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<H1><A NAME="SECTION007800000000000000000">
5.8 <tt class="module">cmath</tt> --
         Mathematical functions for complex numbers</A>
</H1>

<P>
<A NAME="module-cmath"></A>

<P>
This module is always available.  It provides access to mathematical
functions for complex numbers.  The functions are:

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1278' xml:id='l2h-1278' class="function">acos</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the arc cosine of <var>x</var>.
There are two branch cuts:
One extends right from 1 along the real axis to &infin;, continuous
from below.
The other extends left from -1 along the real axis to -&infin;,
continuous from above.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1279' xml:id='l2h-1279' class="function">acosh</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the hyperbolic arc cosine of <var>x</var>.
There is one branch cut, extending left from 1 along the real axis
to -&infin;, continuous from above.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1280' xml:id='l2h-1280' class="function">asin</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the arc sine of <var>x</var>.
This has the same branch cuts as <tt class="function">acos()</tt>.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1281' xml:id='l2h-1281' class="function">asinh</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the hyperbolic arc sine of <var>x</var>.
There are two branch cuts, extending left from &plusmn;<code>1j</code> to
&plusmn;-&infin;<code>j</code>, both continuous from above.
These branch cuts should be considered a bug to be corrected in a
future release.
The correct branch cuts should extend along the imaginary axis,
one from <code>1j</code> up to &infin;<code>j</code> and continuous from the
right, and one from -<code>1j</code> down to -&infin;<code>j</code> and
continuous from the left.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1282' xml:id='l2h-1282' class="function">atan</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the arc tangent of <var>x</var>.
There are two branch cuts:
One extends from <code>1j</code> along the imaginary axis to
&infin;<code>j</code>, continuous from the left.
The other extends from -<code>1j</code> along the imaginary axis to
-&infin;<code>j</code>, continuous from the left.
(This should probably be changed so the upper cut becomes continuous
from the other side.)
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1283' xml:id='l2h-1283' class="function">atanh</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the hyperbolic arc tangent of <var>x</var>.
There are two branch cuts:
One extends from 1 along the real axis to &infin;, continuous
from above.
The other extends from -1 along the real axis to -&infin;,
continuous from above.
(This should probably be changed so the right cut becomes continuous from
the other side.)
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1284' xml:id='l2h-1284' class="function">cos</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the cosine of <var>x</var>.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1285' xml:id='l2h-1285' class="function">cosh</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the hyperbolic cosine of <var>x</var>.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1286' xml:id='l2h-1286' class="function">exp</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the exponential value <code>e**<var>x</var></code>.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1287' xml:id='l2h-1287' class="function">log</tt></b>(</nobr></td>
  <td><var>x</var><big>[</big><var>, base</var><big>]</big><var></var>)</td></tr></table></dt>
<dd>
Returns the logarithm of <var>x</var> to the given <var>base</var>.
If the <var>base</var> is not specified, returns the natural logarithm of <var>x</var>.
There is one branch cut, from 0 along the negative real axis to
-&infin;, continuous from above.

<span class="versionnote">Changed in version 2.4:
<var>base</var> argument added.</span>

</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1288' xml:id='l2h-1288' class="function">log10</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the base-10 logarithm of <var>x</var>.
This has the same branch cut as <tt class="function">log()</tt>.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1289' xml:id='l2h-1289' class="function">sin</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the sine of <var>x</var>.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1290' xml:id='l2h-1290' class="function">sinh</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the hyperbolic sine of <var>x</var>.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1291' xml:id='l2h-1291' class="function">sqrt</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the square root of <var>x</var>.
This has the same branch cut as <tt class="function">log()</tt>.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1292' xml:id='l2h-1292' class="function">tan</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the tangent of <var>x</var>.
</dl>

<P>
<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
  <td><nobr><b><tt id='l2h-1293' xml:id='l2h-1293' class="function">tanh</tt></b>(</nobr></td>
  <td><var>x</var>)</td></tr></table></dt>
<dd>
Return the hyperbolic tangent of <var>x</var>.
</dl>

<P>
The module also defines two mathematical constants:

<P>
<dl><dt><b><tt id='l2h-1294' xml:id='l2h-1294'>pi</tt></b></dt>
<dd>
The mathematical constant <em>pi</em>, as a real.
</dd></dl>

<P>
<dl><dt><b><tt id='l2h-1295' xml:id='l2h-1295'>e</tt></b></dt>
<dd>
The mathematical constant <em>e</em>, as a real.
</dd></dl>

<P>
Note that the selection of functions is similar, but not identical, to
that in module <tt class="module"><a href="module-math.html">math</a></tt><a id='l2h-1296' xml:id='l2h-1296'></a>.  The reason for having
two modules is that some users aren't interested in complex numbers,
and perhaps don't even know what they are.  They would rather have
<code>math.sqrt(-1)</code> raise an exception than return a complex number.
Also note that the functions defined in <tt class="module">cmath</tt> always return a
complex number, even if the answer can be expressed as a real number
(in which case the complex number has an imaginary part of zero).

