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# Copyright (C) 2001-2019, Python Software Foundation
# This file is distributed under the same license as the Python package.
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"Project-Id-Version: Python 3.7\n"
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"POT-Creation-Date: 2019-05-06 11:59-0400\n"
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#: ../Doc/library/cmath.rst:2
msgid ":mod:`cmath` --- Mathematical functions for complex numbers"
#: ../Doc/library/cmath.rst:9
"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."
#: ../Doc/library/cmath.rst:19
"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."
#: ../Doc/library/cmath.rst:27
msgid "Conversions to and from polar coordinates"
#: ../Doc/library/cmath.rst:29
"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::"
#: ../Doc/library/cmath.rst:36
"*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 "
#: ../Doc/library/cmath.rst:43
"The following functions can be used to convert from the native rectangular "
"coordinates to polar coordinates and back."
#: ../Doc/library/cmath.rst:48
"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::"
#: ../Doc/library/cmath.rst:65
"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."
#: ../Doc/library/cmath.rst:72
"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))``."
#: ../Doc/library/cmath.rst:80
"Return the complex number *x* with polar coordinates *r* and *phi*. "
"Equivalent to ``r * (math.cos(phi) + math.sin(phi)*1j)``."
#: ../Doc/library/cmath.rst:85
msgid "Power and logarithmic functions"
#: ../Doc/library/cmath.rst:89
"Return *e* raised to the power *x*, where *e* is the base of natural "
"logarithms."
#: ../Doc/library/cmath.rst:95
"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."
#: ../Doc/library/cmath.rst:102
"Return the base-10 logarithm of *x*. This has the same branch cut as :func:"
#: ../Doc/library/cmath.rst:108
"Return the square root of *x*. This has the same branch cut as :func:`log`."
#: ../Doc/library/cmath.rst:112
msgid "Trigonometric functions"
#: ../Doc/library/cmath.rst:116
"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."
#: ../Doc/library/cmath.rst:123
"Return the arc sine of *x*. This has the same branch cuts as :func:`acos`."
#: ../Doc/library/cmath.rst:128
"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."
#: ../Doc/library/cmath.rst:136
msgid "Return the cosine of *x*."
#: ../Doc/library/cmath.rst:141
msgid "Return the sine of *x*."
#: ../Doc/library/cmath.rst:146
msgid "Return the tangent of *x*."
#: ../Doc/library/cmath.rst:150
msgid "Hyperbolic functions"
#: ../Doc/library/cmath.rst:154
"Return the inverse hyperbolic cosine of *x*. There is one branch cut, "
"extending left from 1 along the real axis to -∞, continuous from above."
#: ../Doc/library/cmath.rst:160
"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."
#: ../Doc/library/cmath.rst:168
"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 "
#: ../Doc/library/cmath.rst:176
msgid "Return the hyperbolic cosine of *x*."
#: ../Doc/library/cmath.rst:181
msgid "Return the hyperbolic sine of *x*."
#: ../Doc/library/cmath.rst:186
msgid "Return the hyperbolic tangent of *x*."
#: ../Doc/library/cmath.rst:190
msgid "Classification functions"
#: ../Doc/library/cmath.rst:194
"Return ``True`` if both the real and imaginary parts of *x* are finite, and "
"``False`` otherwise."
#: ../Doc/library/cmath.rst:202
"Return ``True`` if either the real or the imaginary part of *x* is an "
"infinity, and ``False`` otherwise."
#: ../Doc/library/cmath.rst:208
"Return ``True`` if either the real or the imaginary part of *x* is a NaN, "
"and ``False`` otherwise."
#: ../Doc/library/cmath.rst:214
"Return ``True`` if the values *a* and *b* are close to each other and "
"``False`` otherwise."
#: ../Doc/library/cmath.rst:217
"Whether or not two values are considered close is determined according to "
"given absolute and relative tolerances."
#: ../Doc/library/cmath.rst:220
"*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."
#: ../Doc/library/cmath.rst:226
"*abs_tol* is the minimum absolute tolerance -- useful for comparisons near "
"zero. *abs_tol* must be at least zero."
#: ../Doc/library/cmath.rst:229
"If no errors occur, the result will be: ``abs(a-b) <= max(rel_tol * "
"max(abs(a), abs(b)), abs_tol)``."
#: ../Doc/library/cmath.rst:232
"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."
#: ../Doc/library/cmath.rst:241
msgid ":pep:`485` -- A function for testing approximate equality"
#: ../Doc/library/cmath.rst:245
msgid "Constants"
#: ../Doc/library/cmath.rst:249
msgid "The mathematical constant *π*, as a float."
#: ../Doc/library/cmath.rst:254
msgid "The mathematical constant *e*, as a float."
#: ../Doc/library/cmath.rst:259
msgid "The mathematical constant *τ*, as a float."
#: ../Doc/library/cmath.rst:266
msgid "Floating-point positive infinity. Equivalent to ``float('inf')``."
#: ../Doc/library/cmath.rst:273
"Complex number with zero real part and positive infinity imaginary part. "
"Equivalent to ``complex(0.0, float('inf'))``."
#: ../Doc/library/cmath.rst:281
"A floating-point \"not a number\" (NaN) value.  Equivalent to "
"``float('nan')``."
#: ../Doc/library/cmath.rst:289
"Complex number with zero real part and NaN imaginary part. Equivalent to "
"``complex(0.0, float('nan'))``."
#: ../Doc/library/cmath.rst:297
"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 "
#: ../Doc/library/cmath.rst:305
"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:"
#: ../Doc/library/cmath.rst:315
"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."
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