270 lines
19 KiB
HTML
270 lines
19 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>catanhf, catanh, catanhl</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-c_numeric_complex_catanh skin-cppreference2 action-view cpp-navbar">
|
|
<div id="cpp-content-base">
|
|
<div id="content"><a id="top"></a>
|
|
<h1 id="firstHeading" class="firstHeading">catanhf, catanh, catanhl</h1>
|
|
<div id="bodyContent">
|
|
<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>
|
|
<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><complex.h></code></div>
|
|
</td>
|
|
<td></td>
|
|
<td></td>
|
|
</tr>
|
|
<tr class="t-dcl t-since-c99">
|
|
<td>
|
|
<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> catanhf<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>
|
|
</td>
|
|
<td>(1)</td>
|
|
<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
|
|
</tr>
|
|
<tr class="t-dcl t-since-c99">
|
|
<td>
|
|
<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> catanh<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>
|
|
</td>
|
|
<td>(2)</td>
|
|
<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
|
|
</tr>
|
|
<tr class="t-dcl t-since-c99">
|
|
<td>
|
|
<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> catanhl<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>
|
|
</td>
|
|
<td>(3)</td>
|
|
<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
|
|
</tr>
|
|
<tr class="t-dsc-header">
|
|
<td>
|
|
<div>定义于头文件 <code><tgmath.h></code></div>
|
|
</td>
|
|
<td></td>
|
|
<td></td>
|
|
</tr>
|
|
<tr class="t-dcl t-since-c99">
|
|
<td>
|
|
<div><span class="mw-geshi c source-c"><span class="co2">#define atanh( z )</span></span></div>
|
|
</td>
|
|
<td>(4)</td>
|
|
<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
|
|
</tr>
|
|
<tr class="t-dcl-sep">
|
|
<td></td>
|
|
<td></td>
|
|
<td></td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<div class="t-li1"><span class="t-li">1-3)</span> 计算 <code>z</code> 的复反双曲正切，其分支切割为沿实轴的 <span class="texhtml" style="white-space: nowrap;">[−1; +1]</span> 区间外部。</div>
|
|
<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>catanhl</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>catanh</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>catanhf</code> 。若 <code>z</code> 为实数或整数，则该宏调用对应的实函数（ <span class="t-c"><span class="mw-geshi c source-c">atanhf</span></span> 、 <span class="t-c"><span class="mw-geshi c source-c"><a href="c-numeric-math-atanh.html"><span class="kw681">atanh</span></a></span></span> 、 <span class="t-c"><span class="mw-geshi c source-c">atanhl</span></span> ）。若为虚数 <code>z</code> ，则该宏调用 <span class="t-c"><span class="mw-geshi c source-c"><a href="c-numeric-math-atan.html"><span class="kw674">atan</span></a></span></span> 的对应实数版本，实现等式 <span class="texhtml" style="white-space: nowrap;">atanh(iy) = i atan(y)</span> ，而返回类型是虚数。</div>
|
|
<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>复参数</td>
|
|
</tr>
|
|
</table>
|
|
<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">返回值</span></h3>
|
|
<p>若不发生错误，则返回 <code>z</code> 的复反双曲正切，范围在数学上为沿着实轴无界的半条，沿着虚轴为区间 <span class="texhtml" style="white-space: nowrap;">[−iπ/2; +iπ/2]</span> 。</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="c-numeric-math-math_errhandling.html">math_errhandling</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 c source-c">catanh<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>catanh<span class="br0">(</span>z<span class="br0">)</span><span class="br0">)</span></span></span></li>
|
|
