cmath.po

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#: ../../library/cmath.rst:2
msgid ":mod:`!cmath` --- Mathematical functions for complex numbers"
msgstr ":mod:`!cmath` --- 複數的數學函式"

#: ../../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:`~object.__complex__` or a :meth:`~object.__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 ""
"本模組提供一些適用於複數的數學函式。本模組中的函式接受整數、浮點數或複數作為"
"引數。它們也接受任何具有 :meth:`~object.__complex__` 或 :meth:`~object."
"__float__` 方法的 Python 物件：這些方法分別用於將物件轉換為複數或浮點數，然後"
"再將函式應用於轉換後的結果。"

#: ../../library/cmath.rst:18
msgid ""
"For functions involving branch cuts, we have the problem of deciding how to "
"define those functions on the cut itself. Following Kahan's \"Branch cuts "
"for complex elementary functions\" paper, as well as Annex G of C99 and "
"later C standards, we use the sign of zero to distinguish one side of the "
"branch cut from the other: for a branch cut along (a portion of) the real "
"axis we look at the sign of the imaginary part, while for a branch cut along "
"the imaginary axis we look at the sign of the real part."
msgstr ""
"對於涉及分枝切割 (branch cut) 的函式，我們面臨的問題是決定如何定義在切割本身"
"上的這些函式。遵循 Kahan 的論文 \"Branch cuts for complex elementary "
"functions\"，以及 C99 的附錄 G 和後來的 C 標準，我們使用零符號來區分分枝切割"
"的兩側：對於沿著（一部分）實數軸的分枝切割，我們查看虛部的符號，而對於沿虛軸"
"的分枝切割，我們則查看實部的符號。"

#: ../../library/cmath.rst:26
msgid ""
"For example, the :func:`cmath.sqrt` function has a branch cut along the "
"negative real axis. An argument of ``complex(-2.0, -0.0)`` is treated as "
"though it lies *below* the branch cut, and so gives a result on the negative "
"imaginary axis::"
msgstr ""
"例如 :func:`cmath.sqrt` 函式具有一條沿負實軸的分枝切割。 引數 "
"``complex(-2.0, -0.0)`` 被視為位於分枝切割 *下方* 處理，因此給出的結果在負虛"
"軸上： ::"

#: ../../library/cmath.rst:31
msgid ""
">>> cmath.sqrt(complex(-2.0, -0.0))\n"
"-1.4142135623730951j"
msgstr ""
">>> cmath.sqrt(complex(-2.0, -0.0))\n"
"-1.4142135623730951j"

#: ../../library/cmath.rst:34
msgid ""
"But an argument of ``complex(-2.0, 0.0)`` is treated as though it lies above "
"the branch cut::"
msgstr "但是引數 ``complex(-2.0, 0.0)`` 會被當成位於分枝切割上方處理： ::"

#: ../../library/cmath.rst:37
msgid ""
">>> cmath.sqrt(complex(-2.0, 0.0))\n"
"1.4142135623730951j"
msgstr ""
">>> cmath.sqrt(complex(-2.0, 0.0))\n"
"1.4142135623730951j"

#: ../../library/cmath.rst:42
msgid "Conversions to and from polar coordinates"
msgstr "轉換到極座標和從極座標做轉換"

#: ../../library/cmath.rst:44
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``."
msgstr ""
"Python 複數 ``z`` 是用 *直角坐標* 或 *笛卡爾坐標* 儲存在內部的。它完全是由其 "
"*實部* ``z.real`` 和 *虛部* ``z.imag`` 所決定。"

#: ../../library/cmath.rst:48
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 ""
"*極座標* 提供了另一種表示複數的方法。在極座標中，複數 *z* 由絕對值 (modulus) "
"*r* 和相位角 (phase) *phi* 定義。絕對值 *r* 是從 *z* 到原點的距離，而相位角 "
"*phi* 是從正 x 軸到連接原點到 *z* 的線段的逆時針角度（以弧度為單位）。"

#: ../../library/cmath.rst:55
msgid ""
"The following functions can be used to convert from the native rectangular "
"coordinates to polar coordinates and back."
msgstr "以下的函式可用於原始直角座標與極座標之間的相互轉換。"

