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iv.c

/*                                        iv.c
 *
 *    Modified Bessel function of noninteger order
 *
 *
 *
 * SYNOPSIS:
 *
 * double v, x, y, iv();
 *
 * y = iv( v, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of order v of the
 * argument.  If x is negative, v must be integer valued.
 *
 * The function is defined as Iv(x) = Jv( ix ).  It is
 * here computed in terms of the confluent hypergeometric
 * function, according to the formula
 *
 *              v  -x
 * Iv(x) = (x/2)  e   hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1)
 *
 * If v is a negative integer, then v is replaced by -v.
 *
 *
 * ACCURACY:
 *
 * Tested at random points (v, x), with v between 0 and
 * 30, x between 0 and 28.
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    DEC       0,30          2000      3.1e-15     5.4e-16
 *    IEEE      0,30         10000      1.7e-14     2.7e-15
 *
 * Accuracy is diminished if v is near a negative integer.
 *
 * See also hyperg.c.
 *
 */

/* iv.c     */
/* Modified Bessel function of noninteger order */
/* If x < 0, then v must be an integer. */

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

double cephes_bessel_Iv (double v, double x)
{
    double ax, t = floor(v);
    int sign;

    /* If v is a negative integer, invoke symmetry */
    if (v < 0.0 && t == v) {
      t = v = -v;
    }

    /* If x is negative, require v to be an integer */
    sign = 1;
    if (x < 0.0) {
      if (t != v) {
          mtherr("iv", DOMAIN);
          return 0.0;
      }
      if (v != 2.0 * floor(v/2.0))
          sign = -1;
    }

    /* Avoid logarithm singularity */
    if (x == 0.0) {
      if (v == 0.0)
          return 1.0;
      if (v < 0.0) {
          mtherr("iv", CEPHES_OVERFLOW);
          return MAXNUM;
      } else
          return 0.0;
    }

    ax = fabs(x);
    t = v * log(0.5 * ax) - x;
    t = sign * exp(t) / cephes_gamma(v + 1.0);
    ax = v + 0.5;

    return t * hyperg(ax, 2*ax, 2*x);
}


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