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msgstr ""

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#: ../../library/cmath.rst:2

msgid ":mod:`cmath` --- Mathematical functions for complex numbers"

msgstr ":mod:`cmath` --- 关于复数的数学函数"

#: ../../library/cmath.rst:9

msgid ""

"This module 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 ""

"这一模块提供了一些关于复数的数学函数。 该模块的函数的参数为整数、浮点数或复数。 这些函数的参数也可为一个拥有 :meth:`__complex__` "

"或 :meth:`__float__` 方法的 Python 对象，这些方法分别用于将对象转换为复数和浮点数，这些函数作用于转换后的结果。"

#: ../../library/cmath.rst:18

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 ""

"在具有对于有符号零的硬件和系统级支持的平台上，涉及支割线的函数在支割线的 *两侧* 都是连续的：零的符号可用来区别支割线的一侧和另一侧。 "

"在不支持有符号零的平台上，连续性的规则见下文。"

#: ../../library/cmath.rst:26

msgid "Conversions to and from polar coordinates"

msgstr "到极坐标和从极坐标的转换"

#: ../../library/cmath.rst:28

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 ""

"使用 *矩形坐标* 或 *笛卡尔坐标* 在内部存储 Python 复数 ``z``。 这完全取决于它的 *实部* ``z.real`` 和 *虚部* "

"``z.imag``。 换句话说::"

#: ../../library/cmath.rst:35

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 ""

"*极坐标* 提供了另一种复数的表示方法。在极坐标中，一个复数 *z* 由模量 *r* 和相位角 *phi* 来定义。模量 *r* 是从 *z* "

"到坐标原点的距离，而相位角 *phi* 是以弧度为单位的，逆时针的，从正X轴到连接原点和 *z* 的线段间夹角的角度。"

#: ../../library/cmath.rst:42

msgid ""

"The following functions can be used to convert from the native rectangular "

"coordinates to polar coordinates and back."

msgstr "下面的函数可用于原生直角坐标与极坐标的相互转换。"

#: ../../library/cmath.rst:47

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 ""

"将 *x* 的相位 (也称为 *x* 的 *参数*) 返回为一个浮点数。``phase(x)`` 相当于 ``math.atan2(x.imag, "

"x.real)``。 结果处于 [-\\ *π*, *π*] 之间，以及这个操作的分支切断处于负实轴上，从上方连续。 "

"在支持有符号零的系统上（这包涵大多数当前的常用系统），这意味着结果的符号与 ``x.imag`` 的符号相同，即使 ``x.imag`` 的值是 0::"

#: ../../library/cmath.rst:64

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 ""

"一个复数 *x* 的模数（绝对值）可以通过内置函数 :func:`abs` 计算。没有单独的 :mod:`cmath` 模块函数用于这个操作。"

#: ../../library/cmath.rst:71

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 ""

"在极坐标中返回 *x* 的表达方式。返回一个数对 ``(r, phi)``，*r* 是 *x* 的模数，*phi* 是 *x* 的相位角。 "

"``polar(x)`` 相当于 ``(abs(x), phase(x))``。"

#: ../../library/cmath.rst:79

msgid ""

"Return the complex number *x* with polar coordinates *r* and *phi*. "

"Equivalent to ``r * (math.cos(phi) + math.sin(phi)*1j)``."

msgstr ""

"通过极坐标的 *r* 和 *phi* 返回复数 *x*。相当于 ``r * (math.cos(phi) + math.sin(phi)*1j)``。"

#: ../../library/cmath.rst:84

msgid "Power and logarithmic functions"

msgstr "幂函数与对数函数"

#: ../../library/cmath.rst:88

msgid ""

"Return *e* raised to the power *x*, where *e* is the base of natural "

"logarithms."

msgstr "返回 *e* 的 *x* 次方，*e* 是自然对数的底数。"

#: ../../library/cmath.rst:94

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 ""

"返回给定 *base* 的 *x* 的对数。如果没有给定 *base*，返回 *x* 的自然对数。 从 0 到 -∞ 存在一条支割线，沿负实轴之上连续。"

#: ../../library/cmath.rst:101

msgid ""

"Return the base-10 logarithm of *x*. This has the same branch cut as "

":func:`log`."

msgstr "返回底数为 10 的 *x* 的对数。它具有与 :func:`log` 相同的支割线。"

