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2to3.html
63.71
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__builtin__.html
11.79
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__future__.html
15.58
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__main__.html
8.33
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_winreg.html
73.99
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abc.html
26.82
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aepack.html
15.47
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aetools.html
18.79
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aetypes.html
27.08
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aifc.html
29.59
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al.html
22.13
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allos.html
35.48
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anydbm.html
19.31
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archiving.html
10.64
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argparse.html
269.59
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array.html
35.35
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ast.html
42.42
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asynchat.html
35.84
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asyncore.html
43.22
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atexit.html
19.23
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audioop.html
41.32
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autogil.html
9.77
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base64.html
24.8
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basehttpserver.html
40.6
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bastion.html
13.02
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bdb.html
50.06
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binascii.html
26.64
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binhex.html
12.66
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bisect.html
28.52
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bsddb.html
32.02
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bz2.html
31.92
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calendar.html
50.16
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carbon.html
54.14
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cd.html
35.24
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cgi.html
58.4
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cgihttpserver.html
15.12
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cgitb.html
13.5
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chunk.html
17.41
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cmath.html
32.41
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cmd.html
32.69
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code.html
28.92
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codecs.html
125.92
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codeop.html
16.42
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collections.html
150.36
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colorpicker.html
9.02
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colorsys.html
14.38
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commands.html
16.63
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compileall.html
21.18
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compiler.html
78.14
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configparser.html
73.92
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constants.html
14.68
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contextlib.html
22.91
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cookie.html
45.38
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cookielib.html
102.78
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copy.html
14.21
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copy_reg.html
15.82
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crypt.html
11.62
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crypto.html
8.35
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csv.html
82.79
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ctypes.html
280.57
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curses.ascii.html
28.97
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curses.html
200.94
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curses.panel.html
18.43
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custominterp.html
8.59
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datatypes.html
18.84
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datetime.html
261.02
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dbhash.html
18.4
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dbm.html
14.75
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debug.html
11.28
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decimal.html
239.14
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development.html
15.27
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difflib.html
96.93
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dircache.html
13.66
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dis.html
96.52
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distribution.html
8.27
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distutils.html
10.97
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dl.html
19.27
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doctest.html
193.18
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docxmlrpcserver.html
20.02
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dumbdbm.html
16.74
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dummy_thread.html
11.04
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dummy_threading.html
9.96
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easydialogs.html
38.83
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email-examples.html
47.2
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email.charset.html
32.45
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email.encoders.html
14.09
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email.errors.html
19.2
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email.generator.html
24.78
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email.header.html
31.12
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email.html
55.3
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email.iterators.html
14.22
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email.message.html
74.91
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email.mime.html
34.51
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email.parser.html
37.09
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email.utils.html
30.04
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ensurepip.html
20.21
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errno.html
52.31
