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this series however is not sufficiently convergent for purposes of calculation.

Since however log (1+n) = n-ö +5-āt ... ; therefore by addition, logr(1+n) = -log(1+n)+(1-)+ (8,-1)", -(8-1) + ..., (323) which is a series sufficiently convergent. Also another, more convergent, may be deduced from it.

Replace n by -n; then log r(l-n) = -log(1 –n)–(1 –E) + (8,-1)".+(8,-1) + ... (324) and subtracting this from (323), we have logr(1+n) – log r(1-0) = log17m +2{(1–2) – (891) (851) .}. (325)

Now, by (279), log r(1+n) + logr(1 –n) = log in 1,; therefore, adding this to (325), and dividing the result by 2, we have, logr(1+n) = log in my + 5log 171 +(1-2)

–(8) – 1).- (85– 1) -... (326) Since now it is only necessary, as before observed, to calculate r(n) for all values of n between two numbers whose difference is ə, it will be sufficient to apply the preceding formula for all values of n between 0 and; or by reason of the third fundamental theorem, between 0 and a quantity less than ; thus, as n is a small quantity, (326) will give the required value with great facility. These are also convenient numbers, because they give those values of r(n) near to which it has its minimum value.

It is necessary however to determine the value of E.

Let n = 1 in (326); then log r(1+n) = log r(2) = 0; and 1. NA(1-n) _111 . (1+n) sin na

, when n = 1; so that

ni

sin ns

me

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whence it is found that E = .57721566 ...... ; = .07721566 ......;

(328) which is the arithmetical value of Euler's Constant*.

135.] We may also hereby determine the value of n for which r(1+n) is a minimum, and thus complete the remarks as to the course of the function made in Art. 124. When r(1 + n) is a minimum, "= 0; and consequently

• dn d.log r(1+n) -0. Now taking the n-differential of (323), we have

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0=-7+1-E+ (S,–1)n- ($3–1)n? +(84–1)n?... ; and if the coefficients are calculated, it will be found by a process of approximation that n lies between .4 and .5; and that 1+1 = 1.461632.......

(329) We can also hence deduce an expression for the sum of a series of terms in harmonic progression : for, from (300) we have, d.log r(n) +8 = (?="dz dn

Jo 1-2
= [ {1+z+z2+...+2*-2}dz, if n is an integer ;

1 1 1
ī + 7 +3+...+n-Ti ,

(330) which gives an analytical expression for the sum of the harmonic series. d.log r(n)

? may of course be replaced by either of its values given in Art. 131.

136.] If we accept Gauss' definition of the Gamma-function, given in (277), the fundamental theorems may be deduced from it; thus we have

1.2.3 ... (m-1) s(n) =

M", when m = 0; ? (+1)(n+2)... (a +m–1)"

1.2.3 ... (m-1) .. r(n+1) =

ma+1, when m=0; (331) (n+1)(n+2)... (n + m)"

• On certain discrepancies existing in the calculated values of E, see an Article by Oettinger, Crelle's Journal, Vol. LX. p. 375.

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r(n+1) nm

= = n, when m = 0;
r(n) n + m
in f(n+1) = nr(n);

(332) which is the first fundamental theorem of the Gamma-function. Again, replacing n by — n in (331),

1.2.3 ... (m-1)

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therefore

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) (1 - 1) ... (1

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*(1-) (1

) in nam , by (135), Art. 89, Vol. I; (333) which is the second fundamental theorem of the Gamma-function. The third theorem may be proved by a similar process.

137.1 A remarkable value of r(n+1), when n = 00, may also be deduced from Wallis' value of t, which is given in (15), Art. 82. Since r(n+1) = 1.2.3 ... N, r(n+1) 1 2 3 n-1n

(334) maannn . n ñ which is a small fraction of 1, and evidently is infinitesimal when n is infinite; let this quantity = f(n); so that 1.2.3 ... n = 1" f(n);

(335) where f(n) is an infinitesimal, when n = 0. Now by Wallis' Theorem, when n is an integer,

76 22.42.62 ... (2n-2)22n
2 =

2, when n = 0;
1.32.52. ... (2n-1)2
2.4.6 ... (2n-2)2n

1.3.5... (2n-1)
22.42.62 ... (2n-2)2 (2n)2

1.2.3.4... (2n-1)2n
{1.2.3... (n-1)n}a 22

1.2.3 ... (2n-1)2n
na"{f(n)}2 22n
(2n)2nf (2n)

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(27) r(m)

nom+n-te-n,...; (339)

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