176 lines
16 KiB
HTML
176 lines
16 KiB
HTML
<!DOCTYPE html>
|
|
<html lang="zh" dir="ltr" class="client-nojs" xmlns="http://www.w3.org/1999/xhtml" xml:lang="zh">
|
|
<head><meta http-equiv="x-ua-compatible" content="ie=edge">
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
|
<title>std::acosh(std::complex)</title>
|
|
<link rel="stylesheet" href="ext.css" />
|
|
<link rel="stylesheet" href="site_modules.css" />
|
|
</head>
|
|
<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-cpp_numeric_complex_acosh skin-cppreference2 action-view cpp-navbar">
|
|
<div id="cpp-content-base">
|
|
<div id="content"><a id="top"></a>
|
|
<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>acosh<span style="font-size:0.7em; line-height:130%">(std::complex)</span></h1>
|
|
<div id="bodyContent">
|
|
<div id="contentSub"><span class="subpages">< <a href="cpp.html">cpp</a>‎ | <a href="cpp-numeric.html">numeric</a>‎ | <a href="cpp-numeric-complex.html">complex</a></span></div>
|
|
<div id="mw-content-text" lang="zh" dir="ltr" class="mw-content-ltr" xml:lang="zh">
|
|
<table class="t-dcl-begin">
|
|
<tbody>
|
|
<tr class="t-dsc-header">
|
|
<td>
|
|
<div>定义于头文件 <code><a href="cpp-header-complex.html"><complex></a></code></div>
|
|
</td>
|
|
<td></td>
|
|
<td></td>
|
|
</tr>
|
|
<tr class="t-dcl t-since-cxx11">
|
|
<td>
|
|
<div><span class="mw-geshi cpp source-cpp"><span class="kw1">template</span><span class="sy1"><</span> <span class="kw1">class</span> T <span class="sy1">></span><br />
|
|
complex<span class="sy1"><</span>T<span class="sy1">></span> acosh<span class="br0">(</span> <span class="kw4">const</span> complex<span class="sy1"><</span>T<span class="sy1">></span><span class="sy3">&</span> z <span class="br0">)</span><span class="sy4">;</span></span></div>
|
|
</td>
|
|
<td class="t-dcl-nopad"></td>
|
|
<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
|
|
</tr>
|
|
<tr class="t-dcl-sep">
|
|
<td></td>
|
|
<td></td>
|
|
<td></td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>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。</p>
|
|
<h3><span class="mw-headline" id=".E5.8F.82.E6.95.B0">参数</span></h3>
|
|
<table class="t-par-begin">
|
|
<tr class="t-par">
|
|
<td>z</td>
|
|
<td>-</td>
|
|
<td>复数值</td>
|
|
</tr>
|
|
</table>
|
|
<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">返回值</span></h3>
|
|
<p>若不出现错误，则返回 <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>
|
|
<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>
|
|
<p>报告的错误与 <a href="cpp-numeric-math-math_errhandling.html"><tt>math_errhandling</tt></a> 一致。</p>
|
|
<p>若实现支持 IEEE 浮点算术，则</p>
|
|
<ul>
|
|
<li><span class="t-c"><span class="mw-geshi cpp source-cpp"><a href="cpp-numeric-math-acosh.html"><span class="kw1344">std::<span class="me2">acosh</span></span></a><span class="br0">(</span>std<span class="sy4">::</span><span class="me2">conj</span><span class="br0">(</span>z<span class="br0">)</span><span class="br0">)</span> <span class="sy1">==</span> std<span class="sy4">::</span><span class="me2">conj</span><span class="br0">(</span><a href="cpp-numeric-math-acosh.html"><span class="kw1344">std::<span class="me2">acosh</span></span></a><span class="br0">(</span>z<span class="br0">)</span><span class="br0">)</span></span></span></li>
|
|
<li>若 <code>z</code> 为 <code>(±0,+0)</code> ，则结果为 <code>(+0,π/2)</code></li>
|
|
<li>若 <code>z</code> 为 <code>(x,+∞)</code> （对于任何有限 x ），则结果为 <code>(+∞,π/2)</code></li>
|
|
<li>若 <code>z</code> 为 <code>(x,NaN)</code> （对于任何<sup id="cite_ref-1" class="reference"><a href="cpp-numeric-complex-acosh.html#cite_note-1">[1]</a></sup>有限 x ），则结果为 <code>(NaN,NaN)</code> 并可能引发 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span> 。</li>
|
|
<li>若 <code>z</code> 为 <code>(-∞,y)</code> （对于任何有限正 y ），则结果为 <code>(+∞,π)</code></li>
|
|
<li>若 <code>z</code> 为 <code>(+∞,y)</code> （对于任何有限正 y ），则结果为 <code>(+∞,+0)</code></li>
|
|
