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2to3.rst.txt	15898	0644	editdlrm
abc.rst.txt	11805	0644	editdlrm
aifc.rst.txt	7503	0644	editdlrm
allos.rst.txt	678	0644	editdlrm
archiving.rst.txt	440	0644	editdlrm
argparse.rst.txt	77480	0644	editdlrm
array.rst.txt	10812	0644	editdlrm
ast.rst.txt	10159	0644	editdlrm
asynchat.rst.txt	8528	0644	editdlrm
asyncio-dev.rst.txt	13398	0644	editdlrm
asyncio-eventloop.rst.txt	33701	0644	editdlrm
asyncio-eventloops.rst.txt	7660	0644	editdlrm
asyncio-protocol.rst.txt	25170	0644	editdlrm
asyncio-queue.rst.txt	4328	0644	editdlrm
asyncio-stream.rst.txt	14363	0644	editdlrm
asyncio-subprocess.rst.txt	15544	0644	editdlrm
asyncio-sync.rst.txt	9078	0644	editdlrm
asyncio-task.rst.txt	25406	0644	editdlrm
asyncio.rst.txt	2272	0644	editdlrm
asyncore.rst.txt	13624	0644	editdlrm
atexit.rst.txt	3669	0644	editdlrm
audioop.rst.txt	10712	0644	editdlrm
base64.rst.txt	10371	0644	editdlrm
bdb.rst.txt	12985	0644	editdlrm
binary.rst.txt	654	0644	editdlrm
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bisect.rst.txt	5398	0644	editdlrm
builtins.rst.txt	1465	0644	editdlrm
bz2.rst.txt	8825	0644	editdlrm
calendar.rst.txt	10988	0644	editdlrm
cgi.rst.txt	22650	0644	editdlrm
cgitb.rst.txt	2918	0644	editdlrm
chunk.rst.txt	5094	0644	editdlrm
cmath.rst.txt	9326	0644	editdlrm
cmd.rst.txt	13795	0644	editdlrm
code.rst.txt	7817	0644	editdlrm
codecs.rst.txt	75564	0644	editdlrm
codeop.rst.txt	3021	0644	editdlrm
collections.abc.rst.txt	12861	0644	editdlrm
collections.rst.txt	46063	0644	editdlrm
colorsys.rst.txt	1820	0644	editdlrm
compileall.rst.txt	8975	0644	editdlrm
concurrency.rst.txt	673	0644	editdlrm
concurrent.futures.rst.txt	16910	0644	editdlrm
concurrent.rst.txt	171	0644	editdlrm
configparser.rst.txt	48756	0644	editdlrm
constants.rst.txt	3331	0644	editdlrm
contextlib.rst.txt	28175	0644	editdlrm
copy.rst.txt	3378	0644	editdlrm
copyreg.rst.txt	2182	0644	editdlrm
crypt.rst.txt	5122	0644	editdlrm
crypto.rst.txt	411	0644	editdlrm
csv.rst.txt	19902	0644	editdlrm
ctypes.rst.txt	88613	0644	editdlrm
curses.ascii.rst.txt	8920	0644	editdlrm
curses.panel.rst.txt	2764	0644	editdlrm
curses.rst.txt	76397	0644	editdlrm
custominterp.rst.txt	569	0644	editdlrm
datatypes.rst.txt	751	0644	editdlrm
datetime.rst.txt	90524	0644	editdlrm
dbm.rst.txt	14293	0644	editdlrm
debug.rst.txt	471	0644	editdlrm
decimal.rst.txt	74869	0644	editdlrm
development.rst.txt	704	0644	editdlrm
difflib.rst.txt	29861	0644	editdlrm
dis.rst.txt	32567	0644	editdlrm
distribution.rst.txt	452	0644	editdlrm
distutils.rst.txt	1974	0644	editdlrm
doctest.rst.txt	71743	0644	editdlrm
dummy_threading.rst.txt	784	0644	editdlrm
email.charset.rst.txt	9293	0644	editdlrm
email.compat32-message.rst.txt	33493	0644	editdlrm
email.contentmanager.rst.txt	9112	0644	editdlrm
email.encoders.rst.txt	2715	0644	editdlrm
email.errors.rst.txt	4792	0644	editdlrm
email.examples.rst.txt	1915	0644	editdlrm
email.generator.rst.txt	13637	0644	editdlrm
email.header.rst.txt	9188	0644	editdlrm
email.headerregistry.rst.txt	17957	0644	editdlrm
email.iterators.rst.txt	2795	0644	editdlrm
email.message.rst.txt	33008	0644	editdlrm
email.mime.rst.txt	11721	0644	editdlrm
email.parser.rst.txt	14092	0644	editdlrm
email.policy.rst.txt	27022	0644	editdlrm
email.rst.txt	6785	0644	editdlrm
email.util.rst.txt	9214	0644	editdlrm
ensurepip.rst.txt	4987	0644	editdlrm