<li><span class="t-c"><span class="mw-geshi c source-c">catanh<span class="br0">(</span><span class="sy2">-</span>z<span class="br0">)</span> <span class="sy1">==</span> <span class="sy2">-</span>catanh<span class="br0">(</span>z<span class="br0">)</span></span></span></li>
|
|
<li>若 <code>z</code> 为 <code>+0+0i</code> ，则结果为 <code>+0+0i</code></li>
|
|
<li>若 <code>z</code> 为 <code>+0+NaNi</code> ，则结果为 <code>+0+NaNi</code></li>
|
|
<li>若 <code>z</code> 为 <code>+1+0i</code> ，则结果为 <code>+∞+0i</code> 并引发 <span class="t-lc"><a href="c-numeric-fenv-FE_exceptions.html">FE_DIVBYZERO</a></span></li>
|
|
<li>若 <code>z</code> 为 <code>x+∞i</code> （对于任何有限正 x ） ，则结果为 <code>+0+iπ/2</code></li>
|
|
<li>若 <code>z</code> 为 <code>x+NaNi</code> （对于任何有限非零的 x ），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>
|
|
<li>若 <code>z</code> 为 <code>+∞+yi</code> （对于任何有限的 y ），则结果为 <code>+0+iπ/2</code></li>
|
|
<li>若 <code>z</code> 为 <code>+∞+∞i</code> ，则结果为 <code>+0+iπ/2</code></li>
|
|
<li>若 <code>z</code> 为 <code>+∞+NaNi</code>, the result is <code>+0+NaNi</code></li>
|
|
<li>若 <code>z</code> 为 <code>NaN+yi</code> （对于任何有限的 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>
|
|
<li>若 <code>z</code> 为 <code>NaN+∞i</code> ，则结果为 <code>±0+iπ/2</code> （实部的符号未指定）</li>
|
|
<li>若 <code>z</code> 为 <code>NaN+NaNi</code> ，则结果为 <code>NaN+NaNi</code></li>
|
|
</ul>
|
|
<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“复弧双曲正切”，双曲函数的反函数却是面积函数。其参数是双曲扇形的面积，而非弧长。正确的名称是“复反双曲正切”，和较少见的“复面积双曲正切”。</p>
|
|
<p>反双曲正切是多值函数，并要求复平面上的分支切割。我们约定将分支切割置于实轴的划分线 <span class="texhtml" style="white-space: nowrap;">(-∞,-1]</span> 和 <span class="texhtml" style="white-space: nowrap;">[+1,+∞)</span> 。</p>
|
|
反双曲正切的主值的数学定义是 <span class="texhtml" style="white-space: nowrap;">atanh z = <span class="t-mfrac"></span></span>
|
|
<table>
|
|
<tr>
|
|
<td>ln(1+z)-ln(z-1)</td>
|
|
</tr>
|
|
<tr>
|
|
<td>2</td>
|
|
</tr>
|
|
</table>
|
|
。
|
|
<p><br /></p>
|
|
对于任何 z ， <span class="texhtml" style="white-space: nowrap;">atanh(z) = <span class="t-mfrac"></span></span>
|
|
<table>
|
|
<tr>
|
|
<td>atan(iz)</td>
|
|
</tr>
|
|
<tr>
|
|
<td>i</td>
|
|
</tr>
|
|
</table>
|
|
。
|
|
<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="c source-c">
|
|
<pre class="de1"><span class="co2">#include <stdio.h></span>
|
|
<span class="co2">#include <complex.h></span>
|
|
|
|
<span class="kw4">int</span> main<span class="br0">(</span><span class="kw4">void</span><span class="br0">)</span>
|
|
<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="sy1">=</span> catanh<span class="br0">(</span><span class="nu0">2</span><span class="br0">)</span><span class="sy4">;</span>
|
|
<a href="c-io-fprintf.html"><span class="kw848">printf</span></a><span class="br0">(</span><span class="st0">"catanh(+2+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>
|
|
|
|
<span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z2 <span class="sy1">=</span> catanh<span class="br0">(</span><a href="c-numeric-complex-conj.html"><span class="kw760">conj</span></a><span class="br0">(</span><span class="nu0">2</span><span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span> <span class="co1">// 或 C11 中的 catanh(CMPLX(2, -0.0))</span>
|
|
<a href="c-io-fprintf.html"><span class="kw848">printf</span></a><span class="br0">(</span><span class="st0">"catanh(+2-0i) (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>
|
|
|
|
<span class="co1">// 对于任何 z ， atanh(z) = atan(iz)/i</span>
|
|
<span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z3 <span class="sy1">=</span> catanh<span class="br0">(</span><span class="nu0">1</span><span class="sy2">+</span><span class="nu0">2</span><span class="sy2">*</span>I<span class="br0">)</span><span class="sy4">;</span>