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exceptions.html
73.4
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fcntl.html
27.85
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filecmp.html
26.49
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fileformats.html
10.21
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fileinput.html
29.36
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filesys.html
11.55
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fl.html
71.58
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fm.html
15
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fnmatch.html
17.78
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formatter.html
43.91
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fpectl.html
17.84
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fpformat.html
12.65
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fractions.html
27.17
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framework.html
44.24
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frameworks.html
8.11
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ftplib.html
56.56
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functions.html
216.84
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functools.html
31.06
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future_builtins.html
15.9
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gc.html
30.74
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gdbm.html
19.35
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gensuitemodule.html
14.54
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getopt.html
26.56
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getpass.html
12.5
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gettext.html
91.32
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gl.html
26.71
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glob.html
15.42
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grp.html
12.33
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gzip.html
21.95
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hashlib.html
27.4
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heapq.html
36.72
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hmac.html
15.97
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hotshot.html
22.69
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htmllib.html
29.9
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htmlparser.html
45.11
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httplib.html
78.55
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i18n.html
10.61
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ic.html
22.03
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idle.html
45.74
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imageop.html
20.94
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imaplib.html
69.9
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imgfile.html
14.91
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imghdr.html
13.43
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imp.html
41.48
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importlib.html
10.27
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imputil.html
36.11
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index.html
82.54
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inspect.html
64.31
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internet.html
26.77
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intro.html
9.76
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io.html
122.3
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ipc.html
17.18
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itertools.html
129.91
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jpeg.html
15.36
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json.html
79.09
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keyword.html
9.27
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language.html
12.35
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linecache.html
12.7
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locale.html
68.54
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logging.config.html
82.23
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logging.handlers.html
91.96
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logging.html
123.15
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mac.html
24.49
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macos.html
18.51
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macosa.html
14.55
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macostools.html
19.67
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macpath.html
9.19
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mailbox.html
185.53
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mailcap.html
15.3
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markup.html
20.37
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marshal.html
19.88
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math.html
50.52
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md5.html
16.65
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mhlib.html
30.04
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mimetools.html
23.84
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mimetypes.html
35.82
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mimewriter.html
18.19
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mimify.html
16.96
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miniaeframe.html
14.5
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misc.html
7.77
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mm.html
10.19
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mmap.html
33.52
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modulefinder.html
19.85
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modules.html
9.68
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msilib.html
72.78
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msvcrt.html
24.1
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multifile.html
28.82
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multiprocessing.html
423.97
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mutex.html
13.56
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netdata.html
19.28
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netrc.html
15.51
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new.html
16.1
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nis.html
13.19
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nntplib.html
51.08
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numbers.html
40.91
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numeric.html
14.85
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operator.html
118.32
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optparse.html
251.08
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os.html
270.49