enum.rst.txt	32751	0644	editdlrm
errno.rst.txt	6811	0644	editdlrm
exceptions.rst.txt	25570	0644	editdlrm
faulthandler.rst.txt	6136	0644	editdlrm
fcntl.rst.txt	7152	0644	editdlrm
filecmp.rst.txt	5623	0644	editdlrm
fileformats.rst.txt	287	0644	editdlrm
fileinput.rst.txt	8051	0644	editdlrm
filesys.rst.txt	961	0644	editdlrm
fnmatch.rst.txt	3103	0644	editdlrm
formatter.rst.txt	13242	0644	editdlrm
fpectl.rst.txt	4180	0644	editdlrm
fractions.rst.txt	6364	0644	editdlrm
frameworks.rst.txt	391	0644	editdlrm
ftplib.rst.txt	17552	0644	editdlrm
functional.rst.txt	365	0644	editdlrm
functions.rst.txt	71621	0644	editdlrm
functools.rst.txt	18503	0644	editdlrm
gc.rst.txt	9694	0644	editdlrm
getopt.rst.txt	6557	0644	editdlrm
getpass.rst.txt	1882	0644	editdlrm
gettext.rst.txt	25588	0644	editdlrm
glob.rst.txt	3508	0644	editdlrm
grp.rst.txt	2417	0644	editdlrm
gzip.rst.txt	8301	0644	editdlrm
hashlib.rst.txt	26523	0644	editdlrm
heapq.rst.txt	13402	0644	editdlrm
hmac.rst.txt	4020	0644	editdlrm
html.entities.rst.txt	1319	0644	editdlrm
html.parser.rst.txt	11275	0644	editdlrm
html.rst.txt	1304	0644	editdlrm
http.client.rst.txt	18870	0644	editdlrm
http.cookiejar.rst.txt	27832	0644	editdlrm
http.cookies.rst.txt	8645	0644	editdlrm
http.rst.txt	7096	0644	editdlrm
http.server.rst.txt	17639	0644	editdlrm
i18n.rst.txt	408	0644	editdlrm
idle.rst.txt	25834	0644	editdlrm
imaplib.rst.txt	20367	0644	editdlrm
imghdr.rst.txt	2938	0644	editdlrm
imp.rst.txt	15548	0644	editdlrm
importlib.rst.txt	53563	0644	editdlrm
index.rst.txt	2258	0644	editdlrm
inspect.rst.txt	53928	0644	editdlrm
internet.rst.txt	992	0644	editdlrm
intro.rst.txt	2645	0644	editdlrm
io.rst.txt	40328	0644	editdlrm
ipaddress.rst.txt	32340	0644	editdlrm
ipc.rst.txt	649	0644	editdlrm
itertools.rst.txt	38459	0644	editdlrm
json.rst.txt	27876	0644	editdlrm
keyword.rst.txt	617	0644	editdlrm
language.rst.txt	522	0644	editdlrm
linecache.rst.txt	2408	0644	editdlrm
locale.rst.txt	25470	0644	editdlrm
logging.config.rst.txt	33886	0644	editdlrm
logging.handlers.rst.txt	42497	0644	editdlrm
logging.rst.txt	60479	0644	editdlrm
lzma.rst.txt	17361	0644	editdlrm
macpath.rst.txt	707	0644	editdlrm
mailbox.rst.txt	63055	0644	editdlrm
mailcap.rst.txt	3691	0644	editdlrm
markup.rst.txt	679	0644	editdlrm
marshal.rst.txt	4855	0644	editdlrm
math.rst.txt	14375	0644	editdlrm
mimetypes.rst.txt	9839	0644	editdlrm
misc.rst.txt	247	0644	editdlrm
mm.rst.txt	431	0644	editdlrm
mmap.rst.txt	11232	0644	editdlrm
modulefinder.rst.txt	3240	0644	editdlrm
modules.rst.txt	355	0644	editdlrm
msilib.rst.txt	18612	0644	editdlrm
msvcrt.rst.txt	4392	0644	editdlrm
multiprocessing.rst.txt	104494	0644	editdlrm
netdata.rst.txt	339	0644	editdlrm
netrc.rst.txt	3007	0644	editdlrm
nis.rst.txt	1993	0644	editdlrm
nntplib.rst.txt	21821	0644	editdlrm
numbers.rst.txt	8089	0644	editdlrm
numeric.rst.txt	696	0644	editdlrm
operator.rst.txt	18989	0644	editdlrm
optparse.rst.txt	76996	0644	editdlrm
os.path.rst.txt	16419	0644	editdlrm
os.rst.txt	137534	0644	editdlrm
ossaudiodev.rst.txt	17845	0644	editdlrm
othergui.rst.txt	2822	0644	editdlrm
parser.rst.txt	15148	0644	editdlrm
pathlib.rst.txt	30045	0644	editdlrm
pdb.rst.txt	19454	0644	editdlrm
persistence.rst.txt	591	0644	editdlrm
pickle.rst.txt	37440	0644	editdlrm
pickletools.rst.txt	3729	0644	editdlrm
pipes.rst.txt	2557	0644	editdlrm
pkgutil.rst.txt	8730	0644	editdlrm
platform.rst.txt	9624	0644	editdlrm
