269 lines
19 KiB
HTML
269 lines
19 KiB
HTML
<!DOCTYPE html>
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<html lang="zh" dir="ltr" class="client-nojs" xmlns="http://www.w3.org/1999/xhtml" xml:lang="zh">
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<title>cacoshf, cacosh, cacoshl</title>
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<link rel="stylesheet" href="ext.css" />
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<link rel="stylesheet" href="site_modules.css" />
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<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-c_numeric_complex_cacosh skin-cppreference2 action-view cpp-navbar">
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<div id="cpp-content-base">
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<div id="content"><a id="top"></a>
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<h1 id="firstHeading" class="firstHeading">cacoshf, cacosh, cacoshl</h1>
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<div id="bodyContent">
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<div id="contentSub"><span class="subpages">< <a href="c.html">c</a>‎ | <a href="c-numeric.html">numeric</a>‎ | <a href="c-numeric-complex.html">complex</a></span></div>
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<div id="mw-content-text" lang="zh" dir="ltr" class="mw-content-ltr" xml:lang="zh">
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<table class="t-dcl-begin">
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<td>
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<div>定义于头文件 <code><complex.h></code></div>
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</td>
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<td></td>
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<td></td>
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<div><span class="mw-geshi c source-c"><span class="kw4">float</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> cacoshf<span class="br0">(</span> <span class="kw4">float</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z <span class="br0">)</span><span class="sy4">;</span></span></div>
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</td>
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<td>(1)</td>
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<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
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<tr class="t-dcl t-since-c99">
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<td>
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<div><span class="mw-geshi c source-c"><span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> cacosh<span class="br0">(</span> <span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z <span class="br0">)</span><span class="sy4">;</span></span></div>
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</td>
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<td>(2)</td>
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<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
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</tr>
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<div><span class="mw-geshi c source-c"><span class="kw4">long</span> <span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> cacoshl<span class="br0">(</span> <span class="kw4">long</span> <span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z <span class="br0">)</span><span class="sy4">;</span></span></div>
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</td>
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<td>(3)</td>
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<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
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</tr>
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<tr class="t-dsc-header">
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<td>
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<div>定义于头文件 <code><tgmath.h></code></div>
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</td>
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<td></td>
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<td></td>
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<div><span class="mw-geshi c source-c"><span class="co2">#define acosh( z )</span></span></div>
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</td>
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<td>(4)</td>
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<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
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</tr>
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<tr class="t-dcl-sep">
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<td></td>
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<td></td>
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<td></td>
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</table>
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<div class="t-li1"><span class="t-li">1-3)</span> 计算复数值 <code>z</code> 的复反双曲余弦，分支切割在沿实轴小于 1 的值上。</div>
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<div class="t-li1"><span class="t-li">4)</span> 泛型宏：若 <code>z</code> 拥有 <span class="t-c"><span class="mw-geshi c source-c"><span class="kw4">long</span> <span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a></span></span> 类型，则调用 <code>cacoshl</code> 。若 <code>z</code> 拥有 <span class="t-c"><span class="mw-geshi c source-c"><span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a></span></span> 类型，则调用 <code>cacosh</code> 。若 <code>z</code> 拥有 <span class="t-c"><span class="mw-geshi c source-c"><span class="kw4">float</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a></span></span> 类型，则调用 <code>cacoshf</code> 。若 <code>z</code> 为实数或整数，则宏调用对应的实函数（ <span class="t-c"><span class="mw-geshi c source-c">acoshf</span></span> 、 <span class="t-c"><span class="mw-geshi c source-c"><a href="c-numeric-math-acosh.html"><span class="kw680">acosh</span></a></span></span> 、 <span class="t-c"><span class="mw-geshi c source-c">acoshl</span></span> ）。若 <code>z</code> 为虚数，则宏调用对应的复数版本，而返回类型为复数。</div>
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<h3><span class="mw-headline" id=".E5.8F.82.E6.95.B0">参数</span></h3>
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<table class="t-par-begin">
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<tr class="t-par">
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<td>z</td>
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<td>-</td>
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<td>复参数</td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">返回值</span></h3>
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<p><code>z</code> 的复反双曲余弦，沿实轴在区间 <span class="texhtml" style="white-space: nowrap;">[0; ∞)</span> 中，而沿虚轴在区间 <span class="texhtml" style="white-space: nowrap;">[−iπ; +iπ]</span> 中。</p>
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<h3><span class="mw-headline" id=".E9.94.99.E8.AF.AF.E5.A4.84.E7.90.86.E5.8F.8A.E7.89.B9.E6.AE.8A.E5.80.BC">错误处理及特殊值</span></h3>
