« PreviousContinue »
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)r(n-1)
= (n-1)(n-2)... (n-m) r(n—m). (274) And replacing n by n+m
r(n+ m) = (n+m− 1)(n+m-2)... (n+1) n r(n): . (275) so that r(n + m) depends on r(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,
r(n) = (n-1)(n-2)... 3.2 r(2)
= (n-1)(n-2)... 3.2.1 r(1)
(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 1-1, 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) = *(!?
= 0, since r(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
12.22.32. ... (n-1)2 = 1.2.3. ... (2n-1)
i 1.2.3 ... (n-1)
= 2n-11.3.5 ... (2n-1) 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.
when m = 00,
1.2.3 ... (m-1)
1 m", when m = cc, (277) 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); thus
r(n+1) = | e-*x" dx
= [ -e ***** +n /*--4-1dx
= nr(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; po xn-1 dx Jo (1 + x)*+*
n-1 | 2012 dx
= [-m+h-1(1+*w=]+ m7716*1*die
* Commentationes recentiores Societatis Scientiarum Gottingensis ; Vol. I, Gottingen, 1812.
po an-ı dx (n-1) po 21–2 dx Jo (1 + x)m+n =
(n-1)(n-2)... 3.2.1 poo da (m+n-1) (m+n-2)... (m+1)J. (1+x)"+1
(n-1)(n-2) ... 3.2.1 (m+n-1)(m+n-2) ... (m+1) m (n-1)(n—2) ... 2.1.(m-1) (m—2) ... 2.1
(m + n-1) (m+n-2) ............2.1. r(m) r(n)
r(m + n) 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
= imm, by reason of (72) Art. 94; (278) 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 for all values of n from 0 to 5, it is also known for all values of 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 5, it is also known for all other values of n. The following are deductions from the preceding. (1) Since nr(n) = r(n+1), r(1+n) r(1-n) =
(279) sin ni
(2) Let n = 3; then from (278),
In this equation let æ be replaced by 22; 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 = 5; then
2 po dx
then taking the products of all the right-hand members and of all the left-hand members separately, we have in-11) 2
nn Now, by (52), Art. 64, Vol. I., substituting 2n for n, we have
1=(20–20008 + 1) (-–20 cos* + 1...
...(12 – 2.cos (1.1)* + 1). (282)
N In this equation let +1 and -1 be successively substituted for X; then
therefore, taking the product of these two and extracting the square root,
so that substituting in the denominator of (281), and extracting the square root,
130.] The form of the preceding equation suggests the means
of determining the value of | ** log f(x) dx.
Let us first consider / log r(x) dx, and suppose the range of inte
gration to be 1-i, where i = - and is an infinitesimal; so that the value of log f(x) at the inferior limit, being oo, may be excluded; then, if n = 00 ,