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  197.   <section id="module-cmath">

  198. <span id="cmath-mathematical-functions-for-complex-numbers"></span><h1><a class="reference internal" href="#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code></a> — Mathematical functions for complex numbers<a class="headerlink" href="#module-cmath" title="Link to this heading">¶</a></h1>

  199. <hr class="docutils" />

  200. <p>This module provides access to mathematical functions for complex numbers.  The

  201. functions in this module accept integers, floating-point numbers or complex

  202. numbers as arguments. They will also accept any Python object that has either a

  203. <a class="reference internal" href="../reference/datamodel.html#object.__complex__" title="object.__complex__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__complex__()</span></code></a> or a <a class="reference internal" href="../reference/datamodel.html#object.__float__" title="object.__float__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__float__()</span></code></a> method: these methods are used to

  204. convert the object to a complex or floating-point number, respectively, and

  205. the function is then applied to the result of the conversion.</p>

  206. <div class="admonition note">

  207. <p class="admonition-title">Note</p>

  208. <p>For functions involving branch cuts, we have the problem of deciding how to

  209. define those functions on the cut itself. Following Kahan’s “Branch cuts for

  210. complex elementary functions” paper, as well as Annex G of C99 and later C

  211. standards, we use the sign of zero to distinguish one side of the branch cut

  212. from the other: for a branch cut along (a portion of) the real axis we look

  213. at the sign of the imaginary part, while for a branch cut along the

  214. imaginary axis we look at the sign of the real part.</p>

  215. <p>For example, the <a class="reference internal" href="#cmath.sqrt" title="cmath.sqrt"><code class="xref py py-func docutils literal notranslate"><span class="pre">cmath.sqrt()</span></code></a> function has a branch cut along the

  216. negative real axis. An argument of <code class="docutils literal notranslate"><span class="pre">complex(-2.0,</span> <span class="pre">-0.0)</span></code> is treated as

  217. though it lies <em>below</em> the branch cut, and so gives a result on the negative

  218. imaginary axis:</p>

  219. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">cmath</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0</span><span class="p">))</span>

  220. <span class="go">-1.4142135623730951j</span>

  221. </pre></div>

  222. </div>

  223. <p>But an argument of <code class="docutils literal notranslate"><span class="pre">complex(-2.0,</span> <span class="pre">0.0)</span></code> is treated as though it lies above

  224. the branch cut:</p>

  225. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">cmath</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">))</span>

  226. <span class="go">1.4142135623730951j</span>

  227. </pre></div>

  228. </div>

  229. </div>

  230. <section id="conversions-to-and-from-polar-coordinates">

  231. <h2>Conversions to and from polar coordinates<a class="headerlink" href="#conversions-to-and-from-polar-coordinates" title="Link to this heading">¶</a></h2>

  232. <p>A Python complex number <code class="docutils literal notranslate"><span class="pre">z</span></code> is stored internally using <em>rectangular</em>

  233. or <em>Cartesian</em> coordinates.  It is completely determined by its <em>real

  234. part</em> <code class="docutils literal notranslate"><span class="pre">z.real</span></code> and its <em>imaginary part</em> <code class="docutils literal notranslate"><span class="pre">z.imag</span></code>.  In other

  235. words:</p>

  236. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="n">z</span> <span class="o">==</span> <span class="n">z</span><span class="o">.</span><span class="n">real</span> <span class="o">+</span> <span class="n">z</span><span class="o">.</span><span class="n">imag</span><span class="o">*</span><span class="mi">1</span><span class="n">j</span>

  237. </pre></div>

  238. </div>

  239. <p><em>Polar coordinates</em> give an alternative way to represent a complex

  240. number.  In polar coordinates, a complex number <em>z</em> is defined by the

  241. modulus <em>r</em> and the phase angle <em>phi</em>. The modulus <em>r</em> is the distance

  242. from <em>z</em> to the origin, while the phase <em>phi</em> is the counterclockwise

  243. angle, measured in radians, from the positive x-axis to the line

  244. segment that joins the origin to <em>z</em>.</p>

  245. <p>The following functions can be used to convert from the native

  246. rectangular coordinates to polar coordinates and back.</p>

  247. <dl class="py function">

  248. <dt class="sig sig-object py" id="cmath.phase">

  249. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">phase</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.phase" title="Link to this definition">¶</a></dt>

  250. <dd><p>Return the phase of <em>x</em> (also known as the <em>argument</em> of <em>x</em>), as a float.