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os.path.html
48.87
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ossaudiodev.html
49.79
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othergui.html
9.98
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parser.html
44.52
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pdb.html
41.44
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persistence.html
16.25
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pickle.html
107.94
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pickletools.html
13.21
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pipes.html
21.85
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pkgutil.html
30.59
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platform.html
40.14
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plistlib.html
20.96
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popen2.html
30.13
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poplib.html
27.93
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posix.html
17.27
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posixfile.html
23.68
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pprint.html
36.42
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profile.html
76.65
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pty.html
11.42
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pwd.html
13.53
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py_compile.html
13.33
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pyclbr.html
18.14
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pydoc.html
14.33
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pyexpat.html
93.28
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python.html
13.53
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queue.html
29.64
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quopri.html
14.5
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random.html
47.78
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re.html
163.11
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readline.html
41.2
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repr.html
24.57
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resource.html
30.86
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restricted.html
12.69
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rexec.html
43.59
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rfc822.html
51.25
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rlcompleter.html
15.29
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robotparser.html
15.01
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runpy.html
23.55
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sched.html
21.76
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scrolledtext.html
10.67
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select.html
50.17
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sets.html
39.92
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sgi.html
11.09
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sgmllib.html
38.77
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sha.html
14.55
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shelve.html
32.93
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shlex.html
38.43
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shutil.html
49.49
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signal.html
36.52
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simplehttpserver.html
21.57
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simplexmlrpcserver.html
40.41
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site.html
29
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smtpd.html
15.64
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smtplib.html
51.86
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sndhdr.html
12.05
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socket.html
126.94
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socketserver.html
81.38
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someos.html
17.09
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spwd.html
12.25
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sqlite3.html
154.12
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ssl.html
214.5
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stat.html
41.11
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statvfs.html
13.24
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stdtypes.html
309.83
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string.html
129.58
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stringio.html
21.1
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stringprep.html
21.4
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strings.html
16.42
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struct.html
47.68
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subprocess.html
117
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sun.html
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sunau.html
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sunaudio.html
21.05
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symbol.html
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symtable.html
29.03
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sys.html
118.12
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sysconfig.html
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syslog.html
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tabnanny.html
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tarfile.html
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telnetlib.html
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tempfile.html
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termios.html
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test.html
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textwrap.html
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thread.html
22.67
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threading.html
92.04
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time.html
65.55
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timeit.html
44.38
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tix.html
56.73
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tk.html
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tkinter.html
86.67
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token.html
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tokenize.html
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trace.html
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traceback.html
44.22
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ttk.html
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tty.html
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turtle.html
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types.html
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undoc.html
