shichi

Hyperbolic sine and cosine integrals.

SYNOPSIS:

double x, Chi, Shi, shichi();
shichi(x, &Chi, &Shi);

DESCRIPTION:

Approximates the integrals

x
-
| | cosh t - 1
Chi(x) = eul + ln x + | ----------- dt,
| | t
-
0

x
-
| | sinh t
Shi(x) = | ------ dt
| | t
-
0

where eul = 0.57721566490153286061 is Euler's constant. The integrals are evaluated by power series for x < 8 and by Chebyshev expansions for x between 8 and 88. For large x, both functions approach exp(x)/2x. Arguments greater than 88 in magnitude return MAXNUM.

ACCURACY:

Test interval 0 to 88.

Relative error:
arithmetic function # trials peak rms
DEC Shi 3000 9.1e-17
IEEE Shi 30000 6.9e-16 1.6e-16

Absolute error, except relative when |Chi| > 1:
DEC Chi 2500 9.3e-17
IEEE Chi 30000 8.4e-16 1.4e-16