  251. <code class="docutils literal notranslate"><span class="pre">phase(x)</span></code> is equivalent to <code class="docutils literal notranslate"><span class="pre">math.atan2(x.imag,</span> <span class="pre">x.real)</span></code>.  The result

  252. lies in the range [-<em>π</em>, <em>π</em>], and the branch cut for this operation lies

  253. along the negative real axis.  The sign of the result is the same as the

  254. sign of <code class="docutils literal notranslate"><span class="pre">x.imag</span></code>, even when <code class="docutils literal notranslate"><span class="pre">x.imag</span></code> is zero:</p>

  255. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">phase</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">))</span>

  256. <span class="go">3.141592653589793</span>

  257. <span class="gp">&gt;&gt;&gt; </span><span class="n">phase</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0</span><span class="p">))</span>

  258. <span class="go">-3.141592653589793</span>

  259. </pre></div>

  260. </div>

  261. </dd></dl>

  262. 

  263. <div class="admonition note">

  264. <p class="admonition-title">Note</p>

  265. <p>The modulus (absolute value) of a complex number <em>x</em> can be

  266. computed using the built-in <a class="reference internal" href="functions.html#abs" title="abs"><code class="xref py py-func docutils literal notranslate"><span class="pre">abs()</span></code></a> function.  There is no

  267. separate <a class="reference internal" href="#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code></a> module function for this operation.</p>

  268. </div>

  269. <dl class="py function">

  270. <dt class="sig sig-object py" id="cmath.polar">

  271. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">polar</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.polar" title="Link to this definition">¶</a></dt>

  272. <dd><p>Return the representation of <em>x</em> in polar coordinates.  Returns a

  273. pair <code class="docutils literal notranslate"><span class="pre">(r,</span> <span class="pre">phi)</span></code> where <em>r</em> is the modulus of <em>x</em> and phi is the

  274. phase of <em>x</em>.  <code class="docutils literal notranslate"><span class="pre">polar(x)</span></code> is equivalent to <code class="docutils literal notranslate"><span class="pre">(abs(x),</span>

  275. <span class="pre">phase(x))</span></code>.</p>

  276. </dd></dl>

  277. 

  278. <dl class="py function">

  279. <dt class="sig sig-object py" id="cmath.rect">

  280. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">rect</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">r</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.rect" title="Link to this definition">¶</a></dt>

  281. <dd><p>Return the complex number <em>x</em> with polar coordinates <em>r</em> and <em>phi</em>.

  282. Equivalent to <code class="docutils literal notranslate"><span class="pre">r</span> <span class="pre">*</span> <span class="pre">(math.cos(phi)</span> <span class="pre">+</span> <span class="pre">math.sin(phi)*1j)</span></code>.</p>

  283. </dd></dl>

  284. 

  285. </section>

  286. <section id="power-and-logarithmic-functions">

  287. <h2>Power and logarithmic functions<a class="headerlink" href="#power-and-logarithmic-functions" title="Link to this heading">¶</a></h2>

  288. <dl class="py function">

  289. <dt class="sig sig-object py" id="cmath.exp">

  290. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">exp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.exp" title="Link to this definition">¶</a></dt>

  291. <dd><p>Return <em>e</em> raised to the power <em>x</em>, where <em>e</em> is the base of natural

  292. logarithms.</p>

  293. </dd></dl>

  294. 

  295. <dl class="py function">

  296. <dt class="sig sig-object py" id="cmath.log">

  297. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">log</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="optional">[</span>, <em class="sig-param"><span class="n"><span class="pre">base</span></span></em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#cmath.log" title="Link to this definition">¶</a></dt>

  298. <dd><p>Returns the logarithm of <em>x</em> to the given <em>base</em>. If the <em>base</em> is not

  299. specified, returns the natural logarithm of <em>x</em>. There is one branch cut,

  300. from 0 along the negative real axis to -∞.</p>

  301. </dd></dl>

  302. 