#: ../../library/cmath.rst:107

msgid ""

"Return the square root of *x*. This has the same branch cut as :func:`log`."

msgstr "返回 *x* 的平方根。 它具有与 :func:`log` 相同的支割线。"

#: ../../library/cmath.rst:111

msgid "Trigonometric functions"

msgstr "三角函数"

#: ../../library/cmath.rst:115

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 ""

"返回 *x* 的反余弦。这里有两条支割线：一条沿着实轴从 1 向右延伸到 ∞，从下面连续延伸。另外一条沿着实轴从 -1 向左延伸到 "

"-∞，从上面连续延伸。"

#: ../../library/cmath.rst:122

msgid ""

"Return the arc sine of *x*. This has the same branch cuts as :func:`acos`."

msgstr "返回 *x* 的反正弦。它与 :func:`acos` 有相同的支割线。"

#: ../../library/cmath.rst:127

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 ""

"返回 *x* 的反正切。它具有两条支割线：一条沿着虚轴从 ``1j`` 延伸到 ``∞j``，向右持续延伸。另一条是沿着虚轴从 ``-1j`` 延伸到"

" ``-∞j`` ，向左持续延伸。"

#: ../../library/cmath.rst:135

msgid "Return the cosine of *x*."

msgstr "返回 *x* 的余弦。"

#: ../../library/cmath.rst:140

msgid "Return the sine of *x*."

msgstr "返回 *x* 的正弦。"

#: ../../library/cmath.rst:145

msgid "Return the tangent of *x*."

msgstr "返回 *x* 的正切。"

#: ../../library/cmath.rst:149

msgid "Hyperbolic functions"

msgstr "双曲函数"

#: ../../library/cmath.rst:153

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 "返回 *x* 的反双曲余弦。它有一条支割线沿着实轴从 1 到 -∞ 向左延伸，从上方持续延伸。"

#: ../../library/cmath.rst:159

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 ""

"返回 *x* 的反双曲正弦。它有两条支割线：一条沿着虚轴从 ``1j`` 向右持续延伸到 ``∞j``。另一条是沿着虚轴从 ``-1j`` "

"向左持续延伸到 ``-∞j``。"

#: ../../library/cmath.rst:167

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 ""

"返回 *x* 的反双曲正切。它有两条支割线：一条是沿着实轴从 ``1`` 延展到 ``∞``，从下面持续延展。另一条是沿着实轴从 ``-1`` 延展到 "

"``-∞``，从上面持续延展。"

#: ../../library/cmath.rst:175

msgid "Return the hyperbolic cosine of *x*."

msgstr "返回 *x* 的双曲余弦值。"

#: ../../library/cmath.rst:180

msgid "Return the hyperbolic sine of *x*."

msgstr "返回 *x* 的双曲正弦值。"

#: ../../library/cmath.rst:185

msgid "Return the hyperbolic tangent of *x*."

msgstr "返回 *x* 的双曲正切值。"

#: ../../library/cmath.rst:189

msgid "Classification functions"

msgstr "分类函数"

#: ../../library/cmath.rst:193

msgid ""

"Return ``True`` if both the real and imaginary parts of *x* are finite, and "

"``False`` otherwise."

msgstr "如果 *x* 的实部和虚部都是有限的，则返回 ``True``，否则返回 ``False``。"

#: ../../library/cmath.rst:201

msgid ""

"Return ``True`` if either the real or the imaginary part of *x* is an "

"infinity, and ``False`` otherwise."

msgstr "如果 *x* 的实部或者虚部是无穷大的，则返回 ``True``，否则返回 ``False``。"

#: ../../library/cmath.rst:207

msgid ""

"Return ``True`` if either the real or the imaginary part of *x* is a NaN, "

"and ``False`` otherwise."

msgstr "如果 *x* 的实部或者虚部是 NaN，则返回 ``True`` ，否则返回 ``False``。"

#: ../../library/cmath.rst:213

msgid ""

"Return ``True`` if the values *a* and *b* are close to each other and "

"``False`` otherwise."

msgstr "若 *a* 和 *b* 的值比较接近则返回 ``True``，否则返回 ``False``。"

#: ../../library/cmath.rst:216

msgid ""

"Whether or not two values are considered close is determined according to "

