751] 
235 
751. 
NOTE ON RIEMANN’S PAPER “VERSUCH EINER ALLGEMEINEN 
AUFFASSUNG DER INTEGRATION UND DIFFERENTIATION 
[From the Matliematische Annalen, t. xvi. (1880), pp. 81, 82.] 
The Editors of Kiemann’s works remark that the paper in question was contained 
in a MS. of his student time (dated 14 Jan. 1847) and was probably never intended 
for publication: indeed that he would not in later years have recognised the validity 
of the principles upon which it is founded. The idea is however a noticeable one: 
Riemann considers z x+ll , a function of x + h, expanded in a doubly infinite, necessarily 
divergent, series of integer or fractional powers of h, according to the law 
%x+h = —' Jc$ v x z. h v , (2) 
where the meaning is explained to be that the exponents differ from each other by 
integer values, in effect, that v has all the values a + p, a a given integer or fractional 
value, and p any integer number from — oo to + qo , zero included. 
Riemann deduces a theory of fractional differentiation: but without considering 
the question which has always appeared to me to be the great difficulty in such a 
theory: what is the real meaning of a complementary function containing an infinity 
of arbitrary constants ? or, in other words, what is the arbitrariness of the complemen 
tary function of this nature which presents itself in the theory ? 
I wish to point out the relation between the paper referred to, and a shoit 
paper of my own “On a doubly infinite Series/ Quart. Math. Journ. t. VI. (1851), 
pp. 45—47 } [102] : this commences with the remark “ The following completely para 
doxical investigation of the properties of the function T (which I have been in possession 
* Werke, pp. 331—344. 
30—2