  303. <dl class="py function">

  304. <dt class="sig sig-object py" id="cmath.log10">

  305. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">log10</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.log10" title="Link to this definition">¶</a></dt>

  306. <dd><p>Return the base-10 logarithm of <em>x</em>. This has the same branch cut as

  307. <a class="reference internal" href="#cmath.log" title="cmath.log"><code class="xref py py-func docutils literal notranslate"><span class="pre">log()</span></code></a>.</p>

  308. </dd></dl>

  309. 

  310. <dl class="py function">

  311. <dt class="sig sig-object py" id="cmath.sqrt">

  312. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">sqrt</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.sqrt" title="Link to this definition">¶</a></dt>

  313. <dd><p>Return the square root of <em>x</em>. This has the same branch cut as <a class="reference internal" href="#cmath.log" title="cmath.log"><code class="xref py py-func docutils literal notranslate"><span class="pre">log()</span></code></a>.</p>

  314. </dd></dl>

  315. 

  316. </section>

  317. <section id="trigonometric-functions">

  318. <h2>Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Link to this heading">¶</a></h2>

  319. <dl class="py function">

  320. <dt class="sig sig-object py" id="cmath.acos">

  321. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">acos</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.acos" title="Link to this definition">¶</a></dt>

  322. <dd><p>Return the arc cosine of <em>x</em>. There are two branch cuts: One extends right

  323. from 1 along the real axis to ∞. The other extends left from -1 along the

  324. real axis to -∞.</p>

  325. </dd></dl>

  326. 

  327. <dl class="py function">

  328. <dt class="sig sig-object py" id="cmath.asin">

  329. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">asin</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.asin" title="Link to this definition">¶</a></dt>

  330. <dd><p>Return the arc sine of <em>x</em>. This has the same branch cuts as <a class="reference internal" href="#cmath.acos" title="cmath.acos"><code class="xref py py-func docutils literal notranslate"><span class="pre">acos()</span></code></a>.</p>

  331. </dd></dl>

  332. 

  333. <dl class="py function">

  334. <dt class="sig sig-object py" id="cmath.atan">

  335. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">atan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.atan" title="Link to this definition">¶</a></dt>

  336. <dd><p>Return the arc tangent of <em>x</em>. There are two branch cuts: One extends from

  337. <code class="docutils literal notranslate"><span class="pre">1j</span></code> along the imaginary axis to <code class="docutils literal notranslate"><span class="pre">∞j</span></code>. The other extends from <code class="docutils literal notranslate"><span class="pre">-1j</span></code>

  338. along the imaginary axis to <code class="docutils literal notranslate"><span class="pre">-∞j</span></code>.</p>

  339. </dd></dl>

  340. 

  341. <dl class="py function">

  342. <dt class="sig sig-object py" id="cmath.cos">

  343. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">cos</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.cos" title="Link to this definition">¶</a></dt>

  344. <dd><p>Return the cosine of <em>x</em>.</p>

  345. </dd></dl>

  346. 

  347. <dl class="py function">

  348. <dt class="sig sig-object py" id="cmath.sin">

  349. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">sin</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.sin" title="Link to this definition">¶</a></dt>

  350. <dd><p>Return the sine of <em>x</em>.</p>

  351. </dd></dl>

  352. 

  353. <dl class="py function">

  354. <dt class="sig sig-object py" id="cmath.tan">

  355. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">tan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.tan" title="Link to this definition">¶</a></dt>

  356. <dd><p>Return the tangent of <em>x</em>.</p>

  357. </dd></dl>

  358. 

  359. </section>

  360. <section id="hyperbolic-functions">

  361. <h2>Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Link to this heading">¶</a></h2>

  362. <dl class="py function">

  363. <dt class="sig sig-object py" id="cmath.acosh">

  364. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">acosh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.acosh" title="Link to this definition">¶</a></dt>

  365. <dd><p>Return the inverse hyperbolic cosine of <em>x</em>. There is one branch cut,

  366. extending left from 1 along the real axis to -∞.</p>

  367. </dd></dl>

  368. 

