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| <h3 class="section">19.5 Hyperbolic Functions</h3> |
| |
| <p><a name="index-hyperbolic-functions-2200"></a> |
| The functions in this section are related to the exponential functions; |
| see <a href="Exponents-and-Logarithms.html#Exponents-and-Logarithms">Exponents and Logarithms</a>. |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>sinh</b> (<var>double x</var>)<var><a name="index-sinh-2201"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>sinhf</b> (<var>float x</var>)<var><a name="index-sinhf-2202"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>sinhl</b> (<var>long double x</var>)<var><a name="index-sinhl-2203"></a></var><br> |
| <blockquote><p>These functions return the hyperbolic sine of <var>x</var>, defined |
| mathematically as <code>(exp (</code><var>x</var><code>) - exp (-</code><var>x</var><code>)) / 2</code><!-- /@w -->. They |
| may signal overflow if <var>x</var> is too large. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>cosh</b> (<var>double x</var>)<var><a name="index-cosh-2204"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>coshf</b> (<var>float x</var>)<var><a name="index-coshf-2205"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>coshl</b> (<var>long double x</var>)<var><a name="index-coshl-2206"></a></var><br> |
| <blockquote><p>These function return the hyperbolic cosine of <var>x</var>, |
| defined mathematically as <code>(exp (</code><var>x</var><code>) + exp (-</code><var>x</var><code>)) / 2</code><!-- /@w -->. |
| They may signal overflow if <var>x</var> is too large. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>tanh</b> (<var>double x</var>)<var><a name="index-tanh-2207"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>tanhf</b> (<var>float x</var>)<var><a name="index-tanhf-2208"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>tanhl</b> (<var>long double x</var>)<var><a name="index-tanhl-2209"></a></var><br> |
| <blockquote><p>These functions return the hyperbolic tangent of <var>x</var>, |
| defined mathematically as <code>sinh (</code><var>x</var><code>) / cosh (</code><var>x</var><code>)</code><!-- /@w -->. |
| They may signal overflow if <var>x</var> is too large. |
| </p></blockquote></div> |
| |
| <p><a name="index-hyperbolic-functions-2210"></a> |
| There are counterparts for the hyperbolic functions which take |
| complex arguments. |
| |
| <!-- complex.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: complex double <b>csinh</b> (<var>complex double z</var>)<var><a name="index-csinh-2211"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>csinhf</b> (<var>complex float z</var>)<var><a name="index-csinhf-2212"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>csinhl</b> (<var>complex long double z</var>)<var><a name="index-csinhl-2213"></a></var><br> |
| <blockquote><p>These functions return the complex hyperbolic sine of <var>z</var>, defined |
| mathematically as <code>(exp (</code><var>z</var><code>) - exp (-</code><var>z</var><code>)) / 2</code><!-- /@w -->. |
| </p></blockquote></div> |
| |
| <!-- complex.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: complex double <b>ccosh</b> (<var>complex double z</var>)<var><a name="index-ccosh-2214"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>ccoshf</b> (<var>complex float z</var>)<var><a name="index-ccoshf-2215"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>ccoshl</b> (<var>complex long double z</var>)<var><a name="index-ccoshl-2216"></a></var><br> |
| <blockquote><p>These functions return the complex hyperbolic cosine of <var>z</var>, defined |
| mathematically as <code>(exp (</code><var>z</var><code>) + exp (-</code><var>z</var><code>)) / 2</code><!-- /@w -->. |
| </p></blockquote></div> |
| |
| <!-- complex.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: complex double <b>ctanh</b> (<var>complex double z</var>)<var><a name="index-ctanh-2217"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>ctanhf</b> (<var>complex float z</var>)<var><a name="index-ctanhf-2218"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>ctanhl</b> (<var>complex long double z</var>)<var><a name="index-ctanhl-2219"></a></var><br> |
| <blockquote><p>These functions return the complex hyperbolic tangent of <var>z</var>, |
| defined mathematically as <code>csinh (</code><var>z</var><code>) / ccosh (</code><var>z</var><code>)</code><!-- /@w -->. |