25.54
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unicodedata.html
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unittest.html
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unix.html
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urllib.html
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urllib2.html
128.77
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urlparse.html
49.04
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user.html
13.41
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userdict.html
32.96
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uu.html
13.25
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uuid.html
33.55
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warnings.html
51.86
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wave.html
29.08
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weakref.html
41.14
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webbrowser.html
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whichdb.html
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windows.html
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winsound.html
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wsgiref.html
94.38
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xdrlib.html
38.05
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xml.dom.html
110.39
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xml.dom.minidom.html
45.23
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xml.dom.pulldom.html
16.32
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xml.etree.elementtree.html
124.2
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xml.html
18.76
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xml.sax.handler.html
47.5
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xml.sax.html
26.04
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xml.sax.reader.html
51.4
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xml.sax.utils.html
18.11
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xmlrpclib.html
71.79
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zipfile.html
67.26
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zipimport.html
24.32
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zlib.html
34.41
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<!DOCTYPE html> <html> <head> <meta charset="utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> <title>9.3. cmath — Mathematical functions for complex numbers — Python 2.7.18 documentation</title> <link rel="stylesheet" type="text/css" href="../_static/pygments.css" /> <link rel="stylesheet" type="text/css" href="../_static/classic.css" /> <script data-url_root="../" id="documentation_options" src="../_static/documentation_options.js"></script> <script src="../_static/jquery.js"></script> <script src="../_static/underscore.js"></script> <script src="../_static/doctools.js"></script> <script src="../_static/sidebar.js"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within Python 2.7.18 documentation" href="../_static/opensearch.xml"/> <link rel="author" title="About these documents" href="../about.html" /> <link rel="index" title="Index" href="../genindex.html" /> <link rel="search" title="Search" href="../search.html" /> <link rel="copyright" title="Copyright" href="../copyright.html" /> <link rel="next" title="9.4. decimal — Decimal fixed point and floating point arithmetic" href="decimal.html" /> <link rel="prev" title="9.2. math — Mathematical functions" href="math.html" /> <link rel="shortcut icon" type="image/png" href="../_static/py.png" /> <link rel="canonical" href="file:///usr/share/doc/python2.7/html/library/cmath.html" /> <script type="text/javascript" src="../_static/copybutton.js"></script> </head><body> <div id="outdated-warning" style="padding: .5em; text-align: center; background-color: #FFBABA; color: #6A0E0E;"> This document is for an old version of Python that is <a href="https://devguide.python.org/devcycle/#end-of-life-branches">no longer supported</a>. You should install the python3 and python3-doc packages and read the <a href="file:///usr/share/doc/python3-doc/html/library/cmath.html"> Python documentation for the Python3 version packaged in this release</a>. </div> <div class="related" role="navigation" aria-label="related navigation"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" accesskey="I">index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="decimal.html" title="9.4. decimal — Decimal fixed point and floating point arithmetic" accesskey="N">next</a> |</li> <li class="right" > <a href="math.html" title="9.2. math — Mathematical functions" accesskey="P">previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="https://www.python.org/">Python</a> »</li> <li> <a href="../index.html">Python 2.7.18 documentation</a> » </li> <li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> »</li> <li class="nav-item nav-item-2"><a href="numeric.html" accesskey="U"><span class="section-number">9. </span>Numeric and Mathematical Modules</a> »</li> <li class="nav-item nav-item-this"><a href=""><span class="section-number">9.3. </span><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> — Mathematical functions for complex numbers</a></li> </ul> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body" role="main"> <section id="module-cmath"> <span id="cmath-mathematical-functions-for-complex-numbers"></span><h1><span class="section-number">9.3. </span><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="Permalink to this headline">¶</a></h1> <p>This module is always available. It provides access to mathematical functions for complex numbers. The functions in this module accept integers, floating-point numbers or complex numbers as arguments. They will also accept any Python object that has either a <code class="xref py py-meth docutils literal notranslate"><span class="pre">__complex__()</span></code> or a <code class="xref py py-meth docutils literal notranslate"><span class="pre">__float__()</span></code> method: these methods are used to convert the object to a complex or floating-point number, respectively, and the function is then applied to the result of the conversion.</p> <div class="admonition note"> <p class="admonition-title">Note</p> <p>On platforms with hardware and system-level support for signed zeros, functions involving branch cuts are continuous on <em>both</em> sides of the branch cut: the sign of the zero distinguishes one side of the branch cut from the other. On platforms that do not support signed zeros the continuity is as specified below.