  369. <dl class="py function">

  370. <dt class="sig sig-object py" id="cmath.asinh">

  371. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">asinh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.asinh" title="Link to this definition">¶</a></dt>

  372. <dd><p>Return the inverse hyperbolic sine of <em>x</em>. There are two branch cuts:

  373. One extends from <code class="docutils literal notranslate"><span class="pre">1j</span></code> along the imaginary axis to <code class="docutils literal notranslate"><span class="pre">∞j</span></code>.  The other

  374. extends from <code class="docutils literal notranslate"><span class="pre">-1j</span></code> along the imaginary axis to <code class="docutils literal notranslate"><span class="pre">-∞j</span></code>.</p>

  375. </dd></dl>

  376. 

  377. <dl class="py function">

  378. <dt class="sig sig-object py" id="cmath.atanh">

  379. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">atanh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.atanh" title="Link to this definition">¶</a></dt>

  380. <dd><p>Return the inverse hyperbolic tangent of <em>x</em>. There are two branch cuts: One

  381. extends from <code class="docutils literal notranslate"><span class="pre">1</span></code> along the real axis to <code class="docutils literal notranslate"><span class="pre">∞</span></code>. The other extends from

  382. <code class="docutils literal notranslate"><span class="pre">-1</span></code> along the real axis to <code class="docutils literal notranslate"><span class="pre">-∞</span></code>.</p>

  383. </dd></dl>

  384. 

  385. <dl class="py function">

  386. <dt class="sig sig-object py" id="cmath.cosh">

  387. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">cosh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.cosh" title="Link to this definition">¶</a></dt>

  388. <dd><p>Return the hyperbolic cosine of <em>x</em>.</p>

  389. </dd></dl>

  390. 

  391. <dl class="py function">

  392. <dt class="sig sig-object py" id="cmath.sinh">

  393. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">sinh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.sinh" title="Link to this definition">¶</a></dt>

  394. <dd><p>Return the hyperbolic sine of <em>x</em>.</p>

  395. </dd></dl>

  396. 

  397. <dl class="py function">

  398. <dt class="sig sig-object py" id="cmath.tanh">

  399. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">tanh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.tanh" title="Link to this definition">¶</a></dt>

  400. <dd><p>Return the hyperbolic tangent of <em>x</em>.</p>

  401. </dd></dl>

  402. 

  403. </section>

  404. <section id="classification-functions">

  405. <h2>Classification functions<a class="headerlink" href="#classification-functions" title="Link to this heading">¶</a></h2>

  406. <dl class="py function">

  407. <dt class="sig sig-object py" id="cmath.isfinite">

  408. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">isfinite</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isfinite" title="Link to this definition">¶</a></dt>

  409. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if both the real and imaginary parts of <em>x</em> are finite, and

  410. <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>

  411. <div class="versionadded">

  412. <p><span class="versionmodified added">New in version 3.2.</span></p>

  413. </div>

  414. </dd></dl>

  415. 

  416. <dl class="py function">

  417. <dt class="sig sig-object py" id="cmath.isinf">

  418. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">isinf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isinf" title="Link to this definition">¶</a></dt>

  419. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if either the real or the imaginary part of <em>x</em> is an

  420. infinity, and <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>

  421. </dd></dl>

  422. 

  423. <dl class="py function">

  424. <dt class="sig sig-object py" id="cmath.isnan">

  425. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">isnan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isnan" title="Link to this definition">¶</a></dt>

  426. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if either the real or the imaginary part of <em>x</em> is a NaN,

  427. and <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>

  428. </dd></dl>

  429. 

  430. <dl class="py function">

  431. <dt class="sig sig-object py" id="cmath.isclose">

  432. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">isclose</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rel_tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-09</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">abs_tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#cmath.isclose" title="Link to this definition">¶</a></dt>

  433. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if the values <em>a</em> and <em>b</em> are close to each other and

  434. <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>

  435. <p>Whether or not two values are considered close is determined according to

  436. given absolute and relative tolerances.</p>

  437. <p><em>rel_tol</em> is the relative tolerance – it is the maximum allowed difference

  438. between <em>a</em> and <em>b</em>, relative to the larger absolute value of <em>a</em> or <em>b</em>.

