REPORT ON CERTAIN BRANCHES OF ANALYSIS. 21 1 



It is easy to construct formulae which may exhibit the possi- 

 bility of their thus degenerating into others of a much more 

 simple form, when one or more of the independent variables be- 

 come whole numbers : of this kind is the formula 



« + <S sin (2 r TT + ^) + y sin (g rir + flp + &c. ,. 



a + /3sin9 + ysinfl' +,&c. ^ fv)> 



which is, or is not, identical with 4) (r), according as r is a whole 

 or a fractional number : such functions are termed nticlulating 

 functions by Legendre *. We can conceive also the possible 

 existence of many other transcendents amongst the unknown 

 and undiscovered results of algebra, which may possess a simi- 

 lar property. 



The transcendent 



X '^ ^ O^s D' 



mentioned above, possesses many properties which give it an 

 uncommon importance in analysis, and most of all from its fur- 

 nishing the connecting link in the transition from integral 

 and positive to general indices of differentiation in algebraical 

 functions. If we designate, as Legendre has done, 



we shall readily derive the fundamental equation 



r(l +r) = rr(r) t (1) 



which is in a form which admits of all values of r. It appears 



* Traits des Fonctions Elliptiques, torn. xi. p. 476. 

 t In as much as 



t^"-" ,„ _ r (1 + n) 



-x^ = 



X'' = A a;^ 



d j;"-' r (1 + r) 



and 



dn-r+l r (1 + W) 



(lx«-r + 1 * — r (r) '^ — D X —TAX , 



it follows that r A = B, and therefore also that 



"which is the equation (l) : and it is obvious that the transition from 



dn-r d»-r+ 1 



.x" to 



(which is equivalent to the simple differentiation of A .V, when A is a 

 constant coefficient), will lead to the same relation between T (1 + r) and 

 r(r), ivhateiJe)-he the value of »•, whether positive or negative, whole or frac- 

 tional. Legendre has apparently limited this equation to positive values of r. 



p2 