|
|
<a href="c-io-fprintf.html"><span class="kw848">printf</span></a><span class="br0">(</span><span class="st0">"catanh(1+2i) = %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>
|
|
<span class="kw4">double</span> <a href="c-numeric-complex-complex.html"><span class="kw743">complex</span></a> z4 <span class="sy1">=</span> <a href="c-numeric-complex-catan.html"><span class="kw787">catan</span></a><span class="br0">(</span><span class="br0">(</span><span class="nu0">1</span><span class="sy2">+</span><span class="nu0">2</span><span class="sy2">*</span>I<span class="br0">)</span><span class="sy2">*</span>I<span class="br0">)</span><span class="sy2">/</span>I<span class="sy4">;</span>
|
|
<a href="c-io-fprintf.html"><span class="kw848">printf</span></a><span class="br0">(</span><span class="st0">"catan(i * (1+2i))/i = %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>
|
|
<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">catanh(+2+0i) = 0.549306+1.570796i
|
|
catanh(+2-0i) (the other side of the cut) = 0.549306-1.570796i
|
|
catanh(1+2i) = 0.173287+1.178097i
|
|
catan(i * (1+2i))/i = 0.173287+1.178097i</pre></div>
|
|
</div>
|
|
</div>
|
|
<h3><span class="mw-headline" id=".E5.BC.95.E7.94.A8">引用</span></h3>
|
|
<div class="t-ref-std-11">
|
|
<ul>
|
|
<li>C11 标准（ISO/IEC 9899:2011）：</li>
|
|
</ul>
|
|
<dl>
|
|
<dd>
|
|
<ul>
|
|
<li>7.3.6.3 The catanh functions (p: 193)</li>
|
|
</ul>
|
|
</dd>
|
|
</dl>
|
|
<dl>
|
|
<dd>
|
|
<ul>
|
|
<li>7.25 Type-generic math <tgmath.h> (p: 373-375)</li>
|
|
</ul>
|
|
</dd>
|
|
</dl>
|
|
<dl>
|
|
<dd>
|
|
<ul>
|
|
<li>G.6.2.3 The catanh functions (p: 540-541)</li>
|
|
</ul>
|
|
</dd>
|
|
</dl>
|
|
<dl>
|
|
<dd>
|
|
<ul>
|
|
<li>G.7 Type-generic math <tgmath.h> (p: 545)</li>
|
|
</ul>
|
|
</dd>
|
|
</dl>
|
|
<div class="t-ref-std-c99">
|
|
<ul>
|
|
<li>C99 标准（ISO/IEC 9899:1999）：</li>
|
|
</ul>
|
|
<dl>
|
|
<dd>
|
|
<ul>
|
|
<li>7.3.6.3 The catanh functions (p: 175)</li>
|
|
</ul>
|
|
</dd>
|
|
</dl>
|
|
<dl>
|
|
<dd>
|
|
<ul>
|
|
<li>7.22 Type-generic math <tgmath.h> (p: 335-337)</li>
|
|
</ul>
|
|
</dd>
|
|
</dl>
|
|
<dl>
|
|
<dd>
|
|
<ul>
|
|
<li>G.6.2.3 The catanh functions (p: 475-476)</li>
|
|
</ul>
|
|
</dd>
|
|
</dl>
|
|
<dl>
|
|
<dd>
|
|
<ul>
|
|
<li>G.7 Type-generic math <tgmath.h> (p: 480)</li>
|
|
</ul>
|
|
</dd>
|
|
</dl>
|
|
</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="c-numeric-complex-casinh.html"><span class="t-lines"><span>casinh</span><span>casinhf</span><span>casinhl</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>计算复数反双曲正弦<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-complex-cacosh.html"><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>
|
|
</div>
|
|
</td>
|
|
<td>计算复数反双曲余弦<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-complex-ctanh.html"><span class="t-lines"><span>ctanh</span><span>ctanhf</span><span>ctanhl</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>计算复数双曲正切<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-atanh.html"><span class="t-lines"><span>atanh</span><span>atanhf</span><span>atanhl</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>计算反双曲正切（ <span class="mjax" style="display:none">\({\small\operatorname{artanh}{x} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">artanh(x)</span> ）<br />
|
|
<span class="t-mark">(函数)</span></td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
<div class="visualClear"></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</body>
|
|
</html>
|