  439. For example, to set a tolerance of 5%, pass <code class="docutils literal notranslate"><span class="pre">rel_tol=0.05</span></code>.  The default

  440. tolerance is <code class="docutils literal notranslate"><span class="pre">1e-09</span></code>, which assures that the two values are the same

  441. within about 9 decimal digits.  <em>rel_tol</em> must be greater than zero.</p>

  442. <p><em>abs_tol</em> is the minimum absolute tolerance – useful for comparisons near

  443. zero. <em>abs_tol</em> must be at least zero.</p>

  444. <p>If no errors occur, the result will be:

  445. <code class="docutils literal notranslate"><span class="pre">abs(a-b)</span> <span class="pre">&lt;=</span> <span class="pre">max(rel_tol</span> <span class="pre">*</span> <span class="pre">max(abs(a),</span> <span class="pre">abs(b)),</span> <span class="pre">abs_tol)</span></code>.</p>

  446. <p>The IEEE 754 special values of <code class="docutils literal notranslate"><span class="pre">NaN</span></code>, <code class="docutils literal notranslate"><span class="pre">inf</span></code>, and <code class="docutils literal notranslate"><span class="pre">-inf</span></code> will be

  447. handled according to IEEE rules.  Specifically, <code class="docutils literal notranslate"><span class="pre">NaN</span></code> is not considered

  448. close to any other value, including <code class="docutils literal notranslate"><span class="pre">NaN</span></code>.  <code class="docutils literal notranslate"><span class="pre">inf</span></code> and <code class="docutils literal notranslate"><span class="pre">-inf</span></code> are only

  449. considered close to themselves.</p>

  450. <div class="versionadded">

  451. <p><span class="versionmodified added">New in version 3.5.</span></p>

  452. </div>

  453. <div class="admonition seealso">

  454. <p class="admonition-title">See also</p>

  455. <p><span class="target" id="index-0"></span><a class="pep reference external" href="https://peps.python.org/pep-0485/"><strong>PEP 485</strong></a> – A function for testing approximate equality</p>

  456. </div>

  457. </dd></dl>

  458. 

  459. </section>

  460. <section id="constants">

  461. <h2>Constants<a class="headerlink" href="#constants" title="Link to this heading">¶</a></h2>

  462. <dl class="py data">

  463. <dt class="sig sig-object py" id="cmath.pi">

  464. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">pi</span></span><a class="headerlink" href="#cmath.pi" title="Link to this definition">¶</a></dt>

  465. <dd><p>The mathematical constant <em>π</em>, as a float.</p>

  466. </dd></dl>

  467. 

  468. <dl class="py data">

  469. <dt class="sig sig-object py" id="cmath.e">

  470. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">e</span></span><a class="headerlink" href="#cmath.e" title="Link to this definition">¶</a></dt>

  471. <dd><p>The mathematical constant <em>e</em>, as a float.</p>

  472. </dd></dl>

  473. 

  474. <dl class="py data">

  475. <dt class="sig sig-object py" id="cmath.tau">

  476. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">tau</span></span><a class="headerlink" href="#cmath.tau" title="Link to this definition">¶</a></dt>

  477. <dd><p>The mathematical constant <em>τ</em>, as a float.</p>

  478. <div class="versionadded">

  479. <p><span class="versionmodified added">New in version 3.6.</span></p>

  480. </div>

  481. </dd></dl>

  482. 

  483. <dl class="py data">

  484. <dt class="sig sig-object py" id="cmath.inf">

  485. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">inf</span></span><a class="headerlink" href="#cmath.inf" title="Link to this definition">¶</a></dt>

  486. <dd><p>Floating-point positive infinity. Equivalent to <code class="docutils literal notranslate"><span class="pre">float('inf')</span></code>.</p>

  487. <div class="versionadded">

  488. <p><span class="versionmodified added">New in version 3.6.</span></p>

  489. </div>

  490. </dd></dl>

  491. 

  492. <dl class="py data">

  493. <dt class="sig sig-object py" id="cmath.infj">

  494. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">infj</span></span><a class="headerlink" href="#cmath.infj" title="Link to this definition">¶</a></dt>

  495. <dd><p>Complex number with zero real part and positive infinity imaginary

  496. part. Equivalent to <code class="docutils literal notranslate"><span class="pre">complex(0.0,</span> <span class="pre">float('inf'))</span></code>.</p>

  497. <div class="versionadded">

  498. <p><span class="versionmodified added">New in version 3.6.</span></p>

  499. </div>

  500. </dd></dl>

  501. 