#: ../../library/cmath.rst:60
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.  The sign of the result is the same as the "
"sign of ``x.imag``, even when ``x.imag`` is zero::"
msgstr ""
"以浮點數的形式回傳 *x* 的相位角（也稱為 *x* 的 *引數* ）。 ``phase(x)`` 等價"
"於 ``math.atan2(x.imag, x.real)``。結果將位於 [-\\ *π*, *π*] 的範圍內，且此操"
"作的分枝切割將位於負實軸上。結果的符號會與 ``x.imag`` 的符號相同，即使 ``x."
"imag`` 為零： ::"

#: ../../library/cmath.rst:66
msgid ""
">>> phase(complex(-1.0, 0.0))\n"
"3.141592653589793\n"
">>> phase(complex(-1.0, -0.0))\n"
"-3.141592653589793"
msgstr ""
">>> phase(complex(-1.0, 0.0))\n"
"3.141592653589793\n"
">>> phase(complex(-1.0, -0.0))\n"
"-3.141592653589793"

#: ../../library/cmath.rst:74
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 ""
"複數 *x* 的絕對值可以使用內建的 :func:`abs` 函式計算。沒有單獨的 :mod:"
"`cmath` 模組函式適用於此操作。"

#: ../../library/cmath.rst:81
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 ""
"回傳 *x* 在極座標中的表達方式。回傳一組數對 ``(r, phi)``， *r* 是 *x* 的絕對"
"值， *phi* 是 *x* 的相位角。 ``polar(x)`` 相當於 ``(abs(x), phase(x))``。"

#: ../../library/cmath.rst:89
msgid ""
"Return the complex number *x* with polar coordinates *r* and *phi*. "
"Equivalent to ``complex(r * math.cos(phi), r * math.sin(phi))``."
msgstr ""
"透過極座標 *r* 和 *phi* 回傳複數 *x*。相當於 ``complex(r * math.cos(phi), r "
"* math.sin(phi))``。"

#: ../../library/cmath.rst:94
msgid "Power and logarithmic functions"
msgstr "冪函數和對數函數"

#: ../../library/cmath.rst:98
msgid ""
"Return *e* raised to the power *x*, where *e* is the base of natural "
"logarithms."
msgstr "回傳 *e* 的 *x* 次方，其中 *e* 是自然對數的底數。"

#: ../../library/cmath.rst:104
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 -∞."
msgstr ""
"回傳 *x* 給定 *base* 的對數。如果未指定 *base*，則傳回 *x* 的自然對數。存在一"
"條分枝切割，從 0 沿負實數軸到 -∞。"

#: ../../library/cmath.rst:111
msgid ""
"Return the base-10 logarithm of *x*. This has the same branch cut as :func:"
"`log`."
msgstr "回傳 *x* 以 10 為底的對數。它與 :func:`log` 具有相同的分枝切割。"

#: ../../library/cmath.rst:117
msgid ""
"Return the square root of *x*. This has the same branch cut as :func:`log`."
msgstr "回傳 *x* 的平方根。它與 :func:`log` 具有相同的分枝切割。"

#: ../../library/cmath.rst:121
msgid "Trigonometric functions"
msgstr "三角函數"

#: ../../library/cmath.rst:125
msgid ""
"Return the arc cosine of *x*. There are two branch cuts: One extends right "
"from 1 along the real axis to ∞. The other extends left from -1 along the "
"real axis to -∞."
msgstr ""
"回傳 *x* 的反餘弦值。存在兩條分枝切割：一條是從 1 沿著實數軸向右延伸到 ∞。另"
"一條從 -1 沿實數軸向左延伸到 -∞。"

#: ../../library/cmath.rst:132
msgid ""
"Return the arc sine of *x*. This has the same branch cuts as :func:`acos`."
msgstr "回傳 *x* 的反正弦值。它與 :func:`acos` 具有相同的分枝切割。"

#: ../../library/cmath.rst:137
msgid ""
"Return the arc tangent of *x*. There are two branch cuts: One extends from "
"``1j`` along the imaginary axis to ``∞j``. The other extends from ``-1j`` "
"along the imaginary axis to ``-∞j``."
msgstr ""
"回傳 *x* 的反正切值。有兩條分枝切割：一條是從 ``1j`` 沿著虛軸延伸到 ``∞j``。"
"另一條從 ``-1j`` 沿著虛軸延伸到 ``-∞j``。"

