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 1-1 r(n) = [' (log 1)" ̄' dx ; and since log(-) = m (1 − x), when m = ∞ ; -1 r(n) = [ ' m"-1 (1 − x=)"-1d.x, when m = ∞ . Let x be replaced by am: then 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., г(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 。 (1+x)m+ (m+n−1)(m+n-2)... (m+1) (1+x)+1 (n-1) (n-2)...3.2.1 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 for all values of n from 0 to n from 1 2 1 2 , it is also known for all values of 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 1, it is also known for all other values of n. 2 The following are deductions from the preceding. (1) Since nr(n) = r(n+1), In this equation let a 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 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 = (281) 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) +.. |