  502. <dl class="py data">

  503. <dt class="sig sig-object py" id="cmath.nan">

  504. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">nan</span></span><a class="headerlink" href="#cmath.nan" title="Link to this definition">¶</a></dt>

  505. <dd><p>A floating-point “not a number” (NaN) value.  Equivalent to

  506. <code class="docutils literal notranslate"><span class="pre">float('nan')</span></code>.</p>

  507. <div class="versionadded">

  508. <p><span class="versionmodified added">New in version 3.6.</span></p>

  509. </div>

  510. </dd></dl>

  511. 

  512. <dl class="py data">

  513. <dt class="sig sig-object py" id="cmath.nanj">

  514. <span class="sig-prename descclassname"><span class="pre">cmath.</span></span><span class="sig-name descname"><span class="pre">nanj</span></span><a class="headerlink" href="#cmath.nanj" title="Link to this definition">¶</a></dt>

  515. <dd><p>Complex number with zero real part and NaN imaginary part. Equivalent to

  516. <code class="docutils literal notranslate"><span class="pre">complex(0.0,</span> <span class="pre">float('nan'))</span></code>.</p>

  517. <div class="versionadded">

  518. <p><span class="versionmodified added">New in version 3.6.</span></p>

  519. </div>

  520. </dd></dl>

  521. 

  522. <p id="index-1">Note that the selection of functions is similar, but not identical, to that in

  523. module <a class="reference internal" href="math.html#module-math" title="math: Mathematical functions (sin() etc.)."><code class="xref py py-mod docutils literal notranslate"><span class="pre">math</span></code></a>.  The reason for having two modules is that some users aren’t

  524. interested in complex numbers, and perhaps don’t even know what they are.  They

  525. would rather have <code class="docutils literal notranslate"><span class="pre">math.sqrt(-1)</span></code> raise an exception than return a complex

  526. number. Also note that the functions defined in <a class="reference internal" href="#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code></a> always return a

  527. complex number, even if the answer can be expressed as a real number (in which

  528. case the complex number has an imaginary part of zero).</p>

  529. <p>A note on branch cuts: They are curves along which the given function fails to

  530. be continuous.  They are a necessary feature of many complex functions.  It is

  531. assumed that if you need to compute with complex functions, you will understand

  532. about branch cuts.  Consult almost any (not too elementary) book on complex

  533. variables for enlightenment.  For information of the proper choice of branch

  534. cuts for numerical purposes, a good reference should be the following:</p>

  535. <div class="admonition seealso">

  536. <p class="admonition-title">See also</p>

  537. <p>Kahan, W:  Branch cuts for complex elementary functions; or, Much ado about

  538. nothing’s sign bit.  In Iserles, A., and Powell, M. (eds.), The state of the art

  539. in numerical analysis. Clarendon Press (1987) pp165–211.</p>

  540. </div>

  541. </section>

  542. </section>

  543. 

  544. 

  545.             <div class="clearer"></div>

  546.           </div>

  547.         </div>

  548.       </div>

  549.       <div class="sphinxsidebar" role="navigation" aria-label="main navigation">

  550.         <div class="sphinxsidebarwrapper">

  551.   <div>

  552.     <h3><a href="../contents.html">Table of Contents</a></h3>

  553.     <ul>

  554. <li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> — Mathematical functions for complex numbers</a><ul>

  555. <li><a class="reference internal" href="#conversions-to-and-from-polar-coordinates">Conversions to and from polar coordinates</a></li>

  556. <li><a class="reference internal" href="#power-and-logarithmic-functions">Power and logarithmic functions</a></li>

  557. <li><a class="reference internal" href="#trigonometric-functions">Trigonometric functions</a></li>

  558. <li><a class="reference internal" href="#hyperbolic-functions">Hyperbolic functions</a></li>

  559. <li><a class="reference internal" href="#classification-functions">Classification functions</a></li>

  560. <li><a class="reference internal" href="#constants">Constants</a></li>

  561. </ul>

  562. </li>

  563. </ul>

  564. 

  565.   </div>

  566.   <div>

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