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# Copyright (C) 2001-2020, Python Software Foundation
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#: ../Doc/library/cmath.rst:2
msgid ":mod:`!cmath` --- Mathematical functions for complex numbers"
msgstr ":mod:`cmath` -- Función matemática para números complejos"
#: ../Doc/library/cmath.rst:9
"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."
"Este modulo proporciona acceso a funciones matemáticas para números "
"complejos. Las funciones de este módulo aceptan números enteros, números de "
"coma flotante o números complejos como argumentos. Aceptarán además "
"cualquier objeto de Python que tenga tanto un método :meth:`__complex__`  o "
"un método :meth:`__float__` : estos métodos son usados para convertir el "
"objeto a un número complejo o de coma flotante, respectivamente, y la "
"función es entonces aplicada al resultado de la conversión."
#: ../Doc/library/cmath.rst:18
"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."
#: ../Doc/library/cmath.rst:26
"For example, the :func:`cmath.sqrt` function has a branch cut along the "
"negative real axis. An argument of ``-2-0j`` is treated as though it lies "
"*below* the branch cut, and so gives a result on the negative imaginary "
#: ../Doc/library/cmath.rst:31
">>> cmath.sqrt(-2-0j)\n"
"-1.4142135623730951j"
#: ../Doc/library/cmath.rst:34
"But an argument of ``-2+0j`` is treated as though it lies above the branch "
#: ../Doc/library/cmath.rst:37
">>> cmath.sqrt(-2+0j)\n"
#: ../Doc/library/cmath.rst:42
msgid "**Conversions to and from polar coordinates**"
msgstr "Conversión a y desde coordenadas polares"
#: ../Doc/library/cmath.rst:44
msgid ":func:`phase(z) <phase>`"
#: ../Doc/library/cmath.rst:44
msgid "Return the phase of *z*"
msgstr "Retorna el seno de *x*."
#: ../Doc/library/cmath.rst:45
msgid ":func:`polar(z) <polar>`"
#: ../Doc/library/cmath.rst:45
msgid "Return the representation of *z* in polar coordinates"
msgstr "Conversión a y desde coordenadas polares"
#: ../Doc/library/cmath.rst:46
msgid ":func:`rect(r, phi) <rect>`"
#: ../Doc/library/cmath.rst:46
msgid "Return the complex number *z* with polar coordinates *r* and *phi*"
"Retorna el número complejo *x* con coordenadas polares *r* y *phi*. Esto es "
"equivalente a ``r*(math.cos(phi) + math.sin(phi)*1j)``."
#: ../Doc/library/cmath.rst:48
msgid "**Power and logarithmic functions**"
msgstr "Funciones logarítmicas y de potencias"
#: ../Doc/library/cmath.rst:50
msgid ":func:`exp(z) <exp>`"
#: ../Doc/library/cmath.rst:50
msgid "Return *e* raised to the power *z*"
#: ../Doc/library/cmath.rst:51
msgid ":func:`log(z[, base]) <log>`"
#: ../Doc/library/cmath.rst:51
msgid "Return the logarithm of *z* to the given *base* (*e* by default)"
#: ../Doc/library/cmath.rst:52
msgid ":func:`log10(z) <log10>`"
#: ../Doc/library/cmath.rst:52
msgid "Return the base-10 logarithm of *z*"
msgstr "Retorna el seno de *x*."
#: ../Doc/library/cmath.rst:53
msgid ":func:`sqrt(z) <sqrt>`"
#: ../Doc/library/cmath.rst:53
msgid "Return the square root of *z*"
msgstr "Retorna el seno de *x*."
#: ../Doc/library/cmath.rst:55
msgid "**Trigonometric functions**"
msgstr "Funciones trigonométricas"
#: ../Doc/library/cmath.rst:57
msgid ":func:`acos(z) <acos>`"
#: ../Doc/library/cmath.rst:57
msgid "Return the arc cosine of *z*"
msgstr "Retorna el coseno de *x*."
#: ../Doc/library/cmath.rst:58
msgid ":func:`asin(z) <asin>`"
#: ../Doc/library/cmath.rst:58
msgid "Return the arc sine of *z*"
msgstr "Retorna el coseno de *x*."
#: ../Doc/library/cmath.rst:59
msgid ":func:`atan(z) <atan>`"
#: ../Doc/library/cmath.rst:59
msgid "Return the arc tangent of *z*"
msgstr "Retorna la tangente de *x*."
#: ../Doc/library/cmath.rst:60
msgid ":func:`cos(z) <cos>`"
#: ../Doc/library/cmath.rst:60
msgid "Return the cosine of *z*"
msgstr "Retorna el coseno de *x*."
