# SOME DESCRIPTIVE TITLE. # Copyright (C) 2001-2019, Python Software Foundation # This file is distributed under the same license as the Python package. # FIRST AUTHOR , YEAR. # #, fuzzy msgid "" msgstr "" "Project-Id-Version: Python 3.7\n" "Report-Msgid-Bugs-To: \n" "POT-Creation-Date: 2019-05-06 11:59-0400\n" "PO-Revision-Date: YEAR-MO-DA HO:MI+ZONE\n" "Last-Translator: FULL NAME \n" "Language-Team: LANGUAGE \n" "MIME-Version: 1.0\n" "Content-Type: text/plain; charset=UTF-8\n" "Content-Transfer-Encoding: 8bit\n" #: ../Doc/library/cmath.rst:2 msgid ":mod:`cmath` --- Mathematical functions for complex numbers" msgstr "" #: ../Doc/library/cmath.rst:9 msgid "" "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 :meth:`__complex__` or a :" "meth:`__float__` 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." msgstr "" #: ../Doc/library/cmath.rst:19 msgid "" "On platforms with hardware and system-level support for signed zeros, " "functions involving branch cuts are continuous on *both* 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." msgstr "" #: ../Doc/library/cmath.rst:27 msgid "Conversions to and from polar coordinates" msgstr "" #: ../Doc/library/cmath.rst:29 msgid "" "A Python complex number ``z`` is stored internally using *rectangular* or " "*Cartesian* coordinates. It is completely determined by its *real part* ``z." "real`` and its *imaginary part* ``z.imag``. In other words::" msgstr "" #: ../Doc/library/cmath.rst:36 msgid "" "*Polar coordinates* give an alternative way to represent a complex number. " "In polar coordinates, a complex number *z* is defined by the modulus *r* and " "the phase angle *phi*. The modulus *r* is the distance from *z* to the " "origin, while the phase *phi* is the counterclockwise angle, measured in " "radians, from the positive x-axis to the line segment that joins the origin " "to *z*." msgstr "" #: ../Doc/library/cmath.rst:43 msgid "" "The following functions can be used to convert from the native rectangular " "coordinates to polar coordinates and back." msgstr "" #: ../Doc/library/cmath.rst:48 msgid "" "Return the phase of *x* (also known as the *argument* of *x*), as a float. " "``phase(x)`` is equivalent to ``math.atan2(x.imag, x.real)``. 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 ``x.imag``, " "even when ``x.imag`` is zero::" msgstr "" #: ../Doc/library/cmath.rst:65 msgid "" "The modulus (absolute value) of a complex number *x* can be computed using " "the built-in :func:`abs` function. There is no separate :mod:`cmath` module " "function for this operation." msgstr "" #: ../Doc/library/cmath.rst:72 msgid "" "Return the representation of *x* in polar coordinates. Returns a pair ``(r, " "phi)`` where *r* is the modulus of *x* and phi is the phase of *x*. " "``polar(x)`` is equivalent to ``(abs(x), phase(x))``." msgstr "" #: ../Doc/library/cmath.rst:80 msgid "" "Return the complex number *x* with polar coordinates *r* and *phi*. " "Equivalent to ``r * (math.cos(phi) + math.sin(phi)*1j)``." msgstr "" #: ../Doc/library/cmath.rst:85 msgid "Power and logarithmic functions" msgstr "" #: ../Doc/library/cmath.rst:89 msgid "" "Return *e* raised to the power *x*, where *e* is the base of natural " "logarithms." msgstr "" #: ../Doc/library/cmath.rst:95 msgid "" "Returns the logarithm of *x* to the given *base*. If the *base* is not " "specified, returns the natural logarithm of *x*. There is one branch cut, " "from 0 along the negative real axis to -∞, continuous from above." msgstr "" #: ../Doc/library/cmath.rst:102 msgid "" "Return the base-10 logarithm of *x*. This has the same branch cut as :func:" "`log`." msgstr "" #: ../Doc/library/cmath.rst:108 msgid "" "Return the square root of *x*. This has the same branch cut as :func:`log`." msgstr "" #: ../Doc/library/cmath.rst:112 msgid "Trigonometric functions" msgstr "" #: ../Doc/library/cmath.rst:116 msgid "" "Return the arc cosine of *x*. 