An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 158-160. Eulerian integrals. 281
the numerator and denominator. We can express the value of this
integral as the difference of two integrals of the form 1), thus:
X rn efXm-1 _ XLm' — 1 = n ~ ) i 1
<o sin rl- sin - - -t
Both sides are continuous functions of a inclusive of the value a = n,
consequently we find:
5) I dx _- -.- - e-n
n^ _ 1| sin m 7 sin - 7
s i3n n
o_ S I 1 -- 511
- cot -- - cot -.
Putting x" =-, - -- a, 1- b this becomes:
GO
6) f Z d1- ds =7 (cota - cot b), (0 < a and b < 1).
Let b - 1- a:
Go
7) dz === 2 n cotcaz.
160. Eulerian integrals (Gamma functions).
09
As found in ~ 154, fxne-xdx has the value j =I. 2... n for
0
every positive integer value of n. Withdrawing this restriction as to
the exponents of x, the problem arises, what value has the integral
00
x a —le- x dx
for an arbitrary value of a? With any finite value for the upper
limit its value can be expressed by a series, the exponential being
replaced by its expansion; but for an infinite limit this procedure
affords no direct solution.
The value of the integral is finite only for a > 0; for, when this
condition is not satisfied, the function to be integrated becomes infinite for x 0 in an order higher than the first (~ 148). After
Legendre, the integral is called the Eulerian integral of the
second kind, and the required value as a function of the exponent a
briefly denoted by F(a) its function Gamma*). Therefore:
') Euler: Inst. calc. integr., P. I, Cap. 4. 8. 9. Also: Nov. Comment. Acad.
Petrop., T. XVI; in Salomon ibidem Vol. IV, Supplement Ill. Legendre: Traitd
des fonctions elliptiques et des intdgrales Euldriennes. Tome II.