</p> </div> <section id="conversions-to-and-from-polar-coordinates"> <h2><span class="section-number">9.3.1. </span>Conversions to and from polar coordinates<a class="headerlink" href="#conversions-to-and-from-polar-coordinates" title="Permalink to this headline">¶</a></h2> <p>A Python complex number <code class="docutils literal notranslate"><span class="pre">z</span></code> is stored internally using <em>rectangular</em> or <em>Cartesian</em> coordinates. It is completely determined by its <em>real 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 words:</p> <div class="highlight-default 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> </pre></div> </div> <p><em>Polar coordinates</em> give an alternative way to represent a complex number. In polar coordinates, a complex number <em>z</em> is defined by the modulus <em>r</em> and the phase angle <em>phi</em>. The modulus <em>r</em> is the distance from <em>z</em> to the origin, while the phase <em>phi</em> is the counterclockwise angle, measured in radians, from the positive x-axis to the line segment that joins the origin to <em>z</em>.</p> <p>The following functions can be used to convert from the native rectangular coordinates to polar coordinates and back.</p> <dl class="py function"> <dt class="sig sig-object py" id="cmath.phase"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the phase of <em>x</em> (also known as the <em>argument</em> of <em>x</em>), as a float. <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 lies in the range [-π, π], and the branch cut for this operation lies along the negative real axis, continuous from above. On systems with support for signed zeros (which includes most systems in current use), this means that the sign of the result is the same as the 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> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </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> <span class="go">3.1415926535897931</span> <span class="gp">>>> </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> <span class="go">-3.1415926535897931</span> </pre></div> </div> <div class="versionadded"> <p><span class="versionmodified added">New in version 2.6.</span></p> </div> </dd></dl> <div class="admonition note"> <p class="admonition-title">Note</p> <p>The modulus (absolute value) of a complex number <em>x</em> can be 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 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> </div> <dl class="py function"> <dt class="sig sig-object py" id="cmath.polar"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the representation of <em>x</em> in polar coordinates. Returns a 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 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> <span class="pre">phase(x))</span></code>.</p> <div class="versionadded"> <p><span class="versionmodified added">New in version 2.6.</span></p> </div> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.rect"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the complex number <em>x</em> with polar coordinates <em>r</em> and <em>phi</em>. 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> <div class="versionadded"> <p><span class="versionmodified added">New in version 2.6.</span></p> </div> </dd></dl> </section> <section id="power-and-logarithmic-functions"> <h2><span class="section-number">9.3.2. </span>Power and logarithmic functions<a class="headerlink" href="#power-and-logarithmic-functions" title="Permalink to this headline">¶</a></h2> <dl class="py function"> <dt class="sig sig-object py" id="cmath.exp"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the exponential value <code class="docutils literal notranslate"><span class="pre">e**x</span></code>.</p> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.log"> <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="Permalink to this definition">¶</a></dt> <dd><p>Returns the logarithm of <em>x</em> to the given <em>base</em>. If the <em>base</em> is not specified, returns the natural logarithm of <em>x</em>. There is one branch cut, from 0 along the negative real axis to -∞, continuous from above.</p> <div class="versionchanged"> <p><span class="versionmodified changed">Changed in version 2.4: </span><em>base</em> argument added.</p> </div> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.log10"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the base-10 logarithm 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> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.sqrt"> <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="Permalink to this definition">¶</a></dt> <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> </dd></dl> </section> <section id="trigonometric-functions"> <h2><span class="section-number">9.3.3. </span>Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Permalink to this headline">¶</a></h2> <dl class="py function"> <dt class="sig sig-object py" id="cmath.acos"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the arc cosine of <em>x</em>. There are two branch cuts: One extends right from 1 along the real axis to ∞, continuous from below. The other extends left from -1 along the real axis to -∞, continuous from above.</p> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.asin"> <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="Permalink to this definition">¶</a></dt> <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> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.atan"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the arc tangent of <em>x</em>. There are two branch cuts: 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>, continuous from the right. The other 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>, continuous from the left.</p> <div class="versionchanged"> <p><span class="versionmodified changed">Changed in version 2.6: </span>direction of continuity of upper cut reversed</p> </div> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.cos"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the cosine of <em>x</em>.</p> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.sin"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the sine of <em>x</em>.</p> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.tan"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the tangent of <em>x</em>.</p> </dd></dl> </section> <section id="hyperbolic-functions"> <h2><span class="section-number">9.3.4. </span>Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Permalink to this headline">¶</a></h2> <dl class="py function"> <dt class="sig sig-object py" id="cmath.acosh"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the inverse hyperbolic cosine of <em>x</em>. There is one branch cut, extending left from 1 along the real axis to -∞, continuous from above.</p> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.asinh"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the inverse hyperbolic sine of <em>x</em>. There are two branch cuts: 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>, continuous from the right. The other 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>, continuous from the left.