plistlib.rst.txt	7398	0644	editdlrm
poplib.rst.txt	8197	0644	editdlrm
posix.rst.txt	3694	0644	editdlrm
pprint.rst.txt	14274	0644	editdlrm
profile.rst.txt	27969	0644	editdlrm
pty.rst.txt	3123	0644	editdlrm
pwd.rst.txt	2739	0644	editdlrm
pyclbr.rst.txt	3297	0644	editdlrm
pydoc.rst.txt	4688	0644	editdlrm
pyexpat.rst.txt	28620	0644	editdlrm
python.rst.txt	475	0644	editdlrm
py_compile.rst.txt	3851	0644	editdlrm
queue.rst.txt	7268	0644	editdlrm
quopri.rst.txt	2571	0644	editdlrm
random.rst.txt	18503	0644	editdlrm
re.rst.txt	64936	0644	editdlrm
readline.rst.txt	11943	0644	editdlrm
reprlib.rst.txt	5162	0644	editdlrm
resource.rst.txt	12291	0644	editdlrm
rlcompleter.rst.txt	2294	0644	editdlrm
runpy.rst.txt	8285	0644	editdlrm
sched.rst.txt	4842	0644	editdlrm
secrets.rst.txt	5930	0644	editdlrm
select.rst.txt	28175	0644	editdlrm
selectors.rst.txt	8925	0644	editdlrm
shelve.rst.txt	8390	0644	editdlrm
shlex.rst.txt	16441	0644	editdlrm
shutil.rst.txt	26014	0644	editdlrm
signal.rst.txt	17218	0644	editdlrm
site.rst.txt	9628	0644	editdlrm
smtpd.rst.txt	10806	0644	editdlrm
smtplib.rst.txt	23439	0644	editdlrm
sndhdr.rst.txt	1994	0644	editdlrm
socket.rst.txt	68202	0644	editdlrm
socketserver.rst.txt	23712	0644	editdlrm
spwd.rst.txt	2958	0644	editdlrm
sqlite3.rst.txt	39215	0644	editdlrm
ssl.rst.txt	96021	0644	editdlrm
stat.rst.txt	9805	0644	editdlrm
statistics.rst.txt	15301	0644	editdlrm
stdtypes.rst.txt	179240	0644	editdlrm
string.rst.txt	35374	0644	editdlrm
stringprep.rst.txt	4279	0644	editdlrm
struct.rst.txt	19581	0644	editdlrm
subprocess.rst.txt	45793	0644	editdlrm
sunau.rst.txt	7390	0644	editdlrm
superseded.rst.txt	258	0644	editdlrm
symbol.rst.txt	975	0644	editdlrm
symtable.rst.txt	4964	0644	editdlrm
sys.rst.txt	58273	0644	editdlrm
sysconfig.rst.txt	8749	0644	editdlrm
syslog.rst.txt	4302	0644	editdlrm
tabnanny.rst.txt	2007	0644	editdlrm
tarfile.rst.txt	31749	0644	editdlrm
telnetlib.rst.txt	7899	0644	editdlrm
tempfile.rst.txt	13857	0644	editdlrm
termios.rst.txt	3751	0644	editdlrm
test.rst.txt	25764	0644	editdlrm
text.rst.txt	584	0644	editdlrm
textwrap.rst.txt	10519	0644	editdlrm
threading.rst.txt	39257	0644	editdlrm
time.rst.txt	32967	0644	editdlrm
timeit.rst.txt	13000	0644	editdlrm
tk.rst.txt	1639	0644	editdlrm
tkinter.rst.txt	33027	0644	editdlrm
tkinter.scrolledtext.rst.txt	1255	0644	editdlrm
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token.rst.txt	2607	0644	editdlrm
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trace.rst.txt	6914	0644	editdlrm
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tracemalloc.rst.txt	22089	0644	editdlrm
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unittest.mock-examples.rst.txt	46206	0644	editdlrm
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unittest.rst.txt	93304	0644	editdlrm
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urllib.error.rst.txt	2199	0644	editdlrm
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winsound.rst.txt	5134	0644	editdlrm
wsgiref.rst.txt	33047	0644	editdlrm
xdrlib.rst.txt	8078	0644	editdlrm
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xml.rst.txt	6071	0644	editdlrm
xml.sax.handler.rst.txt	15399	0644	editdlrm
xml.sax.reader.rst.txt	12133	0644	editdlrm
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xml.sax.utils.rst.txt	3901	0644	editdlrm
xmlrpc.client.rst.txt	23071	0644	editdlrm
