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SCIENCE 



[N. S. Vol. XXXIX. No. 995 



by his series (I) could consist of separate 

 pieces of any sort, not merely having no 

 logical or definitional dependence on one 

 another, but even not connecting succes- 

 sively at their ends. Thus by virtue of Fou- 

 rier's assertion the power of representation 

 through analytic expression is at least as 

 great as the power of geometric picturi- 

 zation. 



When once it was realized that mathemat- 

 ical expression could be adapted to the 

 most diverse and unrelated demands upon 

 it, no logical stopping-point could be seen 

 short of the definition to-day accepted for 

 a function of a real variable, and often re- 

 ferred to as the Dirichlet definition of a 

 function. If, namely, to every value of x 

 in an interval there corresponds a definite 

 value of 2/ (no matter how fixed or deter- 

 mined), y is called a function of x. For 

 example, y may be equal to + 1 at all 

 rational points which are everywhere dense 

 in any interval, and equal to at the irra- 

 tional points which are likewise everywhere 

 dense. The Fourier series has thus neces- 

 sitated a radical reconstruction of the notion 

 of a function. This is the first of its serv- 

 ices which I ivish to emphasize, the devel- 

 opment and complete clarification of the 

 concept of a function. 



Without loss of generality the interval in 

 which the representation of the function by 

 the series is required may be supposed to 

 lie between — tt and + -w. The series has 

 then the form (I.) hitherto assumed. To 

 determine its coefficients from the function 

 Fourier used for the most part the 

 equations, 



1 c" 



Cn = - I fix) COS nxdx, 



1 P" 



6n = - / fix) sin nxdx; 



but this determination, as Fourier himself 

 stated, had been made by Euler before him. 

 Trigonometric series whose coefficients can 



be obtained from the function represented 

 in this manner are now called Fourier's 

 series in distinction from trigonometric se- 

 ries whose coefficients can not be so obtained 

 through integration. I have, however, in 

 the title of my paper used the term "Fou- 

 rier's series" in the older and broader sense 

 as synonymous with all series of the form 

 (I.). 



The consideration of trigonometric series 

 from a strict mathematical standpoint 

 marks a second epoch in their history. This 

 began with Dirichlet in 1829 in a memoir 

 remarkable for its combination of clearness 

 and rigor. Here he first determined accu- 

 rately a set of sufficient conditions for the 

 expansion of a function into a Fourier se- 

 ries. These familiar "Dirichlet conditions" 

 it is scarcely necessary to repeat. 



The extension of his results was at once 

 sought, in particular by Riemann in a 

 Gottingen Habilitations-Dissertation, which 

 bore the title "Ueber die Darstellbarkeit 

 einer Function dureh eine trigonometrisehe 

 Reihe. " Riemann 's aim was, however, to 

 determine the necessary conditions for the 

 representability of the function by the 

 series. Must the function be integrable, as 

 required in the sufficient conditions of Dir- 

 ichlet ? Must it have only a finite number of 

 maxima and minima and of discontinuities? 

 Such questions as these were easily an- 

 swered by him in the negative, and a flood 

 of light was poured upon the problem of 

 representability but without making visible 

 its complete solution. Possibly it was for 

 this reason that this Habilitationsschrift, 

 though delivered in 1854, was not published 

 until thirteen years later, and then only 

 after Riemann 's death. Yet the work is 

 a classic. As has been said of the poet 

 Coleridge, so it could be said of Riemann, 

 he wrote but little, but that little should be 

 bound in gold. 



To put the theory of Fourier's series on 



