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| |
| <h3 class="section">19.6 Special Functions</h3> |
| |
| <p><a name="index-special-functions-2240"></a><a name="index-Bessel-functions-2241"></a><a name="index-gamma-function-2242"></a> |
| These are some more exotic mathematical functions which are sometimes |
| useful. Currently they only have real-valued versions. |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>erf</b> (<var>double x</var>)<var><a name="index-erf-2243"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>erff</b> (<var>float x</var>)<var><a name="index-erff-2244"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>erfl</b> (<var>long double x</var>)<var><a name="index-erfl-2245"></a></var><br> |
| <blockquote><p><code>erf</code> returns the error function of <var>x</var>. The error |
| function is defined as |
| <pre class="smallexample"> erf (x) = 2/sqrt(pi) * integral from 0 to x of exp(-t^2) dt |
| </pre> |
| </blockquote></div> |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>erfc</b> (<var>double x</var>)<var><a name="index-erfc-2246"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>erfcf</b> (<var>float x</var>)<var><a name="index-erfcf-2247"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>erfcl</b> (<var>long double x</var>)<var><a name="index-erfcl-2248"></a></var><br> |
| <blockquote><p><code>erfc</code> returns <code>1.0 - erf(</code><var>x</var><code>)</code>, but computed in a |
| fashion that avoids round-off error when <var>x</var> is large. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>lgamma</b> (<var>double x</var>)<var><a name="index-lgamma-2249"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>lgammaf</b> (<var>float x</var>)<var><a name="index-lgammaf-2250"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>lgammal</b> (<var>long double x</var>)<var><a name="index-lgammal-2251"></a></var><br> |
| <blockquote><p><code>lgamma</code> returns the natural logarithm of the absolute value of |
| the gamma function of <var>x</var>. The gamma function is defined as |
| <pre class="smallexample"> gamma (x) = integral from 0 to &infin; of t^(x-1) e^-t dt |
| </pre> |
| <p><a name="index-signgam-2252"></a>The sign of the gamma function is stored in the global variable |
| <var>signgam</var>, which is declared in <samp><span class="file">math.h</span></samp>. It is <code>1</code> if |
| the intermediate result was positive or zero, or <code>-1</code> if it was |
| negative. |
| |
| <p>To compute the real gamma function you can use the <code>tgamma</code> |
| function or you can compute the values as follows: |
| <pre class="smallexample"> lgam = lgamma(x); |
| gam = signgam*exp(lgam); |
| </pre> |
| <p>The gamma function has singularities at the non-positive integers. |
| <code>lgamma</code> will raise the zero divide exception if evaluated at a |
| singularity. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- XPG --> |
| <div class="defun"> |
| — Function: double <b>lgamma_r</b> (<var>double x, int *signp</var>)<var><a name="index-lgamma_005fr-2253"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- XPG --> |
| — Function: float <b>lgammaf_r</b> (<var>float x, int *signp</var>)<var><a name="index-lgammaf_005fr-2254"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- XPG --> |
| — Function: long double <b>lgammal_r</b> (<var>long double x, int *signp</var>)<var><a name="index-lgammal_005fr-2255"></a></var><br> |
| <blockquote><p><code>lgamma_r</code> is just like <code>lgamma</code>, but it stores the sign of |
| the intermediate result in the variable pointed to by <var>signp</var> |
| instead of in the <var>signgam</var> global. This means it is reentrant. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>gamma</b> (<var>double x</var>)<var><a name="index-gamma-2256"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>gammaf</b> (<var>float x</var>)<var><a name="index-gammaf-2257"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>gammal</b> (<var>long double x</var>)<var><a name="index-gammal-2258"></a></var><br> |
| <blockquote><p>These functions exist for compatibility reasons. They are equivalent to |
| <code>lgamma</code> etc. It is better to use <code>lgamma</code> since for one the |
| name reflects better the actual computation, moreover <code>lgamma</code> is |
| standardized in ISO C99<!-- /@w --> while <code>gamma</code> is not. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- XPG, ISO --> |
| <div class="defun"> |