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xmlrpc.server.rst.txt	14990	0644	editdlrm
zipapp.rst.txt	17392	0644	editdlrm
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zipimport.rst.txt	5805	0644	editdlrm
zlib.rst.txt	13912	0644	editdlrm
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__future__.rst.txt	5245	0644	editdlrm
__main__.rst.txt	904	0644	editdlrm

:mod:`math` --- Mathematical functions ====================================== .. module:: math :synopsis: Mathematical functions (sin() etc.). .. testsetup:: from math import fsum -------------- This module is always available. It provides access to the mathematical functions defined by the C standard. These functions cannot be used with complex numbers; use the functions of the same name from the :mod:`cmath` module if you require support for complex numbers. The distinction between functions which support complex numbers and those which don't is made since most users do not want to learn quite as much mathematics as required to understand complex numbers. Receiving an exception instead of a complex result allows earlier detection of the unexpected complex number used as a parameter, so that the programmer can determine how and why it was generated in the first place. The following functions are provided by this module. Except when explicitly noted otherwise, all return values are floats. Number-theoretic and representation functions --------------------------------------------- .. function:: ceil(x) Return the ceiling of *x*, the smallest integer greater than or equal to *x*. If *x* is not a float, delegates to ``x.__ceil__()``, which should return an :class:`~numbers.Integral` value. .. function:: copysign(x, y) Return a float with the magnitude (absolute value) of *x* but the sign of *y*. On platforms that support signed zeros, ``copysign(1.0, -0.0)`` returns *-1.0*. .. function:: fabs(x) Return the absolute value of *x*. .. function:: factorial(x) Return *x* factorial. Raises :exc:`ValueError` if *x* is not integral or is negative. .. function:: floor(x) Return the floor of *x*, the largest integer less than or equal to *x*. If *x* is not a float, delegates to ``x.__floor__()``, which should return an :class:`~numbers.Integral` value. .. function:: fmod(x, y) Return ``fmod(x, y)``, as defined by the platform C library. Note that the Python expression ``x % y`` may not return the same result. The intent of the C standard is that ``fmod(x, y)`` be exactly (mathematically; to infinite precision) equal to ``x - n*y`` for some integer *n* such that the result has the same sign as *x* and magnitude less than ``abs(y)``. Python's ``x % y`` returns a result with the sign of *y* instead, and may not be exactly computable for float arguments. For example, ``fmod(-1e-100, 1e100)`` is ``-1e-100``, but the result of Python's ``-1e-100 % 1e100`` is ``1e100-1e-100``, which cannot be represented exactly as a float, and rounds to the surprising ``1e100``. For this reason, function :func:`fmod` is generally preferred when working with floats, while Python's ``x % y`` is preferred when working with integers. .. function:: frexp(x) Return the mantissa and exponent of *x* as the pair ``(m, e)``. *m* is a float and *e* is an integer such that ``x == m * 2**e`` exactly. If *x* is zero, returns ``(0.0, 0)``, otherwise ``0.5 <= abs(m) < 1``. This is used