<P>
A note on branch cuts: They are curves along which the given function
fails to be continuous.  They are a necessary feature of many complex
functions.  It is assumed that if you need to compute with complex
functions, you will understand about branch cuts.  Consult almost any
(not too elementary) book on complex variables for enlightenment.  For
information of the proper choice of branch cuts for numerical
purposes, a good reference should be the following:

<P>
<div class="seealso">
  <p class="heading">See Also:</p>

  <div class="seetext"><p>Kahan, W:  Branch cuts for complex elementary functions;
           or, Much ado about nothing's sign bit.  In Iserles, A.,
           and Powell, M. (eds.), <em class="citetitle"
 >The state of the art in
           numerical analysis</em>. Clarendon Press (1987) pp165-211.</p></div>
</div>

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	54
	55	<H1><A NAME="SECTION007800000000000000000">
	56	5.8 <tt class="module">cmath</tt> --
	57	Mathematical functions for complex numbers</A>
	58	</H1>
	59
	60	<P>
	61	<A NAME="module-cmath"></A>
	62
	63	<P>
	64	This module is always available. It provides access to mathematical
65	functions for complex numbers. The functions are:
66
67	<P>
68	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
69	<td><nobr><b><tt id='l2h-1278' xml:id='l2h-1278' class="function">acos</tt></b>(</nobr></td>
70	<td><var>x</var>)</td></tr></table></dt>
71	<dd>
72	Return the arc cosine of <var>x</var>.
73	There are two branch cuts:
74	One extends right from 1 along the real axis to ∞, continuous
75	from below.
76	The other extends left from -1 along the real axis to -∞,
77	continuous from above.
78	</dl>
79
80	<P>
81	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
82	<td><nobr><b><tt id='l2h-1279' xml:id='l2h-1279' class="function">acosh</tt></b>(</nobr></td>
83	<td><var>x</var>)</td></tr></table></dt>
84	<dd>
85	Return the hyperbolic arc cosine of <var>x</var>.
86	There is one branch cut, extending left from 1 along the real axis
87	to -∞, continuous from above.
88	</dl>
89
90	<P>
91	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
92	<td><nobr><b><tt id='l2h-1280' xml:id='l2h-1280' class="function">asin</tt></b>(</nobr></td>
93	<td><var>x</var>)</td></tr></table></dt>
94	<dd>
95	Return the arc sine of <var>x</var>.
96	This has the same branch cuts as <tt class="function">acos()</tt>.
97	</dl>
98
99	<P>
100	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
101	<td><nobr><b><tt id='l2h-1281' xml:id='l2h-1281' class="function">asinh</tt></b>(</nobr></td>
102	<td><var>x</var>)</td></tr></table></dt>
103	<dd>
104	Return the hyperbolic arc sine of <var>x</var>.
105	There are two branch cuts, extending left from ±<code>1j</code> to
106	±-∞<code>j</code>, both continuous from above.
107	These branch cuts should be considered a bug to be corrected in a
108	future release.
109	The correct branch cuts should extend along the imaginary axis,
110	one from <code>1j</code> up to ∞<code>j</code> and continuous from the
111	right, and one from -<code>1j</code> down to -∞<code>j</code> and
112	continuous from the left.
113	</dl>
114
115	<P>
116	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
117	<td><nobr><b><tt id='l2h-1282' xml:id='l2h-1282' class="function">atan</tt></b>(</nobr></td>
118	<td><var>x</var>)</td></tr></table></dt>
119	<dd>
120	Return the arc tangent of <var>x</var>.
121	There are two branch cuts:
122	One extends from <code>1j</code> along the imaginary axis to
123	∞<code>j</code>, continuous from the left.
124	The other extends from -<code>1j</code> along the imaginary axis to
125	-∞<code>j</code>, continuous from the left.
126	(This should probably be changed so the upper cut becomes continuous
127	from the other side.)
128	</dl>
129
130	<P>
131	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
132	<td><nobr><b><tt id='l2h-1283' xml:id='l2h-1283' class="function">atanh</tt></b>(</nobr></td>
133	<td><var>x</var>)</td></tr></table></dt>
134	<dd>
135	Return the hyperbolic arc tangent of <var>x</var>.
136	There are two branch cuts:
137	One extends from 1 along the real axis to ∞, continuous
138	from above.
139	The other extends from -1 along the real axis to -∞,
140	continuous from above.
141	(This should probably be changed so the right cut becomes continuous from
142	the other side.)
143	</dl>
144
145	<P>
146	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
147	<td><nobr><b><tt id='l2h-1284' xml:id='l2h-1284' class="function">cos</tt></b>(</nobr></td>
148	<td><var>x</var>)</td></tr></table></dt>
149	<dd>
150	Return the cosine of <var>x</var>.
151	</dl>
152
153	<P>
154	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
155	<td><nobr><b><tt id='l2h-1285' xml:id='l2h-1285' class="function">cosh</tt></b>(</nobr></td>
156	<td><var>x</var>)</td></tr></table></dt>