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<p>报告的错误与 <a href="c-numeric-math-math_errhandling.html">math_errhandling</a> 一致。</p>
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<p>若实现支持 IEEE 浮点算术，则</p>
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<ul>
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<li><span class="t-c"><span class="mw-geshi c source-c">cacosh<span class="br0">(</span><a href="c-numeric-complex-conj.html"><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="c-numeric-complex-conj.html"><span class="kw760">conj</span></a><span class="br0">(</span>cacosh<span class="br0">(</span>z<span class="br0">)</span><span class="br0">)</span></span></span></li>
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<li>若 <code>z</code> 为 <code>±0+0i</code> ，则结果为 <code>+0+iπ/2</code></li>
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<li>若 <code>z</code> 为 <code>+x+∞i</code> （对于任何有限 x ），则结果为 <code>+∞+iπ/2</code></li>
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<li>若 <code>z</code> 为 <code>+x+NaNi</code> （对于任何非零有限 x ），结果为 <code>NaN+NaNi</code> 并可能引发 <span class="t-lc"><a href="c-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span></li>
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<li>若 <code>z</code> 为 <code>0+NaNi</code> ，则结果为 <code>NaN±iπ/2</code> ，其中虚部符号未指定</li>
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<li>若 <code>z</code> 为 <code>-∞+yi</code> （对于任何有限正 y ），则结果为 <code>+∞+iπ</code></li>
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<li>若 <code>z</code> 为 <code>+∞+yi</code> （对于任何有限正 y ），则结果为 <code>+∞+0i</code></li>
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<li>若 <code>z</code> 为 <code>-∞+∞i</code> ，则结果为 <code>+∞+3iπ/4</code></li>
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<li>若 <code>z</code> 为 <code>±∞+NaNi</code> ，则结果为 <code>+∞+NaNi</code></li>
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<li>若 <code>z</code> 为 <code>NaN+yi</code> （对于任何有限 y ），则结果为 <code>NaN+NaNi</code> 并可能引发 <span class="t-lc"><a href="c-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span> 。</li>
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<li>若 <code>z</code> 为 <code>NaN+∞i</code> ，则结果为 <code>+∞+NaNi</code></li>
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<li>若 <code>z</code> 为 <code>NaN+NaNi</code> ，则结果为 <code>NaN+NaNi</code></li>
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</ul>
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<h3><span class="mw-headline" id=".E6.B3.A8.E6.84.8F">注意</span></h3>
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<p>尽管 C 标准命名此函数为“复弧双曲余弦”，双曲函数的反函数是面积函数。其参数为双曲扇形的面积，而非弧长。正确名称是“复反双曲余弦”，和更少见的“复面积双曲余弦”。</p>
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<p>反双曲余弦是多值函数，在复平面上要求分支切割。约定将分支切割置于实轴的线段 <span class="texhtml" style="white-space: nowrap;">(-∞,+1)</span> 上。</p>
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<p>复反双曲余弦主值的数学定义是 <span class="texhtml" style="white-space: nowrap;">acosh z = ln(z + <span class="t-mrad"><span>√</span><span>z+1</span></span> <span class="t-mrad"><span>√</span><span>z-1</span></span>)</span> 。</p>
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对于任何 z ， <span class="texhtml" style="white-space: nowrap;">acosh(z) = <span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td><span class="t-mrad"><span>√</span><span>z-1</span></span></td>
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</tr>
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<tr>
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<td><span class="t-mrad"><span>√</span><span>1-z</span></span></td>
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</tr>
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</table>
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acos(z) ，或在复平面上半部简单地为 <span class="texhtml" style="white-space: nowrap;">i acos(z)</span> 。
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<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">示例</span></h3>
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<div class="t-example">
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<div class="t-example-live-link"></div>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="c source-c">
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<pre class="de1"><span class="co2">#include <stdio.h></span>
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<span class="co2">#include <complex.h></span>
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<span class="kw4">int</span> main<span class="br0">(</span><span class="kw4">void</span><span class="br0">)</span>
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<span class="br0">{</span>
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<span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z <span class="sy1">=</span> cacosh<span class="br0">(</span><span class="nu16">0.5</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="c-io-fprintf.html"><span class="kw848">printf</span></a><span class="br0">(</span><span class="st0">"cacosh(+0.5+0i) = %f%+fi<span class="es1">\n</span>"</span>, <a href="c-numeric-complex-creal.html"><span class="kw754">creal</span></a><span class="br0">(</span>z<span class="br0">)</span>, <a href="c-numeric-complex-cimag.html"><span class="kw751">cimag</span></a><span class="br0">(</span>z<span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z2 <span class="sy1">=</span> <a href="c-numeric-complex-conj.html"><span class="kw760">conj</span></a><span class="br0">(</span><span class="nu16">0.5</span><span class="br0">)</span><span class="sy4">;</span> <span class="co1">// 或 C11 中的 cacosh(CMPLX(0.5, -0.0))</span>
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<a href="c-io-fprintf.html"><span class="kw848">printf</span></a><span class="br0">(</span><span class="st0">"cacosh(+0.5-0i) (the other side of the cut) = %f%+fi<span class="es1">\n</span>"</span>, <a href="c-numeric-complex-creal.html"><span class="kw754">creal</span></a><span class="br0">(</span>z2<span class="br0">)</span>, <a href="c-numeric-complex-cimag.html"><span class="kw751">cimag</span></a><span class="br0">(</span>z2<span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="co1">// 在上半平面， acosh(z) = i*acos(z) </span>
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<span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z3 <span class="sy1">=</span> <a href="c-numeric-complex-casinh.html"><span class="kw799">casinh</span></a><span class="br0">(</span><span class="nu0">1</span><span class="sy2">+</span>I<span class="br0">)</span><span class="sy4">;</span>
