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<h1 id="firstHeading" class="firstHeading">casinhf, casinh, casinhl</h1> <table class="t-dcl-begin"> <tr class="t-dsc-header"> <th> Defined in header <code><complex.h></code> </th> <th> </th> <th> </th> </tr> <tr class="t-dcl t-since-c99"> <td> <pre data-language="c">float complex casinhf( float complex z );</pre>
</td> <td> (1) </td> <td> <span class="t-mark-rev t-since-c99">(since C99)</span> </td> </tr> <tr class="t-dcl t-since-c99"> <td> <pre data-language="c">double complex casinh( double complex z );</pre>
</td> <td> (2) </td> <td> <span class="t-mark-rev t-since-c99">(since C99)</span> </td> </tr> <tr class="t-dcl t-since-c99"> <td> <pre data-language="c">long double complex casinhl( long double complex z );</pre>
</td> <td> (3) </td> <td> <span class="t-mark-rev t-since-c99">(since C99)</span> </td> </tr> <tr class="t-dsc-header"> <th> Defined in header <code><tgmath.h></code> </th> <th> </th> <th> </th> </tr> <tr class="t-dcl t-since-c99"> <td> <pre data-language="c">#define asinh( z )</pre>
</td> <td> (4) </td> <td> <span class="t-mark-rev t-since-c99">(since C99)</span> </td> </tr> </table> <div class="t-li1">
<span class="t-li">1-3)</span> Computes the complex arc hyperbolic sine of <code>z</code> with branch cuts outside the interval [−i; +i] along the imaginary axis.</div> <div class="t-li1">
<span class="t-li">4)</span> Type-generic macro: If <code>z</code> has type <code><span class="kw4">long</span> <span class="kw4">double</span> <a href="http://en.cppreference.com/w/c/numeric/complex/complex"><span class="kw743">complex</span></a></code>, <code>casinhl</code> is called. if <code>z</code> has type <code><span class="kw4">double</span> <a href="http://en.cppreference.com/w/c/numeric/complex/complex"><span class="kw743">complex</span></a></code>, <code>casinh</code> is called, if <code>z</code> has type <code><span class="kw4">float</span> <a href="http://en.cppreference.com/w/c/numeric/complex/complex"><span class="kw743">complex</span></a></code>, <code>casinhf</code> is called. If <code>z</code> is real or integer, then the macro invokes the corresponding real function (<code>asinhf</code>, <code><a href="http://en.cppreference.com/w/c/numeric/math/asinh"><span class="kw679">asinh</span></a></code>, <code>asinhl</code>). If <code>z</code> is imaginary, then the macro invokes the corresponding real version of the function <code><a href="../math/asin" title="c/numeric/math/asin">asin</a></code>, implementing the formula asinh(iy) = i asin(y), and the return type is imaginary.</div> <h3 id="Parameters"> Parameters</h3> <table class="t-par-begin"> <tr class="t-par"> <td> z </td> <td> - </td> <td> complex argument </td>
</tr>
</table> <h3 id="Return_value"> Return value</h3> <p>If no errors occur, the complex arc hyperbolic sine of <code>z</code> is returned, in the range of a strip mathematically unbounded along the real axis and in the interval [−iπ/2; +iπ/2] along the imaginary axis.</p>
<h3 id="Error_handling_and_special_values"> Error handling and special values</h3> <p>Errors are reported consistent with <a href="../math/math_errhandling" title="c/numeric/math/math errhandling">math_errhandling</a></p>
<p>If the implementation supports IEEE floating-point arithmetic,</p>
<ul>
<li> <code>casinh<span class="br0">(</span><a href="http://en.cppreference.com/w/c/numeric/complex/conj"><span class="kw760">conj</span></a><span class="br0">(</span>z<span class="br0">)</span><span class="br0">)</span> <span class="sy1">==</span> <a href="http://en.cppreference.com/w/c/numeric/complex/conj"><span class="kw760">conj</span></a><span class="br0">(</span>casinh<span class="br0">(</span>z<span class="br0">)</span><span class="br0">)</span></code> </li>
<li> <code>casinh(-z) == -casinh(z)</code> </li>
<li> If <code>z</code> is <code>+0+0i</code>, the result is <code>+0+0i</code> </li>
<li> If <code>z</code> is <code>x+∞i</code> (for any positive finite x), the result is <code>+∞+π/2</code> </li>
<li> If <code>z</code> is <code>x+NaNi</code> (for any finite x), the result is <code>NaN+NaNi</code> and <code><a href="../fenv/fe_exceptions" title="c/numeric/fenv/FE exceptions">FE_INVALID</a></code> may be raised </li>
<li> If <code>z</code> is <code>+∞+yi</code> (for any positive finite y), the result is <code>+∞+0i</code> </li>
<li> If <code>z</code> is <code>+∞+∞i</code>, the result is <code>+∞+iπ/4</code> </li>
<li> If <code>z</code> is <code>+∞+NaNi</code>, the result is <code>+∞+NaNi</code> </li>
<li> If <code>z</code> is <code>NaN+0i</code>, the result is <code>NaN+0i</code> </li>