"given absolute and relative tolerances."

msgstr "根据给定的绝对和相对容差确定两个值是否被认为是接近的。"

#: ../../library/cmath.rst:219

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 ""

"*rel_tol* 是相对容差 —— 它是 *a* 和 *b* 之间允许的最大差值，相对于 *a* 或 *b* "

"的较大绝对值。例如，要设置5％的容差，请传递 ``rel_tol=0.05`` 。默认容差为 ``1e-09``，确保两个值在大约9位十进制数字内相同。"

" *rel_tol* 必须大于零。"

#: ../../library/cmath.rst:225

msgid ""

"*abs_tol* is the minimum absolute tolerance -- useful for comparisons near "

"zero. *abs_tol* must be at least zero."

msgstr "*abs_tol* 是最小绝对容差 —— 对于接近零的比较很有用。 *abs_tol* 必须至少为零。"

#: ../../library/cmath.rst:228

msgid ""

"If no errors occur, the result will be: ``abs(a-b) <= max(rel_tol * "

"max(abs(a), abs(b)), abs_tol)``."

msgstr ""

"如果没有错误发生，结果将是： ``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)`` 。"

#: ../../library/cmath.rst:231

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 ""

"IEEE 754特殊值 ``NaN`` ， ``inf`` 和 ``-inf`` 将根据IEEE规则处理。具体来说， ``NaN`` "

"不被认为接近任何其他值，包括 ``NaN`` 。 ``inf`` 和 ``-inf`` 只被认为接近自己。"

#: ../../library/cmath.rst:240

msgid ":pep:`485` -- A function for testing approximate equality"

msgstr ":pep:`485` —— 用于测试近似相等的函数"

#: ../../library/cmath.rst:244

msgid "Constants"

msgstr "常量"

#: ../../library/cmath.rst:248

msgid "The mathematical constant *π*, as a float."

msgstr "数学常数 *π* ，作为一个浮点数。"

#: ../../library/cmath.rst:253

msgid "The mathematical constant *e*, as a float."

msgstr "数学常数 *e* ，作为一个浮点数。"

#: ../../library/cmath.rst:258

msgid "The mathematical constant *τ*, as a float."

msgstr "数学常数 *τ* ，作为一个浮点数。"

#: ../../library/cmath.rst:265

msgid "Floating-point positive infinity. Equivalent to ``float('inf')``."

msgstr "浮点正无穷大。相当于 ``float('inf')``。"

#: ../../library/cmath.rst:272

msgid ""

"Complex number with zero real part and positive infinity imaginary part. "

"Equivalent to ``complex(0.0, float('inf'))``."

msgstr "具有零实部和正无穷虚部的复数。相当于 ``complex(0.0, float('inf'))``。"

#: ../../library/cmath.rst:280

msgid ""

"A floating-point \"not a number\" (NaN) value. Equivalent to "

"``float('nan')``."

msgstr "浮点“非数字”（NaN）值。相当于 ``float('nan')``。"

#: ../../library/cmath.rst:288

msgid ""

"Complex number with zero real part and NaN imaginary part. Equivalent to "

"``complex(0.0, float('nan'))``."

msgstr "具有零实部和 NaN 虚部的复数。相当于 ``complex(0.0, float('nan'))``。"

#: ../../library/cmath.rst:296

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 ""

"请注意，函数的选择与模块 :mod:`math` 中的函数选择相似，但不完全相同。 "

"拥有两个模块的原因是因为有些用户对复数不感兴趣，甚至根本不知道它们是什么。它们宁愿 ``math.sqrt(-1)`` 引发异常，也不想返回一个复数。 "

"另请注意，被 :mod:`cmath` 定义的函数始终会返回一个复数，尽管答案可以表示为一个实数（在这种情况下，复数的虚数部分为零）。"

#: ../../library/cmath.rst:304

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 ""

"关于支割线的注释：它们是沿着给定函数无法连续的曲线。它们是许多复变函数的必要特征。 假设您需要使用复变函数进行计算，您将会了解支割线的概念。 "

"请参阅几乎所有关于复变函数的（不太基本）的书来获得启发。 对于如何正确地基于数值目的来选择支割线的相关信息，一个良好的参考如下："

#: ../../library/cmath.rst:314

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 ""

"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."