<li>若 <code>z</code> 为 <code>(-∞,+∞)</code> ，则结果为 <code>(+∞,3π/4)</code></li>
|
|
<li>若 <code>z</code> 为 <code>(±∞,NaN)</code> ，则结果为 <code>(+∞,NaN)</code></li>
|
|
<li>若 <code>z</code> 为 <code>(NaN,y)</code> （对于任何有限 y ），则结果为 <code>(NaN,NaN)</code> 并可能引发 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span> 。</li>
|
|
<li>若 <code>z</code> 为 <code>(NaN,+∞)</code> ，则结果为 <code>(+∞,NaN)</code></li>
|
|
<li>若 <code>z</code> 为 <code>(NaN,NaN)</code> ，则结果为 <code>(NaN,NaN)</code></li>
|
|
</ul>
|
|
<ol class="references">
|
|
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="cpp-numeric-complex-acosh.html#cite_ref-1">↑</a></span> <span class="reference-text">由 <a rel="nofollow" class="external text" href="http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1892.htm#dr_471">C11 DR471</a> ，这只对非零 x 成立。若 <code>z</code> 为 <code>(0,NaN)</code> ，则结果应为 <code>(NaN,π/2)</code></span></li>
|
|
</ol>
|
|
<h3><span class="mw-headline" id=".E6.B3.A8.E6.84.8F">注意</span></h3>
|
|
<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>
|
|
<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>
|
|
<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>
|
|
对于任何 z ， <span class="texhtml" style="white-space: nowrap;">acosh(z) = <span class="t-mfrac"></span></span>
|
|
<table>
|
|
<tr>
|
|
<td><span class="t-mrad"><span>√</span><span>z-1</span></span></td>
|
|
</tr>
|
|
<tr>
|
|
<td><span class="t-mrad"><span>√</span><span>1-z</span></span></td>
|
|
</tr>
|
|
</table>
|
|
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> 。
|
|
<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">示例</span></h3>
|
|
<div class="t-example">
|
|
<div class="t-example-live-link"></div>
|
|
<div dir="ltr" class="mw-geshi" style="text-align: left;">
|
|
<div class="cpp source-cpp">
|
|
<pre class="de1"><span class="co2">#include <iostream></span>
|
|
<span class="co2">#include <complex></span>
|
|
|
|
<span class="kw4">int</span> main<span class="br0">(</span><span class="br0">)</span>
|
|
<span class="br0">{</span>
|
|
<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <a href="cpp-io-manip-fixed.html"><span class="kw1784">std::<span class="me2">fixed</span></span></a><span class="sy4">;</span>
|
|
<a href="cpp-numeric-complex.html"><span class="kw1472">std::<span class="me2">complex</span></span></a><span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span> z1<span class="br0">(</span><span class="nu16">0.5</span>, <span class="nu0">0</span><span class="br0">)</span><span class="sy4">;</span>
|
|
<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"acosh"</span> <span class="sy1"><<</span> z1 <span class="sy1"><<</span> <span class="st0">" = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-math-acosh.html"><span class="kw1344">std::<span class="me2">acosh</span></span></a><span class="br0">(</span>z1<span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
|
|
|
|
<a href="cpp-numeric-complex.html"><span class="kw1472">std::<span class="me2">complex</span></span></a><span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span> z2<span class="br0">(</span><span class="nu16">0.5</span>, <span class="sy2">-</span><span class="nu16">0.0</span><span class="br0">)</span><span class="sy4">;</span>
|
|
<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"acosh"</span> <span class="sy1"><<</span> z2 <span class="sy1"><<</span> <span class="st0">" (the other side of the cut) = "</span>
|
|
<span class="sy1"><<</span> <a href="cpp-numeric-math-acosh.html"><span class="kw1344">std::<span class="me2">acosh</span></span></a><span class="br0">(</span>z2<span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
|
|
|
|
<span class="co1">// 在上半平面， acosh = i acos </span>
|
|
<a href="cpp-numeric-complex.html"><span class="kw1472">std::<span class="me2">complex</span></span></a><span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span> z3<span class="br0">(</span><span class="nu0">1</span>, <span class="nu0">1</span><span class="br0">)</span>, i<span class="br0">(</span><span class="nu0">0</span>, <span class="nu0">1</span><span class="br0">)</span><span class="sy4">;</span>
