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) +.. |