to "pick apart" the internal representation of a float in a portable way. .. function:: fsum(iterable) Return an accurate floating point sum of values in the iterable. Avoids loss of precision by tracking multiple intermediate partial sums:: >>> sum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1]) 0.9999999999999999 >>> fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1]) 1.0 The algorithm's accuracy depends on IEEE-754 arithmetic guarantees and the typical case where the rounding mode is half-even. On some non-Windows builds, the underlying C library uses extended precision addition and may occasionally double-round an intermediate sum causing it to be off in its least significant bit. For further discussion and two alternative approaches, see the `ASPN cookbook recipes for accurate floating point summation `_\. .. function:: gcd(a, b) Return the greatest common divisor of the integers *a* and *b*. If either *a* or *b* is nonzero, then the value of ``gcd(a, b)`` is the largest positive integer that divides both *a* and *b*. ``gcd(0, 0)`` returns ``0``. .. versionadded:: 3.5 .. function:: isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0) Return ``True`` if the values *a* and *b* are close to each other and ``False`` otherwise. Whether or not two values are considered close is determined according to given absolute and relative tolerances. *rel_tol* is the relative tolerance -- it is the maximum allowed difference between *a* and *b*, relative to the larger absolute value of *a* or *b*. For example, to set a tolerance of 5%, pass ``rel_tol=0.05``. The default tolerance is ``1e-09``, which assures that the two values are the same within about 9 decimal digits. *rel_tol* must be greater than zero. *abs_tol* is the minimum absolute tolerance -- useful for comparisons near zero. *abs_tol* must be at least zero. If no errors occur, the result will be: ``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)``. The IEEE 754 special values of ``NaN``, ``inf``, and ``-inf`` will be handled according to IEEE rules. Specifically, ``NaN`` is not considered close to any other value, including ``NaN``. ``inf`` and ``-inf`` are only considered close to themselves. .. versionadded:: 3.5 .. seealso:: :pep:`485` -- A function for testing approximate equality .. function:: isfinite(x) Return ``True`` if *x* is neither an infinity nor a NaN, and ``False`` otherwise. (Note that ``0.0`` *is* considered finite.) .. versionadded:: 3.2 .. function:: isinf(x) Return ``True`` if *x* is a positive or negative infinity, and ``False`` otherwise. .. function:: isnan(x) Return ``True`` if *x* is a NaN (not a number), and ``False`` otherwise. .. function:: ldexp(x, i) Return ``x * (2**i)``. This is essentially the inverse of function :func:`frexp`. .. function:: modf(x) Return the fractional and integer parts of *x*. Both results carry the sign of *x* and are floats. .. function:: trunc(x) Return the :class:`~numbers.Real` value *x* truncated to an :class:`~numbers.Integral` (usually an integer). Delegates to :meth:`x.