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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> | |
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