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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)r(n-2)

= (n-1)(n-2)... (n-m) r(nm). (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)
= (n-1)(n-2)... 3.2.1,

(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

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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


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when m = 00,

1.2.3 ... (m-1)
= n(n+ 1)...(n+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
L m +n - 1(1+x)"+n-ido m+n-1 J. (1+x)*+ *-1

= [-m+h-1(1+*w=]+ m7716*1*die


* Commentationes recentiores Societatis Scientiarum Gottingensis ; Vol. I, Gottingen, 1812.

po an-ı dx (n-1) po 212 dx Jo (1 + x)m+n =

m+n-1. (1+x)"+n–1

(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) (m2) ... 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

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= 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

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(2) Let n = 3; then from (278),

In this equation let æ be replaced by 22; then

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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


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then taking the products of all the right-hand members and of all the left-hand members separately, we have in-11) 2


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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

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therefore, taking the product of these two and extracting the square root,

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

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

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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 ,

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