</p> <div class="versionchanged"> <p><span class="versionmodified changed">Changed in version 2.6: </span>branch cuts moved to match those recommended by the C99 standard</p> </div> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.atanh"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the inverse hyperbolic tangent of <em>x</em>. There are two branch cuts: One 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>, continuous from below. The other 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>, continuous from above.</p> <div class="versionchanged"> <p><span class="versionmodified changed">Changed in version 2.6: </span>direction of continuity of right cut reversed</p> </div> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.cosh"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the hyperbolic cosine of <em>x</em>.</p> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.sinh"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the hyperbolic sine of <em>x</em>.</p> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.tanh"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return the hyperbolic tangent of <em>x</em>.</p> </dd></dl> </section> <section id="classification-functions"> <h2><span class="section-number">9.3.5. </span>Classification functions<a class="headerlink" href="#classification-functions" title="Permalink to this headline">¶</a></h2> <dl class="py function"> <dt class="sig sig-object py" id="cmath.isinf"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if the real or the imaginary part of x is positive or negative infinity.</p> <div class="versionadded"> <p><span class="versionmodified added">New in version 2.6.</span></p> </div> </dd></dl> <dl class="py function"> <dt class="sig sig-object py" id="cmath.isnan"> <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="Permalink to this definition">¶</a></dt> <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if the real or imaginary part of x is not a number (NaN).</p> <div class="versionadded"> <p><span class="versionmodified added">New in version 2.6.</span></p> </div> </dd></dl> </section> <section id="constants"> <h2><span class="section-number">9.3.6. </span>Constants<a class="headerlink" href="#constants" title="Permalink to this headline">¶</a></h2> <dl class="py data"> <dt class="sig sig-object py" id="cmath.pi"> <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="Permalink to this definition">¶</a></dt> <dd><p>The mathematical constant <em>π</em>, as a float.</p> </dd></dl> <dl class="py data"> <dt class="sig sig-object py" id="cmath.e"> <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="Permalink to this definition">¶</a></dt> <dd><p>The mathematical constant <em>e</em>, as a float.</p> </dd></dl> <p id="index-0">Note that the selection of functions is similar, but not identical, to that in 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 interested in complex numbers, and perhaps don’t even know what they are. They would rather have <code class="docutils literal notranslate"><span class="pre">math.sqrt(-1)</span></code> raise an exception than return a complex 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 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> <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="admonition seealso"> <p class="admonition-title">See also</p> <p>Kahan, W: Branch cuts for complex elementary functions; or, Much ado about nothing’s sign bit. In Iserles, A., and Powell, M. (eds.), The state of the art in numerical analysis. Clarendon Press (1987) pp165–211.</p> </div> </section> </section> <div class="clearer"></div> </div> </div> </div> <div class="sphinxsidebar" role="navigation" aria-label="main navigation"> <div class="sphinxsidebarwrapper"> <h3><a href="../contents.html">Table of Contents</a></h3> <ul> <li><a class="reference internal" href="#">9.3. <code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> — Mathematical functions for complex numbers</a><ul> <li><a class="reference internal" href="#conversions-to-and-from-polar-coordinates">9.3.1. Conversions to and from polar coordinates</a></li> <li><a class="reference internal" href="#power-and-logarithmic-functions">9.3.2. Power and logarithmic functions</a></li> <li><a class="reference internal" href="#trigonometric-functions">9.3.3. Trigonometric functions</a></li> <li><a class="reference internal" href="#hyperbolic-functions">9.3.4. Hyperbolic functions</a></li> <li><a class="reference internal" href="#classification-functions">9.3.5. Classification functions</a></li> <li><a class="reference internal" href="#constants">9.3.6. Constants</a></li> </ul> </li> </ul> <h4>Previous topic</h4> <p class="topless"><a href="math.html" title="previous chapter"><span class="section-number">9.2. </span><code class="xref py py-mod docutils literal notranslate"><span class="pre">math</span></code> — Mathematical functions</a></p> <h4>Next topic</h4> <p class="topless"><a href="decimal.html" title="next chapter"><span class="section-number">9.4. </span><code class="xref py py-mod docutils literal notranslate"><span class="pre">decimal</span></code> — Decimal fixed point and floating point arithmetic</a></p> <div role="note" aria-label="source link"> <h3>This Page</h3> <ul class="this-page-menu"> <li><a href="../_sources/library/cmath.rst.txt" rel="nofollow">Show Source</a></li> </ul> </div> <div id="searchbox" style="display: none" role="search"> <h3 id="searchlabel">Quick search</h3> <div class="searchformwrapper"> <form class="search" action="../search.html" method="get"> <input type="text" name="q" aria-labelledby="searchlabel" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false"/> <input type="submit" value="Go" /> </form> </div> </div> <script>$('#searchbox').show(0);</script> </div> </div> <div class="clearer"></div> </div> <div class="related" role="navigation" aria-label="related navigation"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" >index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="decimal.html" title="9.4. decimal — Decimal fixed point and floating point arithmetic" >next</a> |</li> <li class="right" > <a href="math.html" title="9.2. math — Mathematical functions" >previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="https://www.python.org/">Python</a> »</li> <li> <a href="../index.html">Python 2.7.18 documentation</a> » </li> <li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> »</li> <li class="nav-item nav-item-2"><a href="numeric.html" ><span class="section-number">9. </span>Numeric and Mathematical Modules</a> »</li> <li class="nav-item nav-item-this"><a href=""><span class="section-number">9.3. </span><code class="xref py py-mod docutils literal notranslate"><span class="pre">cmath</span></code> — Mathematical functions for complex numbers</a></li> </ul> </div> <div class="footer"> © <a href="../copyright.html">Copyright</a> 1990-2024, Python Software Foundation. <br /> The Python Software Foundation is a non-profit corporation. <a href="https://www.python.org/psf/donations/">Please donate.</a> <br /> Last updated on December 09, 2024. <a href="../bugs.html">Found a bug</a>? <br /> Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 4.3.2. </div> </body> </html>
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