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.

284 Examples on the calculation of definite integrals. Bk. 111. ch. VII.
Provided b > a, both sides are continuous functions of c, the
value c = 0 included (~ 152 a). For c 0 and k = 1 the equation
assumes the form:
r (b) = r (a) r - a) (b > a),
that can also be written:
re(a+ b) r Ji + X)+b
This formula shows how to calculate a new integral, of the
binomial class, by means of the functions gamma. Legendre styles
it the Eulerian integral of the first kind. Putting:
_ y_ dx dy
1Y- y? ( x 1
CZ — y --- I + -- X
we have y = 0 for x = 0, y = 1 for x = co; and when y is replaced
again by x or by 1- x, IV) changes into:
1 1
F(a) Fb) = a-l - X)-1 (1 -- - X b-1 dx.*)
0 0
When we put a + b =1, taking a therefore as a proper fraction,
since r() = 1, formula IV) becomes:
0o
0
V) d1 x+x - (a) r(l - a) = sinr > ~ 1]59 Formula 3).
This formula reduces the calculation of all values of F(a) for arguments
greater than I to that of values for arguments between zero and ~.
We have for a = the special value:
r(l) Je-xx- i dx = r'.,o
0
Replacing x by y2, we obtain the integral treated in ~ 158.
162. Expression of the function gamma by an infinite
produ ct.
Integral I) can be differentiated with respect to the parameter a
(~ 152) and we get:
0O
d F (t) _ r(c) =.e- _f-exl()(X.
If we replace l(x) by its value as an integral, ~ 154 Formula 3a):
*) Euler, loc. cit., ~ 25.