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