157	<dd>
158	Return the hyperbolic cosine of <var>x</var>.
159	</dl>
160
161	<P>
162	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
163	<td><nobr><b><tt id='l2h-1286' xml:id='l2h-1286' class="function">exp</tt></b>(</nobr></td>
164	<td><var>x</var>)</td></tr></table></dt>
165	<dd>
166	Return the exponential value <code>e**<var>x</var></code>.
167	</dl>
168
169	<P>
170	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
171	<td><nobr><b><tt id='l2h-1287' xml:id='l2h-1287' class="function">log</tt></b>(</nobr></td>
172	<td><var>x</var><big>[</big><var>, base</var><big>]</big><var></var>)</td></tr></table></dt>
173	<dd>
174	Returns the logarithm of <var>x</var> to the given <var>base</var>.
175	If the <var>base</var> is not specified, returns the natural logarithm of <var>x</var>.
176	There is one branch cut, from 0 along the negative real axis to
177	-∞, continuous from above.
178
179	<span class="versionnote">Changed in version 2.4:
180	<var>base</var> argument added.</span>
181
182	</dl>
183
184	<P>
185	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
186	<td><nobr><b><tt id='l2h-1288' xml:id='l2h-1288' class="function">log10</tt></b>(</nobr></td>
187	<td><var>x</var>)</td></tr></table></dt>
188	<dd>
189	Return the base-10 logarithm of <var>x</var>.
190	This has the same branch cut as <tt class="function">log()</tt>.
191	</dl>
192
193	<P>
194	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
195	<td><nobr><b><tt id='l2h-1289' xml:id='l2h-1289' class="function">sin</tt></b>(</nobr></td>
196	<td><var>x</var>)</td></tr></table></dt>
197	<dd>
198	Return the sine of <var>x</var>.
199	</dl>
200
201	<P>
202	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
203	<td><nobr><b><tt id='l2h-1290' xml:id='l2h-1290' class="function">sinh</tt></b>(</nobr></td>
204	<td><var>x</var>)</td></tr></table></dt>
205	<dd>
206	Return the hyperbolic sine of <var>x</var>.
207	</dl>
208
209	<P>
210	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
211	<td><nobr><b><tt id='l2h-1291' xml:id='l2h-1291' class="function">sqrt</tt></b>(</nobr></td>
212	<td><var>x</var>)</td></tr></table></dt>
213	<dd>
214	Return the square root of <var>x</var>.
215	This has the same branch cut as <tt class="function">log()</tt>.
216	</dl>
217
218	<P>
219	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
220	<td><nobr><b><tt id='l2h-1292' xml:id='l2h-1292' class="function">tan</tt></b>(</nobr></td>
221	<td><var>x</var>)</td></tr></table></dt>
222	<dd>
223	Return the tangent of <var>x</var>.
224	</dl>
225
226	<P>
227	<dl><dt><table cellpadding="0" cellspacing="0"><tr valign="baseline">
228	<td><nobr><b><tt id='l2h-1293' xml:id='l2h-1293' class="function">tanh</tt></b>(</nobr></td>
229	<td><var>x</var>)</td></tr></table></dt>
230	<dd>
231	Return the hyperbolic tangent of <var>x</var>.
232	</dl>
233
234	<P>
235	The module also defines two mathematical constants:
236
237	<P>
238	<dl><dt><b><tt id='l2h-1294' xml:id='l2h-1294'>pi</tt></b></dt>
239	<dd>
240	The mathematical constant <em>pi</em>, as a real.
241	</dd></dl>
242
243	<P>
244	<dl><dt><b><tt id='l2h-1295' xml:id='l2h-1295'>e</tt></b></dt>
245	<dd>
246	The mathematical constant <em>e</em>, as a real.
247	</dd></dl>
248
249	<P>
250	Note that the selection of functions is similar, but not identical, to
251	that in module <tt class="module"><a href="module-math.html">math</a></tt><a id='l2h-1296' xml:id='l2h-1296'></a>. The reason for having
252	two modules is that some users aren't interested in complex numbers,
253	and perhaps don't even know what they are. They would rather have
254	<code>math.sqrt(-1)</code> raise an exception than return a complex number.
255	Also note that the functions defined in <tt class="module">cmath</tt> always return a
256	complex number, even if the answer can be expressed as a real number
257	(in which case the complex number has an imaginary part of zero).
258
259	<P>
260	A note on branch cuts: They are curves along which the given function
261	fails to be continuous. They are a necessary feature of many complex
262	functions. It is assumed that if you need to compute with complex
263	functions, you will understand about branch cuts. Consult almost any
264	(not too elementary) book on complex variables for enlightenment. For
265	information of the proper choice of branch cuts for numerical
266	purposes, a good reference should be the following:
267
268	<P>
269	<div class="seealso">
270	<p class="heading">See Also:</p>
271
272	<div class="seetext"><p>Kahan, W: Branch cuts for complex elementary functions;
273	or, Much ado about nothing's sign bit. In Iserles, A.,
274	and Powell, M. (eds.), <em class="citetitle"
275	>The state of the art in
276	numerical analysis</em>. Clarendon Press (1987) pp165-211.</p></div>
277	</div>
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