#: ../../library/cmath.rst:144
msgid "Return the cosine of *x*."
msgstr "回傳 *x* 的餘弦值。"

#: ../../library/cmath.rst:149
msgid "Return the sine of *x*."
msgstr "回傳 *x* 的正弦值。"

#: ../../library/cmath.rst:154
msgid "Return the tangent of *x*."
msgstr "回傳 *x* 的正切值。"

#: ../../library/cmath.rst:158
msgid "Hyperbolic functions"
msgstr "雙曲函數"

#: ../../library/cmath.rst:162
msgid ""
"Return the inverse hyperbolic cosine of *x*. There is one branch cut, "
"extending left from 1 along the real axis to -∞."
msgstr ""
"回傳 *x* 的反雙曲餘弦值。存在一條分枝切割，從 1 沿實數軸向左延伸到 -∞。"

#: ../../library/cmath.rst:168
msgid ""
"Return the inverse hyperbolic sine of *x*. There are two branch cuts: One "
"extends from ``1j`` along the imaginary axis to ``∞j``.  The other extends "
"from ``-1j`` along the imaginary axis to ``-∞j``."
msgstr ""
"回傳 *x* 的反雙曲正弦值。存在兩條分枝切割：一條是從 ``1j`` 沿著虛軸延伸到 "
"``∞j``。另一條從 ``-1j`` 沿著虛軸延伸到 ``-∞j``。"

#: ../../library/cmath.rst:175
msgid ""
"Return the inverse hyperbolic tangent of *x*. There are two branch cuts: One "
"extends from ``1`` along the real axis to ``∞``. The other extends from "
"``-1`` along the real axis to ``-∞``."
msgstr ""
"回傳 *x* 的反雙曲正切值。存在兩條分枝切割：一條是從 ``1`` 沿著實數軸延伸到 "
"``∞``。另一條從 ``-1`` 沿著實數軸延伸到 ``-∞``。"

#: ../../library/cmath.rst:182
msgid "Return the hyperbolic cosine of *x*."
msgstr "回傳 *x* 的反雙曲餘弦值。"

#: ../../library/cmath.rst:187
msgid "Return the hyperbolic sine of *x*."
msgstr "回傳 *x* 的反雙曲正弦值。"

#: ../../library/cmath.rst:192
msgid "Return the hyperbolic tangent of *x*."
msgstr "回傳 *x* 的反雙曲正切值。"

#: ../../library/cmath.rst:196
msgid "Classification functions"
msgstr "分類函式"

#: ../../library/cmath.rst:200
msgid ""
"Return ``True`` if both the real and imaginary parts of *x* are finite, and "
"``False`` otherwise."
msgstr "如果 *x* 的實部和虛部都是有限的，則回傳 ``True``，否則回傳 ``False``。"

#: ../../library/cmath.rst:208
msgid ""
"Return ``True`` if either the real or the imaginary part of *x* is an "
"infinity, and ``False`` otherwise."
msgstr "如果 *x* 的實部或虛部是無窮大，則回傳 ``True``，否則回傳 ``False``。"

#: ../../library/cmath.rst:214
msgid ""
"Return ``True`` if either the real or the imaginary part of *x* is a NaN, "
"and ``False`` otherwise."
msgstr "如果 *x* 的實部或虛部為 NaN，則回傳 ``True``，否則回傳 ``False``。"

#: ../../library/cmath.rst:220
msgid ""
"Return ``True`` if the values *a* and *b* are close to each other and "
"``False`` otherwise."
msgstr "如果 *a* 和 *b* 的值相互接近，則回傳 ``True``，否則回傳 ``False``。"

#: ../../library/cmath.rst:223
msgid ""
"Whether or not two values are considered close is determined according to "
"given absolute and relative tolerances.  If no errors occur, the result will "
"be: ``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)``."
msgstr ""
"兩數是否足夠接近取決於給定的絕對及相對容許偏差 (tolerance)。如果沒有錯誤發"
"生，結果將為：``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)``。"

