Special Functions

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These are some more exotic mathematical functions which are sometimes useful. Currently they only have real-valued versions.

- Function: double erf (double x)

- Function: float erff (float x)

- Function: long double erfl (long double x)

erf returns the error function of x. The error function is defined as

erf (x) = 2/sqrt(pi) * integral from 0 to x of exp(-t^2) dt

- Function: double erfc (double x)

- Function: float erfcf (float x)

- Function: long double erfcl (long double x)

erfc returns 1.0 - erf(x), but computed in a fashion that avoids round-off error when x is large.

- Function: double lgamma (double x)

- Function: float lgammaf (float x)

- Function: long double lgammal (long double x)

lgamma returns the natural logarithm of the absolute value of the gamma function of x. The gamma function is defined as

gamma (x) = integral from 0 to oo of t^(x-1) e^-t dt

The sign of the gamma function is stored in the global variable signgam, which is declared in math.h. It is 1 if the intermediate result was positive or zero, or -1 if it was negative.

To compute the real gamma function you can use the tgamma function or you can compute the values as follows:

lgam = lgamma(x);
gam  = signgam*exp(lgam);

The gamma function has singularities at the non-positive integers. lgamma will raise the zero divide exception if evaluated at a singularity.

- Function: double lgamma_r (double x, int *signp)

- Function: float lgammaf_r (float x, int *signp)

- Function: long double lgammal_r (long double x, int *signp)

lgamma_r is just like lgamma, but it stores the sign of the intermediate result in the variable pointed to by signp instead of in the signgam global. This means it is reentrant.

- Function: double gamma (double x)

- Function: float gammaf (float x)

- Function: long double gammal (long double x)

These functions exist for compatibility reasons. They are equivalent to lgamma etc. It is better to use lgamma since for one the name reflects better the actual computation, moreover lgamma is standardized in ISO C99 while gamma is not.

- Function: double tgamma (double x)

- Function: float tgammaf (float x)

- Function: long double tgammal (long double x)

tgamma applies the gamma function to x. The gamma function is defined as

gamma (x) = integral from 0 to oo of t^(x-1) e^-t dt

This function was introduced in ISO C99.

- Function: double j0 (double x)

- Function: float j0f (float x)

- Function: long double j0l (long double x)

j0 returns the Bessel function of the first kind of order 0 of x. It may signal underflow if x is too large.

- Function: double j1 (double x)

- Function: float j1f (float x)

- Function: long double j1l (long double x)

j1 returns the Bessel function of the first kind of order 1 of x. It may signal underflow if x is too large.

- Function: double jn (int n, double x)

- Function: float jnf (int n, float x)

- Function: long double jnl (int n, long double x)

jn returns the Bessel function of the first kind of order n of x. It may signal underflow if x is too large.

- Function: double y0 (double x)

- Function: float y0f (float x)

- Function: long double y0l (long double x)

y0 returns the Bessel function of the second kind of order 0 of x. It may signal underflow if x is too large. If x is negative, y0 signals a domain error; if it is zero, y0 signals overflow and returns -oo.

- Function: double y1 (double x)

- Function: float y1f (float x)

- Function: long double y1l (long double x)

y1 returns the Bessel function of the second kind of order 1 of x. It may signal underflow if x is too large. If x is negative, y1 signals a domain error; if it is zero, y1 signals overflow and returns -oo.

- Function: double yn (int n, double x)

- Function: float ynf (int n, float x)

- Function: long double ynl (int n, long double x)

yn returns the Bessel function of the second kind of order n of x. It may signal underflow if x is too large. If x is negative, yn signals a domain error; if it is zero, yn signals overflow and returns -oo.