The following theorems are deductions from it.

(1) Let n-1, n−2, n−3,... n-m, be successively substituted for n in (273); then

r(n) (n-1)г(n-1)

= (n-1) (n-2)г(n-2)

= (n−1)(n-2)... (n—m) г(n—m).

(274)

And replacing n by n+m

(275)

so that r(n+m) depends on r(n).

r(n+m) = (n+m−1)(n+m−2) ... (n + 1) nг(n) :

Consequently if the value of

r(n) is known for all values of n comprised between 0 and 1, or between 1 and 2, or generally between any two numbers the difference between which 1, the value of the function will be known for other real values of the argument.

=

(2) If in (274) n is an integral as well as positive number, = (n-1) (n-2)...3.2г(2)

r(n)

(276)

since by (256), r(1) 1. So that if n is a positive integral number, r(n) is the product of all integral numbers from 1 to n−1, both inclusive. For this reason r(n) has been called the factorial function. Thus r(n) is known for all positive integral

numbers.

(3) If n = 0, then from (272),

r(0)

r(1) =3 0

= ∞, since г(1) = 1;

which confirms the theorem given in (255).

Hence also we have a particular form of B (m, n) which deserves notice. Let m = n = a positive integer; then

The particular form of the Gamma-function given in (276) leads to another definition of it which we shall hereafter find of considerable use. From (248) we have

PRICE, VOL. II.

Y

if m and n are integers. This equivalent has been taken by Gauss as the definition of the Gamma-function; and from it he has derived in his celebrated memoir* all the properties of the function.

128.] I may in passing observe that (273) may be deduced by integration by parts from the definite integral which defines r(n);

because n being positive, the integrated part vanishes at both limits. And if n is also an integer, we shall derive by successive integration the theorem given in (276); viz.,

r(n) = (n-1) (n-2)... 3.2.1.

And I may also observe that this being the case, if m and n are both integers, (262) may be proved as follows by indefinite integration;

*Commentationes recentiores Societatis Scientiarum Gottingensis; Vol. I,

Gottingen, 1812.

=

=

(m+n−1)(m+n−2)... (m+1)。 (1+x)TM+1

=

r(m + n)

2.1

2.1

As the risk of error is great in a subject of so delicate a nature as the evaluation of definite integrals, it is expedient to verify the theorems, whenever verification is possible. Thus, although in the general theorem given in (262), m and n are not necessarily integral numbers, yet the preceding process proves the truth of the theorem when they are integers.

129.] Second fundamental theorem of the Gamma-function. In (262), let m+n = 1; then

which is the second fundamental theorem of the Gamma-function. It is subject to no other condition than that n and 1-n are both positive numbers: so that n is a positive proper fraction.

From this theorem it follows that if the value of r(n) is known

1

for all values of n from 0 to, it is also known for all values of

1

2

n from to 1. And consequently from this theorem taken in connection with the first general theorem we learn that if the value of r(n) is known for all values of n between two numbers whose difference is, it is also known for all other values of n.

The following are deductions from the preceding.

In this equation let r be replaced by x2; then

which value has been already found in (114), Art. 100.

This may be determined from (262) without the intervention

of (278). Thus in (262) let m = n =

1

then

dx if x is replaced by x2,

=

(3) In (278) let n be successively replaced by

then taking the products of all the right-hand members and of all the left-hand members separately, we have

Now, by (52), Art. 64, Vol. I., substituting 2n for n, we have

In this equation let +1 and -1 be successively substituted for r; then

n = 2n−1(1 + cos 87) (1 + cos 2) (1+cos)...

(1+cos

n

̧(n−1)); (284)

n

therefore, taking the product of these two and extracting the

so that substituting in the denominator of (281), and extracting

130.] The form of the preceding equation suggests the means

S

n

Let us first consider log r(x) dx, and suppose the range of inte

1

gration to be 1-i, where i = = and is an infinitesimal; so that

n

the value of log г(x) at the inferior limit, being ∞, may be excluded; then, if n = ∞,

['logr(x) dx = {logr (4) + logr (2) + logr (2)

+..