#: ../../library/cmath.rst:227
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 nonnegative and less than "
"``1.0``."
msgstr ""
"*rel_tol* 為相對容許偏差 ── *a* 與 *b* 兩數差的最大容許值，與 *a* 及 *b* 兩數"
"的絕對值中較大者相關。例如欲設置 5% 的容許偏差，則傳入 ``rel_tol=0.05``。其預"
"設值為 ``1e-09``，該值可確保兩數於大約 9 個十進數位內相同。*rel_tol* 須不為負"
"且小於 ``1.0``。"

#: ../../library/cmath.rst:234
msgid ""
"*abs_tol* is the absolute tolerance; it defaults to ``0.0`` and it must be "
"nonnegative.  When comparing ``x`` to ``0.0``, ``isclose(x, 0)`` is computed "
"as ``abs(x) <= rel_tol  * abs(x)``, which is ``False`` for any ``x`` and "
"rel_tol less than ``1.0``.  So add an appropriate positive abs_tol argument "
"to the call."
msgstr ""

#: ../../library/cmath.rst:240
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 ""
"定義於 IEEE 754 浮點標準中的特殊值 ``NaN``、``inf`` 和 ``-inf`` 會根據該標準"
"處理。更明確地說，``NaN`` 不會與包含自身在內的任何數字足夠接近，而 ``inf`` "
"及 ``-inf`` 皆只與自身接近。"

#: ../../library/cmath.rst:249
msgid ":pep:`485` -- A function for testing approximate equality"
msgstr ":pep:`485` ── 用於測試近似相等的函式"

#: ../../library/cmath.rst:253
msgid "Constants"
msgstr "常數"

#: ../../library/cmath.rst:257
msgid "The mathematical constant *π*, as a float."
msgstr "數學常數 *π*，作為一個浮點數。"

#: ../../library/cmath.rst:262
msgid "The mathematical constant *e*, as a float."
msgstr "數學常數 *e*，作為一個浮點數。"

#: ../../library/cmath.rst:267
msgid "The mathematical constant *τ*, as a float."
msgstr "數學常數 *τ*，作為一個浮點數。"

#: ../../library/cmath.rst:274
msgid "Floating-point positive infinity. Equivalent to ``float('inf')``."
msgstr "正無窮大的浮點數。相當於 ``float('inf')``。"

#: ../../library/cmath.rst:281
msgid ""
"Complex number with zero real part and positive infinity imaginary part. "
"Equivalent to ``complex(0.0, float('inf'))``."
msgstr "實部為零和虛部為正無窮的複數。相當於 ``complex(0.0, float('inf'))``。"

#: ../../library/cmath.rst:289
msgid ""
"A floating-point \"not a number\" (NaN) value.  Equivalent to "
"``float('nan')``."
msgstr "浮點「非數字」 (NaN) 值。相當於 ``float('nan')``。"

#: ../../library/cmath.rst:297
msgid ""
"Complex number with zero real part and NaN imaginary part. Equivalent to "
"``complex(0.0, float('nan'))``."
msgstr "實部為零和虛部為 NaN 的複數。相當於 ``complex(0.0, float('nan'))``。"

#: ../../library/cmath.rst:305
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 ""
"請注意，函式的選擇與模組 :mod:`math` 的類似，但並不完全相同。擁有兩個模組的原"
"因是有些用戶對複數不感興趣，甚至根本就不知道它們是什麼。他們寧願讓 ``math."
"sqrt(-1)`` 引發異常，也不願它回傳複數。另請注意， :mod:`cmath` 中所定義的函式"
"始終都會回傳複數，即使答案可以表示為實數（在這種情況下，複數的虛部為零）。"

#: ../../library/cmath.rst:313
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 ""
"關於分枝切割的註釋：它們是沿著給定的不連續函式的曲線。它們是許多複變函數的必"
"要特徵。假設你需要使用複變函數進行計算，你將會了解分枝切割的概念。請參閱幾乎"
"所有關於複變函數的（不是太初級的）書籍以獲得啟發。對於如何正確地基於數值目的"
"選擇分枝切割的相關訊息，以下內容應該是一個很好的參考："

#: ../../library/cmath.rst:323
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 ""

#: ../../library/cmath.rst:303
msgid "module"
msgstr "module（模組）"

#: ../../library/cmath.rst:303
msgid "math"
msgstr "math（數學）"