| </p></blockquote></div> |
| |
| <p><a name="index-inverse-hyperbolic-functions-2220"></a> |
| <!-- math.h --> |
| <!-- ISO --> |
| |
| <div class="defun"> |
| — Function: double <b>asinh</b> (<var>double x</var>)<var><a name="index-asinh-2221"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>asinhf</b> (<var>float x</var>)<var><a name="index-asinhf-2222"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>asinhl</b> (<var>long double x</var>)<var><a name="index-asinhl-2223"></a></var><br> |
| <blockquote><p>These functions return the inverse hyperbolic sine of <var>x</var>—the |
| value whose hyperbolic sine is <var>x</var>. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>acosh</b> (<var>double x</var>)<var><a name="index-acosh-2224"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>acoshf</b> (<var>float x</var>)<var><a name="index-acoshf-2225"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>acoshl</b> (<var>long double x</var>)<var><a name="index-acoshl-2226"></a></var><br> |
| <blockquote><p>These functions return the inverse hyperbolic cosine of <var>x</var>—the |
| value whose hyperbolic cosine is <var>x</var>. If <var>x</var> is less than |
| <code>1</code>, <code>acosh</code> signals a domain error. |
| </p></blockquote></div> |
| |
| <!-- math.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: double <b>atanh</b> (<var>double x</var>)<var><a name="index-atanh-2227"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: float <b>atanhf</b> (<var>float x</var>)<var><a name="index-atanhf-2228"></a></var><br> |
| <blockquote><!-- math.h --> |
| <!-- ISO --> |
| — Function: long double <b>atanhl</b> (<var>long double x</var>)<var><a name="index-atanhl-2229"></a></var><br> |
| <blockquote><p>These functions return the inverse hyperbolic tangent of <var>x</var>—the |
| value whose hyperbolic tangent is <var>x</var>. If the absolute value of |
| <var>x</var> is greater than <code>1</code>, <code>atanh</code> signals a domain error; |
| if it is equal to 1, <code>atanh</code> returns infinity. |
| </p></blockquote></div> |
| |
| <p><a name="index-inverse-complex-hyperbolic-functions-2230"></a> |
| <!-- complex.h --> |
| <!-- ISO --> |
| |
| <div class="defun"> |
| — Function: complex double <b>casinh</b> (<var>complex double z</var>)<var><a name="index-casinh-2231"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>casinhf</b> (<var>complex float z</var>)<var><a name="index-casinhf-2232"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>casinhl</b> (<var>complex long double z</var>)<var><a name="index-casinhl-2233"></a></var><br> |
| <blockquote><p>These functions return the inverse complex hyperbolic sine of |
| <var>z</var>—the value whose complex hyperbolic sine is <var>z</var>. |
| </p></blockquote></div> |
| |
| <!-- complex.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: complex double <b>cacosh</b> (<var>complex double z</var>)<var><a name="index-cacosh-2234"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>cacoshf</b> (<var>complex float z</var>)<var><a name="index-cacoshf-2235"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>cacoshl</b> (<var>complex long double z</var>)<var><a name="index-cacoshl-2236"></a></var><br> |
| <blockquote><p>These functions return the inverse complex hyperbolic cosine of |
| <var>z</var>—the value whose complex hyperbolic cosine is <var>z</var>. Unlike |
| the real-valued functions, there are no restrictions on the value of <var>z</var>. |
| </p></blockquote></div> |
| |
| <!-- complex.h --> |
| <!-- ISO --> |
| <div class="defun"> |
| — Function: complex double <b>catanh</b> (<var>complex double z</var>)<var><a name="index-catanh-2237"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex float <b>catanhf</b> (<var>complex float z</var>)<var><a name="index-catanhf-2238"></a></var><br> |
| <blockquote><!-- complex.h --> |
| <!-- ISO --> |
| — Function: complex long double <b>catanhl</b> (<var>complex long double z</var>)<var><a name="index-catanhl-2239"></a></var><br> |
| <blockquote><p>These functions return the inverse complex hyperbolic tangent of |
| <var>z</var>—the value whose complex hyperbolic tangent is <var>z</var>. Unlike |
| the real-valued functions, there are no restrictions on the value of |
| <var>z</var>. |
| </p></blockquote></div> |
| |
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