have given a proof for the case where the function becomes infinite, du Bois-Reymond's treat- ment of which Sachse has omitted.
tThis will be the case if they will have a form like that given in the theorem, as is shown in the footnote on page 8.
X This requires of course that/(a) be integrable in the interval ( — tt, n) .
§ Shown in Riemann's Math. Werke, pp. 231-34 : Bull, des Sci. M...ath. , Vol. V, 1873, pp. 41-45 ; Picard, Traite d' Analyse, Vol. I, pp. 240-44.
7 3°. Lim F{x + e)+F(z-*)-2F(x) = Q for yalue of ^ e = £ Let us consider first a function f(x) which is finite and continuous and does not have an infinite number of maxima and minima. We must have necessarily for every value of x between — re and -f- rt, ^\ X dafe(P)dp==f(x). (9) Form the expression 0{x) = F(x)— Pda [/(/?) d/3.
J IT * — It From the property 2° and from (9) it follows that lim 0(* + e)-20(*)+0(*-g) =lim 4*0 {x) _ Q « = o e 2 e=o s 2 and therefore 0(x)=ZC o -\-C 1 X.
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You can find similar books in the "Read Also" column, or choose other free books by Edward Payson Manning to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
, 1880, p. 104). Since this is rather condensed, I have expanded it a little in a few places. In particular, I have given a proof for the case where the function becomes infinite, du Bois-Reymond's treat- ment of which Sachse has omitted.
tThis will be the case if they will have a form like that given in the theorem, as is shown in the footnote on page 8.
X This requires of course that/(a) be integrable in the interval ( — tt, n) .
§ Shown in Riemann's Math. Werke, pp. 231-34 : Bull, des Sci. M...ath. , Vol. V, 1873, pp. 41-45 ; Picard, Traite d' Analyse, Vol. I, pp. 240-44.
7 3°. Lim F{x + e)+F(z-*)-2F(x) = Q for yalue of ^ e = £ Let us consider first a function f(x) which is finite and continuous and does not have an infinite number of maxima and minima. We must have necessarily for every value of x between — re and -f- rt, ^\ X dafe(P)dp==f(x). (9) Form the expression 0{x) = F(x)— Pda [/(/?) d/3.
J IT * — It From the property 2° and from (9) it follows that lim 0(* + e)-20(*)+0(*-g) =lim 4*0 {x) _ Q « = o e 2 e=o s 2 and therefore 0(x)=ZC o -\-C 1 X.
What to read after On the Represenation of a Function By a Trigonometric Series?
You can find similar books in the "Read Also" column, or choose other free books by Edward Payson Manning to read onlineMoreLess
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