2 1

n

3

n+1

logr (n) = (n−1)log ——log + (n-1) log-log +

1

2

......

Let the sum of all the terms after the mth in this series = log (1+i); where i = 0, when m = ∞; m∞; then

2

n

3

log r(n) = (n − 1) log —-log+(n−1) log-log

+log (1+i); (317)

(1+i), when m= ∞, i=0;

m", when m = ∞. (318)

This is Gauss' definition of r(n), and is the equivalent which has already been proved in Art. 127.

From (315) we have the following value of E;

The numerical value of E may be determined by this series; but the calculation is very long and laborious.

134.] The calculation of r (n).

In (311) let n + 1 be substituted for n; then

Now applying Maclaurin's Theorem to log r(1+n) = F(n), say;

F (0) = 0, since r(1) = 1; F′(n) =

d.log r(1+n)
dn

; therefore

'(0)E, by (312); and F"(0),... are given by (320), so that

this series however is not sufficiently convergent for purposes of

n

1

= − log (1+ n) + (1 − E) + (§8,−1) 1,2 − (§3−1) 353

n2

3

which is a series sufficiently convergent. Also another, more convergent, may be deduced from it.

Replace n by -n; then logr (l—n)

Now, by (279), logr (1+n)+ logr(1—n) = log

fore, adding this to (325), and dividing the result by 2, we have,

Since now it is only necessary, as before observed, to calculate r(n) for all values of n between two numbers whose difference is

1

2'

it will be sufficient to apply the preceding formula for all values of n between 0 and or by reason of the third funda

1

1

mental theorem, between 0 and a quantity less than; thus, as

2

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.

11

Let n = 1 in (326); then log r(1+n) = log r(2) = 0; and

log

nπ (1-n)

(1 + n) sin n

=

1

1

log, when n = 1; so that

1-E=

1

1

— log 2 + (8,−1) + ( − 1) + 1 (8,−1) + ... ; (827)

1 3

5

whence it is found that E = .57721566

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

(328)

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. Now taking the n-differential of (323), we have

+1−E+ (S,−1)n—(s,−1) n2 + (§4—1) n3..... ;

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+n = 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,

= √ ] { 1 + 2 + 2 2 + ... + z " ~ 2 } dz, if n is an integer ;

=

which gives an analytical expression for the sum of the harmonic

series.

d. log r(n)
dn

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

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

which is the first fundamental theorem of the Gamma-function.

Again, replacing n by -n in (331),

which is the second fundamental theorem of the Gamma-function. The third theorem may be proved by a similar process.

137.] A remarkable value of г(n + 1), when n = ∞, may also be deduced from Wallis' value of π, which is given in (15), Art. 82. Since г(n+1)= 1.2.3... n,

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... nn" f(n);

(335)

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

which is a functional equation of the form, (2n) = {(n)}2; and of which evidently a solution is (n) = =ean, where a is an

undetermined constant; so that

f(n) (2πn)

=ean, and consequently

(335) becomes

1.2.3... n = n" (2πn) ean, when n∞.

Now to determine a; replace n by n+1; then

1.2.3 ... n (n + 1) = (n+1)"+1 {2π(n+1)} ea(n+1), when n= ∞o; and dividing the latter by the former

1

and, taking logarithms, (1⁄2 + 1⁄2) log (1 + 1) =

(n

=-a; whence, by

(336)

evaluation, when n = ∞, a = −1; and thus,

1.2.3... n = (2π)1n"+3 e ̄”, when n = ∞.

This result enables us to correct the error of the approximate formula (208), Art. 113, and to give a more general theorem. Let us suppose c to be the ratio of the first to the second member of that formula, so that c nearly = 1; then

m (m+1) (m+2) ... (m+n−1) when no.

= cmt em nm+n-he-", (337)

Let m = 1; then 1.2.3 n = ce1¬"n"+3, when n = ∞;

and equating this value to that given in (336), c =

hence from (337),

n

m (m + 1) (m + 2) . . . (m + n − 1) = (2π)3 ( — )m—àn"em-n−1, (338)

when n = co.

m