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." msgstr "" #: ../Doc/library/cmath.rst:123 msgid "" "Return the arc sine of *x*. This has the same branch cuts as :func:`acos`." msgstr "" #: ../Doc/library/cmath.rst:128 msgid "" "Return the arc tangent of *x*. There are two branch cuts: One extends from " "``1j`` along the imaginary axis to ``∞j``, continuous from the right. The " "other extends from ``-1j`` along the imaginary axis to ``-∞j``, continuous " "from the left." msgstr "" #: ../Doc/library/cmath.rst:136 msgid "Return the cosine of *x*." msgstr "" #: ../Doc/library/cmath.rst:141 msgid "Return the sine of *x*." msgstr "" #: ../Doc/library/cmath.rst:146 msgid "Return the tangent of *x*." msgstr "" #: ../Doc/library/cmath.rst:150 msgid "Hyperbolic functions" msgstr "" #: ../Doc/library/cmath.rst:154 msgid "" "Return the inverse hyperbolic cosine of *x*. There is one branch cut, " "extending left from 1 along the real axis to -∞, continuous from above." msgstr "" #: ../Doc/library/cmath.rst:160 msgid "" "Return the inverse hyperbolic sine of *x*. There are two branch cuts: One " "extends from ``1j`` along the imaginary axis to ``∞j``, continuous from the " "right. The other extends from ``-1j`` along the imaginary axis to ``-∞j``, " "continuous from the left." msgstr "" #: ../Doc/library/cmath.rst:168 msgid "" "Return the inverse hyperbolic tangent of *x*. There are two branch cuts: One " "extends from ``1`` along the real axis to ``∞``, continuous from below. The " "other extends from ``-1`` along the real axis to ``-∞``, continuous from " "above." msgstr "" #: ../Doc/library/cmath.rst:176 msgid "Return the hyperbolic cosine of *x*." msgstr "" #: ../Doc/library/cmath.rst:181 msgid "Return the hyperbolic sine of *x*." msgstr "" #: ../Doc/library/cmath.rst:186 msgid "Return the hyperbolic tangent of *x*." msgstr "" #: ../Doc/library/cmath.rst:190 msgid "Classification functions" msgstr "" #: ../Doc/library/cmath.rst:194 msgid "" "Return ``True`` if both the real and imaginary parts of *x* are finite, and " "``False`` otherwise." msgstr "" #: ../Doc/library/cmath.rst:202 msgid "" "Return ``True`` if either the real or the imaginary part of *x* is an " "infinity, and ``False`` otherwise." msgstr "" #: ../Doc/library/cmath.rst:208 msgid "" "Return ``True`` if either the real or the imaginary part of *x* is a NaN, " "and ``False`` otherwise." msgstr "" #: ../Doc/library/cmath.rst:214 msgid "" "Return ``True`` if the values *a* and *b* are close to each other and " "``False`` otherwise." msgstr "" #: ../Doc/library/cmath.rst:217 msgid "" "Whether or not two values are considered close is determined according to " "given absolute and relative tolerances." msgstr "" #: ../Doc/library/cmath.rst:220 msgid "" "*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." msgstr "" #: ../Doc/library/cmath.rst:226 msgid "" "*abs_tol* is the minimum absolute tolerance -- useful for comparisons near " "zero. *abs_tol* must be at least zero." msgstr "" #: ../Doc/library/cmath.rst:229 msgid "" "If no errors occur, the result will be: ``abs(a-b) <= max(rel_tol * " "max(abs(a), abs(b)), abs_tol)``." msgstr "" #: ../Doc/library/cmath.rst:232 msgid "" "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." msgstr "" #: ../Doc/library/cmath.rst:241 msgid ":pep:`485` -- A function for testing approximate equality" msgstr "" #: ../Doc/library/cmath.rst:245 msgid "Constants" msgstr "" #: ../Doc/library/cmath.rst:249 msgid "The mathematical constant *π*, as a float." msgstr "" #: ../Doc/library/cmath.rst:254 msgid "The mathematical constant *e*, as a float." msgstr "" #: ../Doc/library/cmath.rst:259 msgid "The mathematical constant *τ*, as a float." msgstr "" #: ../Doc/library/cmath.rst:266 msgid "Floating-point positive infinity. Equivalent to ``float('inf')``." msgstr "" #: ../Doc/library/cmath.rst:273 msgid "" "Complex number with zero real part and positive infinity imaginary part. " "Equivalent to ``complex(0.0, float('inf'))``." msgstr "" #: ../Doc/library/cmath.rst:281 msgid "" "A floating-point \"not a number\" (NaN) value. Equivalent to " "``float('nan')``." msgstr "" #: ../Doc/library/cmath.rst:289 msgid "" "Complex number with zero real part and NaN imaginary part. Equivalent to " "``complex(0.0, float('nan'))``." msgstr "" #: ../Doc/library/cmath.rst:297 msgid "" "Note that the selection of functions is similar, but not identical, to that " "in module :mod:`math`. 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 ``math.sqrt(-1)`` raise an exception than " "return a complex number. Also note that the functions defined in :mod:" "`cmath` 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)." msgstr "" #: ../Doc/library/cmath.rst:305 msgid "" "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:" msgstr "" #: ../Doc/library/cmath.rst:315 msgid "" "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." msgstr ""