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:: log r(x) log 27 + n(log n–1). (288) 131.] As a primary object of this research however is the determination of the value of r(n) for all positive values of n, we must investigate other general properties of the function, so as to supply the deficiencies of the two former general theorems. They are sufficient for the determination of values of r(n), when the values are known for all numbers lying between two numbers, the difference between which is 5; but they fail to give the latter values. In the course of the inquiry too we shall be led to some new and important theorems of the Gamma-function, and to another proof of Gauss's definition of it.

Taking the n-differential of the Gamma-function, as in (252), we have d.r(n) = |e-***-1 log x dx.

(289) dn T Jo Now by an artifice due, I believe, to M. Cauchy*, which is of great use in the evaluation of definite integrals, we may replace log x by any definite integral which is equivalent to it; and if the limits of this equivalent are constant, the order of the integrations is arbitrary; and they may consequently be taken in that which is most convenient to the problem. Thus, since

= [e-*dz ;

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replacing log x in (289) by this value, we have

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* Exercises d'Analyse, tome II, p. 379. Paris, 1841.

But / -221-1 dx = f(n), by the definition of the Gammafunction; and by (250), / e-(1+2)® zn–1dx = (12)n; so that substituting these in (291), and placing r(n) outside the sign of integration, we have

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Let the n-integral of this equation be taken for the limits n and l; then, bearing in mind that r(1)=1, so that logr(1)=0,

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132.] These equations lead to another fundamental theorem of the Gamma-function, which was discovered by Gauss, and is given by him in his previously cited memoir. The theorem is more commonly called the third fundamental theorem of the Gamma-function; and is useful in reducing the number of particular values of the function which must be determined by direct calculation.

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and let the sum of (300) and of all these several terms be taken; then

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= -r logr. (303) Let the definite integral of this equation be taken for the limits n and; then, since by (286),

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r (m) s (n + 3) =(n + 3) ...(n+";})

= (27) 7 m2="" (rn); (304) which is the third fundamental theorem of the Gamma-function.

The following are particular forms of this general theorem.
Let r= 2;
Let r= 2; f(n)r (n +) = v(2n); (305)

22n-1 r(2n); (305) let r= 3; r(n)r (n + 3) =(n + 3) = 3 2 -(3n); (306) consequently, if n = 3, from (305) we have r () = nt, as we have determined several times heretofore.

We have said that the theorem is useful for the reduction of the number of values of r(n), which must be found by direct calculation. In illustration of this, suppose that the values of r(n) are required for all arguments at an interval of th between two successive integers. By the first general theorem these values depend on those of r(n) at an interval of th between 0 and 1; that is, on r(@), r(ā), +(3), 4(2), 16). Of these the third, viz. (3), = r () = , and is known. Also, by the second general theorem, we have

+6)) = =-= 22 (307) (3)=() =

(309) And by (305), if n = , r(a): (a) = 23 x+r(a). (309) Consequently if (3) is known, all the others are known. In other similar cases the number of the values of the Gamma-function, which must be determined by direct arithmetical summation, may be reduced by means of the theorem contained in (304).


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133.] We now come to the investigation of series for the direct calculation of log r(n). Returning to (297) we have,

d.logor (n) = 1 {"- ", dy. (310)

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= loy{e="v te=(n+1)x+e=(n+2)" +... }dy

1 1 1 1 = + on that in

no "(n1)* *(a +2) + (m +3)3 +...; (311) and this is evidently a convergent series for all values of n, except negative integers. · Let us take the n-integral of (311), between the limits n and 1; then as from (297) and (298) we have d.logr(m) (when n = 1) = 1° -16-}dy=-B,(312) therefore

T(") += (1-7)+(3- n11)+(3-172) + ...; (313) the right-hand member being a convergent series. E is called Euler's Constant, and is of greater importance in analytical investigations than any other constant except those denoted by a and e; the analytical value of it is given in (312); the numerical value will be determined hereafter.

Let us again take the n-integral of (313) between the limits n and l; then, as r(n) = 1, when n=l, and log r(n) = 0, log r (n)=-=(n1) +("77_logņ)+("72-log"?)

+("-log}2) + .... (314) In this series let n = 2; then, since log r (2) = log 1 = 0,

B = (–log) + (–10g) + (3–log) +...; (315) multiplying which by (n—]), and subtracting from (314), we


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