Euler's constant gamma

Euler's constant gamma

This number is defined:
gamma=H.N-LN(N)
where N goes to infinity and the N-th harmonic number is defined to be the sum
H.N=1+(1/2)+(1/3)+...+(1/N)
of the reciprocal of the integers from 1 to N.

ALGORITHM

We use the formula

LN(N)+gamma+error=
X/FACT(1)-X**2/(2*FACT(2))+X**3/(3*FACT(3))...

where FACT is the factorial function and L. Euler proved that error=O(EXP(-X)/X).

IMPLEMENTATION

Unit: program

Interface: the LN function

Result: program displays on the screen first 60 significant decimal digits of the gamma constant

X = 130; W=500; P = 130
numeric digits P
N = X; F = 1; D = 1; S = X
do K = 2 to W
N = - N * X; F = F * K; D = F * K
S = S + N / D
end
say S - LN(X, P)

This program requires about one minute of processing time. There is an another way to obtain the value of the gamma constant: we can compute the gamma number only once and save its value into the text of the function. The following GAMMA_CONST function can be used to compute the first P decimal digits of gamma, for P=1,...,60.

GAMMA_CONST: return,
"0.5772156649015328606065120900" ||,
"824024310421593359399235988058"

CONNECTIONS

Euler's number e
Factorial
Natural logarithm
Exponential function
Technique: Beforehand computed constants

Literature

Brent R. P. Ramanujan and Euler's Constant

Computer Sciences Laboratory, Australian National University, August 1993