| — Function: double <b>tgamma</b> (<var>double x</var>)<var><a name="index-tgamma-2259"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- XPG, ISO --> |
| — Function: float <b>tgammaf</b> (<var>float x</var>)<var><a name="index-tgammaf-2260"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- XPG, ISO --> |
| — Function: long double <b>tgammal</b> (<var>long double x</var>)<var><a name="index-tgammal-2261"></a></var><br> |
| <blockquote><p><code>tgamma</code> applies the gamma function to <var>x</var>. The gamma |
| function is defined as |
| <pre class="smallexample"> gamma (x) = integral from 0 to &infin; of t^(x-1) e^-t dt |
| </pre> |
| <p>This function was introduced in ISO C99<!-- /@w -->. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>j0</b> (<var>double x</var>)<var><a name="index-j0-2262"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>j0f</b> (<var>float x</var>)<var><a name="index-j0f-2263"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>j0l</b> (<var>long double x</var>)<var><a name="index-j0l-2264"></a></var><br> |
| <blockquote><p><code>j0</code> returns the Bessel function of the first kind of order 0 of |
| <var>x</var>. It may signal underflow if <var>x</var> is too large. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>j1</b> (<var>double x</var>)<var><a name="index-j1-2265"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>j1f</b> (<var>float x</var>)<var><a name="index-j1f-2266"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>j1l</b> (<var>long double x</var>)<var><a name="index-j1l-2267"></a></var><br> |
| <blockquote><p><code>j1</code> returns the Bessel function of the first kind of order 1 of |
| <var>x</var>. It may signal underflow if <var>x</var> is too large. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>jn</b> (<var>int n, double x</var>)<var><a name="index-jn-2268"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>jnf</b> (<var>int n, float x</var>)<var><a name="index-jnf-2269"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>jnl</b> (<var>int n, long double x</var>)<var><a name="index-jnl-2270"></a></var><br> |
| <blockquote><p><code>jn</code> returns the Bessel function of the first kind of order |
| <var>n</var> of <var>x</var>. It may signal underflow if <var>x</var> is too large. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>y0</b> (<var>double x</var>)<var><a name="index-y0-2271"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>y0f</b> (<var>float x</var>)<var><a name="index-y0f-2272"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>y0l</b> (<var>long double x</var>)<var><a name="index-y0l-2273"></a></var><br> |
| <blockquote><p><code>y0</code> returns the Bessel function of the second kind of order 0 of |
| <var>x</var>. It may signal underflow if <var>x</var> is too large. If <var>x</var> |
| is negative, <code>y0</code> signals a domain error; if it is zero, |
| <code>y0</code> signals overflow and returns -&infin;. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>y1</b> (<var>double x</var>)<var><a name="index-y1-2274"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>y1f</b> (<var>float x</var>)<var><a name="index-y1f-2275"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>y1l</b> (<var>long double x</var>)<var><a name="index-y1l-2276"></a></var><br> |
| <blockquote><p><code>y1</code> returns the Bessel function of the second kind of order 1 of |
| <var>x</var>. It may signal underflow if <var>x</var> is too large. If <var>x</var> |
| is negative, <code>y1</code> signals a domain error; if it is zero, |
| <code>y1</code> signals overflow and returns -&infin;. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- SVID --> |
| <div class="defun"> |
| — Function: double <b>yn</b> (<var>int n, double x</var>)<var><a name="index-yn-2277"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: float <b>ynf</b> (<var>int n, float x</var>)<var><a name="index-ynf-2278"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- SVID --> |
| — Function: long double <b>ynl</b> (<var>int n, long double x</var>)<var><a name="index-ynl-2279"></a></var><br> |
| <blockquote><p><code>yn</code> returns the Bessel function of the second kind of order <var>n</var> of |
| <var>x</var>. It may signal underflow if <var>x</var> is too large. If <var>x</var> |
| is negative, <code>yn</code> signals a domain error; if it is zero, |
| <code>yn</code> signals overflow and returns -&infin;. |
| </p></blockquote></div> |
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