|
|
<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"acosh"</span> <span class="sy1"><<</span> z3 <span class="sy1"><<</span> <span class="st0">" = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-math-acosh.html"><span class="kw1344">std::<span class="me2">acosh</span></span></a><span class="br0">(</span>z3<span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span>
|
|
<span class="sy1"><<</span> <span class="st0">"i*acos"</span> <span class="sy1"><<</span> z3 <span class="sy1"><<</span> <span class="st0">" = "</span> <span class="sy1"><<</span> i<span class="sy2">*</span><a href="cpp-numeric-math-acos.html"><span class="kw1337">std::<span class="me2">acos</span></span></a><span class="br0">(</span>z3<span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
|
|
<span class="br0">}</span></pre></div>
|
|
</div>
|
|
<p>输出：</p>
|
|
<div dir="ltr" class="mw-geshi" style="text-align: left;">
|
|
<div class="text source-text">
|
|
<pre class="de1">acosh(0.500000,0.000000) = (0.000000,-1.047198)
|
|
acosh(0.500000,-0.000000) (the other side of the cut) = (0.000000,1.047198)
|
|
acosh(1.000000,1.000000) = (1.061275,0.904557)
|
|
i*acos(1.000000,1.000000) = (1.061275,0.904557)</pre></div>
|
|
</div>
|
|
</div>
|
|
<h3><span class="mw-headline" id=".E5.8F.82.E9.98.85">参阅</span></h3>
|
|
<table class="t-dsc-begin">
|
|
<tr class="t-dsc">
|
|
<td>
|
|
<div class="t-dsc-member-div">
|
|
<div><a href="cpp-numeric-complex-acos.html"><span class="t-lines"><span>acos<span class="t-dsc-small">(std::complex)</span></span></span></a></div>
|
|
<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
|
|
</div>
|
|
</td>
|
|
<td>计算复数的反余弦（ <span class="mjax" style="display:none">\({\small\arccos{z} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">arccos(z)</span> ）<br />
|
|
<span class="t-mark">(函数模板)</span></td>
|
|
</tr>
|
|
<tr class="t-dsc">
|
|
<td>
|
|
<div class="t-dsc-member-div">
|
|
<div><a href="cpp-numeric-complex-asinh.html"><span class="t-lines"><span>asinh<span class="t-dsc-small">(std::complex)</span></span></span></a></div>
|
|
<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
|
|
</div>
|
|
</td>
|
|
<td>计算复数的反双曲正弦（ <span class="mjax" style="display:none">\({\small\operatorname{arsinh}{z} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">arsinh(z)</span> ）<br />
|
|
<span class="t-mark">(函数模板)</span></td>
|
|
</tr>
|
|
<tr class="t-dsc">
|
|
<td>
|
|
<div class="t-dsc-member-div">
|
|
<div><a href="cpp-numeric-complex-atanh.html"><span class="t-lines"><span>atanh<span class="t-dsc-small">(std::complex)</span></span></span></a></div>
|
|
<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
|
|
</div>
|
|
</td>
|
|
<td>计算复数的反双曲正切（ <span class="mjax" style="display:none">\({\small\operatorname{artanh}{z} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">artanh(z)</span> ）<br />
|
|
<span class="t-mark">(函数模板)</span></td>
|
|
</tr>
|
|
<tr class="t-dsc">
|
|
<td>
|
|
<div class="t-dsc-member-div">
|
|
<div><a href="cpp-numeric-complex-cosh.html"><span class="t-lines"><span>cosh<span class="t-dsc-small">(std::complex)</span></span></span></a></div>
|
|
</div>
|
|
</td>
|
|
<td>计算复数的双曲余弦（ <span class="mjax" style="display:none">\({\small\cosh{z} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">cosh(z)</span> ）<br />
|
|
<span class="t-mark">(函数模板)</span></td>
|
|
</tr>
|
|
<tr class="t-dsc">
|
|
<td>
|
|
<div class="t-dsc-member-div">
|
|
<div><a href="cpp-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-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</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 class="visualClear"></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</body>
|
|
</html>
|