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FUNCTION. 
[787 
15. We have the function defined by its expression as a hypergeometric series 
F(a, /3, 7, w) = 1 +i— u + ~ i o ri— w +&c., 
1.7 l.z.7.7 + 1 
i.e., this expression of the function serves as a definition, if the series be finite or 
if, being infinite, it is convergent. The function may also be defined as a definite 
integral; in other words, if, in the integral 
f x a ' 1 (1 — x)P' 1 (1 — ux)~y’ dx, 
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we expand the factor (1 — ux) in powers of ux, and then integrate each term 
separately by the formula for the second Eulerian integral, the result is 
which is 
r(a' + /3') r(«' + /3'+l) 1 + * 
Ta'.r/T , a'.7' , a', a' + l. y'.y' + l B , 
r(a' + /3') \ 1 + a'+ /3' .1 U + a.'+ /3'. a'+ /3' + 1.1.2 U + 
or writing a', /3', 7' = a, 7 — a, /3 respectively, this is 
FaT (7 — a) 
so that the new definition is 
r 7 
F (a, /3, 7, u) = - a - t ( 7—— f ¿e a_1 (1 — fl?)* 3-1 (1 — wic) -3 dx; 
17 
i^(a, /3, 7, w), 
but this is in like manner only a definition under the proper limitations as to the 
values of a, /3, 7, ii. It is not here considered how the definition is to be extended 
so as to give a meaning to the function F (a, ¡3, 7, u) for all values, say of the 
parameters a, /3, 7, and of the variable u. There are included a large number of 
special forms which are either algebraic or circular or exponential, for instance 
F (a, ¡3, /3, u) = (1 — u)~ a , &c.; or which are special transcendents which have been 
separately studied, for instance, Bessel’s functions, the Legendrian functions X n presently 
referred to, series occurring in the development of the reciprocal of the distance 
between two planets, &c. 
16. There is a class of functions depending upon a variable or variables x, y, ... 
and a parameter n, say the function for the parameter n is X n such that the product 
of two functions having the same variables, multiplied it may be by a given function 
of the variables, and integrated between given limits, gives a result = 0 or not = 0, 
according as the parameters are unequal or equal; JUX m X n dx = 0, but JUX n 2 dx not 
= 0; the admissible values of the parameters being either any integer values, or 
it may be the roots of a determinate algebraical or transcendental equation; and the 
functions X n may be either algebraical or transcendental. For instance, such a function 
is cos nx; 711 and n being integers, we have f cos mx. cos nx dx = 0, but [ cos 2 nx dx = &r. 
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