__trunc__() `. Note that :func:`frexp` and :func:`modf` have a different call/return pattern than their C equivalents: they take a single argument and return a pair of values, rather than returning their second return value through an 'output parameter' (there is no such thing in Python). For the :func:`ceil`, :func:`floor`, and :func:`modf` functions, note that *all* floating-point numbers of sufficiently large magnitude are exact integers. Python floats typically carry no more than 53 bits of precision (the same as the platform C double type), in which case any float *x* with ``abs(x) >= 2**52`` necessarily has no fractional bits. Power and logarithmic functions ------------------------------- .. function:: exp(x) Return ``e**x``. .. function:: expm1(x) Return ``e**x - 1``. For small floats *x*, the subtraction in ``exp(x) - 1`` can result in a `significant loss of precision `_\; the :func:`expm1` function provides a way to compute this quantity to full precision:: >>> from math import exp, expm1 >>> exp(1e-5) - 1 # gives result accurate to 11 places 1.0000050000069649e-05 >>> expm1(1e-5) # result accurate to full precision 1.0000050000166668e-05 .. versionadded:: 3.2 .. function:: log(x[, base]) With one argument, return the natural logarithm of *x* (to base *e*). With two arguments, return the logarithm of *x* to the given *base*, calculated as ``log(x)/log(base)``. .. function:: log1p(x) Return the natural logarithm of *1+x* (base *e*). The result is calculated in a way which is accurate for *x* near zero. .. function:: log2(x) Return the base-2 logarithm of *x*. This is usually more accurate than ``log(x, 2)``. .. versionadded:: 3.3 .. seealso:: :meth:`int.bit_length` returns the number of bits necessary to represent an integer in binary, excluding the sign and leading zeros. .. function:: log10(x) Return the base-10 logarithm of *x*. This is usually more accurate than ``log(x, 10)``. .. function:: pow(x, y) Return ``x`` raised to the power ``y``. Exceptional cases follow Annex 'F' of the C99 standard as far as possible. In particular, ``pow(1.0, x)`` and ``pow(x, 0.0)`` always return ``1.0``, even when ``x`` is a zero or a NaN. If both ``x`` and ``y`` are finite, ``x`` is negative, and ``y`` is not an integer then ``pow(x, y)`` is undefined, and raises :exc:`ValueError`. Unlike the built-in ``**`` operator, :func:`math.pow` converts both its arguments to type :class:`float`. Use ``**`` or the built-in :func:`pow` function for computing exact integer powers. .. function:: sqrt(x) Return the square root of *x*. Trigonometric functions ----------------------- .. function:: acos(x) Return the arc cosine of *x*, in radians. .. function:: asin(x) Return the arc sine of *x*, in radians. .. function:: atan(x) Return the arc tangent of *x*, in radians. .. function:: atan2(y, x) Return ``atan(y / x)``, in radians. The result is between ``-pi`` and ``pi``. The vector in the plane from the origin to point ``(x, y)`` makes this angle with the positive X axis. The point of :func:`atan2` is that the signs of both inputs are known to it, so it can compute the correct quadrant for the angle. For example, ``atan(1)`` and ``atan2(1, 1)`` are both ``pi/4``, but ``atan2(-1, -1)`` is ``-3*pi/4``. .. function:: cos(x) Return the cosine of *x* radians. .. function:: hypot(x, y) Return the Euclidean norm, ``sqrt(x*x + y*y)``. This is the length of the vector from the origin to point ``(x, y)``. .. function:: sin(x) Return the sine of *x* radians. .. function:: tan(x) Return the tangent of *x* radians. Angular conversion ------------------ .. function:: degrees(x) Convert angle *x* from radians