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<a href="c-io-fprintf.html"><span class="kw848">printf</span></a><span class="br0">(</span><span class="st0">"casinh(1+1i) = %f%+fi<span class="es1">\n</span>"</span>, <a href="c-numeric-complex-creal.html"><span class="kw754">creal</span></a><span class="br0">(</span>z3<span class="br0">)</span>, <a href="c-numeric-complex-cimag.html"><span class="kw751">cimag</span></a><span class="br0">(</span>z3<span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z4 <span class="sy1">=</span> I<span class="sy2">*</span><a href="c-numeric-complex-casin.html"><span class="kw784">casin</span></a><span class="br0">(</span><span class="nu0">1</span><span class="sy2">+</span>I<span class="br0">)</span><span class="sy4">;</span>
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<a href="c-io-fprintf.html"><span class="kw848">printf</span></a><span class="br0">(</span><span class="st0">"I*asin(1+1i) = %f%+fi<span class="es1">\n</span>"</span>, <a href="c-numeric-complex-creal.html"><span class="kw754">creal</span></a><span class="br0">(</span>z4<span class="br0">)</span>, <a href="c-numeric-complex-cimag.html"><span class="kw751">cimag</span></a><span class="br0">(</span>z4<span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="br0">}</span></pre></div>
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</div>
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<p>输出：</p>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="text source-text">
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<pre class="de1">cacosh(+0.5+0i) = 0.000000-1.047198i
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cacosh(+0.5-0i) (the other side of the cut) = 0.500000-0.000000i
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casinh(1+1i) = 1.061275+0.666239i
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I*asin(1+1i) = -1.061275+0.666239i</pre></div>
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</div>
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</div>
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<h3><span class="mw-headline" id=".E5.BC.95.E7.94.A8">引用</span></h3>
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<div class="t-ref-std-11">
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<ul>
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<li>C11 标准（ISO/IEC 9899:2011）：</li>
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</ul>
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<dl>
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<dd>
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<ul>
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<li>7.3.6.1 The cacosh functions (p: 192)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>7.25 Type-generic math <tgmath.h> (p: 373-375)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>G.6.2.1 The cacosh functions (p: 539-540)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>G.7 Type-generic math <tgmath.h> (p: 545)</li>
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</ul>
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</dd>
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</dl>
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<div class="t-ref-std-c99">
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<ul>
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<li>C99 标准（ISO/IEC 9899:1999）：</li>
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</ul>
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<dl>
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<dd>
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<ul>
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<li>7.3.6.1 The cacosh functions (p: 174)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>7.22 Type-generic math <tgmath.h> (p: 335-337)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>G.6.2.1 The cacosh functions (p: 474-475)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>G.7 Type-generic math <tgmath.h> (p: 480)</li>
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</ul>
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</dd>
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</dl>
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</div>
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<h3><span class="mw-headline" id=".E5.8F.82.E9.98.85">参阅</span></h3>
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<table class="t-dsc-begin">
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="c-numeric-complex-cacos.html"><span class="t-lines"><span>cacos</span><span>cacosf</span><span>cacosl</span></span></a></div>
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<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>
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</div>
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</td>
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<td>计算复数反余弦<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="c-numeric-complex-casinh.html"><span class="t-lines"><span>casinh</span><span>casinhf</span><span>casinhl</span></span></a></div>
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<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>
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</div>
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</td>
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<td>计算复数反双曲正弦<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="c-numeric-complex-catanh.html"><span class="t-lines"><span>catanh</span><span>catanhf</span><span>catanhl</span></span></a></div>
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<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>
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</div>
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</td>
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<td>计算复数反双曲正切<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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|
<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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|
<div><a href="c-numeric-complex-ccosh.html"><span class="t-lines"><span>ccosh</span><span>ccoshf</span><span>ccoshl</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>
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|
</div>
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|
</td>
|
|
<td>计算复双曲余弦<br />
|
|
<span class="t-mark">(函数)</span></td>
|
|
</tr>
|
|
<tr class="t-dsc">
|
|
<td>
|
|
<div class="t-dsc-member-div">
|
|
<div><a href="c-numeric-math-acosh.html"><span class="t-lines"><span>acosh</span><span>acoshf</span><span>acoshl</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>
|
|
</div>
|
|
</td>
|
|
<td>计算反双曲余弦（ <span class="mjax" style="display:none">\({\small\operatorname{arcosh}{x} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">arcosh(x)</span> ）<br />
|
|
<span class="t-mark">(函数)</span></td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
</div>
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|
<div class="visualClear"></div>
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</div>
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</div>
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</div>
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</body>
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</html>
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