<li> If <code>z</code> is <code>NaN+yi</code> (for any finite nonzero y), the result is <code>NaN+NaNi</code> and <code><a href="../fenv/fe_exceptions" title="c/numeric/fenv/FE exceptions">FE_INVALID</a></code> may be raised </li>
<li> If <code>z</code> is <code>NaN+∞i</code>, the result is <code>±∞+NaNi</code> (the sign of the real part is unspecified) </li>
<li> If <code>z</code> is <code>NaN+NaNi</code>, the result is <code>NaN+NaNi</code> </li>
</ul> <h3 id="Notes"> Notes</h3> <p>Although the C standard names this function "complex arc hyperbolic sine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic sine", and, less common, "complex area hyperbolic sine".</p>
<p>Inverse hyperbolic sine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments (-<i>i</i>∞,-<i>i</i>) and (<i>i</i>,<i>i</i>∞) of the imaginary axis.</p>
<p>The mathematical definition of the principal value of the inverse hyperbolic sine is asinh z = ln(z + <span class="t-mrad"><span>√</span><span>1+z<sup class="t-su">2</sup></span></span>) For any z, asinh(z) =</p>
<span><span>asin(iz)</span><span>/</span><span>i</span></span> <h3 id="Example"> Example</h3> <div class="t-example"> <div class="c source-c"><pre data-language="c">#include <stdio.h>
#include <complex.h>
int main(void)
{
double complex z = casinh(0+2*I);
printf("casinh(+0+2i) = %f%+fi\n", creal(z), cimag(z));
double complex z2 = casinh(-conj(2*I)); // or casinh(CMPLX(-0.0, 2)) in C11
printf("casinh(-0+2i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));
// for any z, asinh(z) = asin(iz)/i
double complex z3 = casinh(1+2*I);
printf("casinh(1+2i) = %f%+fi\n", creal(z3), cimag(z3));
double complex z4 = casin((1+2*I)*I)/I;
printf("casin(i * (1+2i))/i = %f%+fi\n", creal(z4), cimag(z4));
}</pre></div> <p>Output:</p>
<div class="text source-text"><pre data-language="c">casinh(+0+2i) = 1.316958+1.570796i
casinh(-0+2i) (the other side of the cut) = -1.316958+1.570796i
casinh(1+2i) = 1.469352+1.063440i
casin(i * (1+2i))/i = 1.469352+1.063440i</pre></div> </div> <h3 id="References"> References</h3> <ul>
<li> C11 standard (ISO/IEC 9899:2011): </li>
<ul>
<li> 7.3.6.2 The casinh functions (p: 192-193) </li>
<li> 7.25 Type-generic math <tgmath.h> (p: 373-375) </li>
<li> G.6.2.2 The casinh functions (p: 540) </li>
<li> G.7 Type-generic math <tgmath.h> (p: 545) </li>
</ul>
<li> C99 standard (ISO/IEC 9899:1999): </li>
<ul>
<li> 7.3.6.2 The casinh functions (p: 174-175) </li>
<li> 7.22 Type-generic math <tgmath.h> (p: 335-337) </li>
<li> G.6.2.2 The casinh functions (p: 475) </li>
<li> G.7 Type-generic math <tgmath.h> (p: 480) </li>
</ul>
</ul> <h3 id="See_also"> See also</h3> <table class="t-dsc-begin"> <tr class="t-dsc"> <td> <div><a href="cacosh" title="c/numeric/complex/cacosh"> <span class="t-lines"><span>cacosh</span><span>cacoshf</span><span>cacoshl</span></span></a></div>
<div><span class="t-lines"><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span></span></div> </td> <td> computes the complex arc hyperbolic cosine <br> <span class="t-mark">(function)</span> </td>
</tr> <tr class="t-dsc"> <td> <div><a href="catanh" title="c/numeric/complex/catanh"> <span class="t-lines"><span>catanh</span><span>catanhf</span><span>catanhl</span></span></a></div>
<div><span class="t-lines"><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span></span></div> </td> <td> computes the complex arc hyperbolic tangent <br> <span class="t-mark">(function)</span> </td>
</tr> <tr class="t-dsc"> <td> <div><a href="csinh" title="c/numeric/complex/csinh"> <span class="t-lines"><span>csinh</span><span>csinhf</span><span>csinhl</span></span></a></div>
<div><span class="t-lines"><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span></span></div> </td> <td> computes the complex hyperbolic sine <br> <span class="t-mark">(function)</span> </td>
</tr> <tr class="t-dsc"> <td> <div><a href="../math/asinh" title="c/numeric/math/asinh"> <span class="t-lines"><span>asinh</span><span>asinhf</span><span>asinhl</span></span></a></div>
<div><span class="t-lines"><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span></span></div> </td> <td> computes inverse hyperbolic sine (\({\small\operatorname{arsinh}{x} }\)arsinh(x)) <br> <span class="t-mark">(function)</span> </td>
</tr> <tr class="t-dsc"> <td colspan="2"> <span><a href="https://en.cppreference.com/w/cpp/numeric/complex/asinh" title="cpp/numeric/complex/asinh">C++ documentation</a></span> for <code>asinh</code> </td>
</tr> </table> <div class="_attribution">
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