to degrees. .. function:: radians(x) Convert angle *x* from degrees to radians. Hyperbolic functions -------------------- `Hyperbolic functions `_ are analogs of trigonometric functions that are based on hyperbolas instead of circles. .. function:: acosh(x) Return the inverse hyperbolic cosine of *x*. .. function:: asinh(x) Return the inverse hyperbolic sine of *x*. .. function:: atanh(x) Return the inverse hyperbolic tangent of *x*. .. function:: cosh(x) Return the hyperbolic cosine of *x*. .. function:: sinh(x) Return the hyperbolic sine of *x*. .. function:: tanh(x) Return the hyperbolic tangent of *x*. Special functions ----------------- .. function:: erf(x) Return the `error function `_ at *x*. The :func:`erf` function can be used to compute traditional statistical functions such as the `cumulative standard normal distribution `_:: def phi(x): 'Cumulative distribution function for the standard normal distribution' return (1.0 + erf(x / sqrt(2.0))) / 2.0 .. versionadded:: 3.2 .. function:: erfc(x) Return the complementary error function at *x*. The `complementary error function `_ is defined as ``1.0 - erf(x)``. It is used for large values of *x* where a subtraction from one would cause a `loss of significance `_\. .. versionadded:: 3.2 .. function:: gamma(x) Return the `Gamma function `_ at *x*. .. versionadded:: 3.2 .. function:: lgamma(x) Return the natural logarithm of the absolute value of the Gamma function at *x*. .. versionadded:: 3.2 Constants --------- .. data:: pi The mathematical constant π = 3.141592..., to available precision. .. data:: e The mathematical constant e = 2.718281..., to available precision. .. data:: tau The mathematical constant τ = 6.283185..., to available precision. Tau is a circle constant equal to 2π, the ratio of a circle's circumference to its radius. To learn more about Tau, check out Vi Hart's video `Pi is (still) Wrong `_, and start celebrating `Tau day `_ by eating twice as much pie! .. versionadded:: 3.6 .. data:: inf A floating-point positive infinity. (For negative infinity, use ``-math.inf``.) Equivalent to the output of ``float('inf')``. .. versionadded:: 3.5 .. data:: nan A floating-point "not a number" (NaN) value. Equivalent to the output of ``float('nan')``. .. versionadded:: 3.5 .. impl-detail:: The :mod:`math` module consists mostly of thin wrappers around the platform C math library functions. Behavior in exceptional cases follows Annex F of the C99 standard where appropriate. The current implementation will raise :exc:`ValueError` for invalid operations like ``sqrt(-1.0)`` or ``log(0.0)`` (where C99 Annex F recommends signaling invalid operation or divide-by-zero), and :exc:`OverflowError` for results that overflow (for example, ``exp(1000.0)``). A NaN will not be returned from any of the functions above unless one or more of the input arguments was a NaN; in that case, most functions will return a NaN, but (again following C99 Annex F) there are some exceptions to this rule, for example ``pow(float('nan'), 0.0)`` or ``hypot(float('nan'), float('inf'))``. Note that Python makes no effort to distinguish signaling NaNs from quiet NaNs, and behavior for signaling NaNs remains unspecified. Typical behavior is to treat all NaNs as though they were quiet. .. seealso:: Module :mod:`cmath` Complex number versions of many of these functions.