#: ../Doc/library/cmath.rst:61
msgid ":func:`sin(z) <sin>`"
#: ../Doc/library/cmath.rst:61
msgid "Return the sine of *z*"
msgstr "Retorna el seno de *x*."
#: ../Doc/library/cmath.rst:62
msgid ":func:`tan(z) <tan>`"
#: ../Doc/library/cmath.rst:62
msgid "Return the tangent of *z*"
msgstr "Retorna la tangente de *x*."
#: ../Doc/library/cmath.rst:64
msgid "**Hyperbolic functions**"
msgstr "Funciones hiperbólicas"
#: ../Doc/library/cmath.rst:66
msgid ":func:`acosh(z) <acosh>`"
#: ../Doc/library/cmath.rst:66
msgid "Return the inverse hyperbolic cosine of *z*"
msgstr "Retorna el coseno hiperbólico de *x*."
#: ../Doc/library/cmath.rst:67
msgid ":func:`asinh(z) <asinh>`"
#: ../Doc/library/cmath.rst:67
msgid "Return the inverse hyperbolic sine of *z*"
msgstr "Retorna el seno hiperbólico de *x*."
#: ../Doc/library/cmath.rst:68
msgid ":func:`atanh(z) <atanh>`"
#: ../Doc/library/cmath.rst:68
msgid "Return the inverse hyperbolic tangent of *z*"
msgstr "Retorna la tangente hiperbólica de *x*."
#: ../Doc/library/cmath.rst:69
msgid ":func:`cosh(z) <cosh>`"
#: ../Doc/library/cmath.rst:69
msgid "Return the hyperbolic cosine of *z*"
msgstr "Retorna el coseno hiperbólico de *x*."
#: ../Doc/library/cmath.rst:70
msgid ":func:`sinh(z) <sinh>`"
#: ../Doc/library/cmath.rst:70
msgid "Return the hyperbolic sine of *z*"
msgstr "Retorna el seno hiperbólico de *x*."
#: ../Doc/library/cmath.rst:71
msgid ":func:`tanh(z) <tanh>`"
#: ../Doc/library/cmath.rst:71
msgid "Return the hyperbolic tangent of *z*"
msgstr "Retorna la tangente hiperbólica de *x*."
#: ../Doc/library/cmath.rst:73
msgid "**Classification functions**"
msgstr "Funciones de clasificación"
#: ../Doc/library/cmath.rst:75
msgid ":func:`isfinite(z) <isfinite>`"
#: ../Doc/library/cmath.rst:75
msgid "Check if all components of *z* are finite"
#: ../Doc/library/cmath.rst:76
msgid ":func:`isinf(z) <isinf>`"
#: ../Doc/library/cmath.rst:76
msgid "Check if any component of *z* is infinite"
#: ../Doc/library/cmath.rst:77
msgid ":func:`isnan(z) <isnan>`"
#: ../Doc/library/cmath.rst:77
msgid "Check if any component of *z* is a NaN"
#: ../Doc/library/cmath.rst:78
msgid ":func:`isclose(a, b, *, rel_tol, abs_tol) <isclose>`"
#: ../Doc/library/cmath.rst:78
msgid "Check if the values *a* and *b* are close to each other"
"Retorna ``True`` si los valores *a* y *b* son cercanos el uno al otro y "
"``Falso`` de otro modo."
#: ../Doc/library/cmath.rst:80
msgid "**Constants**"
msgstr "Constantes"
#: ../Doc/library/cmath.rst:82
msgid ":data:`pi`"
#: ../Doc/library/cmath.rst:82
msgid "*π* = 3.141592..."
#: ../Doc/library/cmath.rst:83
msgid ":data:`e`"
#: ../Doc/library/cmath.rst:83
msgid "*e* = 2.718281..."
#: ../Doc/library/cmath.rst:84
msgid ":data:`tau`"
#: ../Doc/library/cmath.rst:84
msgid "*τ* = 2\\ *π* = 6.283185..."
#: ../Doc/library/cmath.rst:85
msgid ":data:`inf`"
#: ../Doc/library/cmath.rst:85
msgid "Positive infinity"
#: ../Doc/library/cmath.rst:86
msgid ":data:`infj`"
#: ../Doc/library/cmath.rst:86
msgid "Pure imaginary infinity"
#: ../Doc/library/cmath.rst:87
msgid ":data:`nan`"
#: ../Doc/library/cmath.rst:87
msgid "\"Not a number\" (NaN)"
#: ../Doc/library/cmath.rst:88
msgid ":data:`nanj`"
#: ../Doc/library/cmath.rst:88
msgid "Pure imaginary NaN"
#: ../Doc/library/cmath.rst:93
msgid "Conversions to and from polar coordinates"
msgstr "Conversión a y desde coordenadas polares"
#: ../Doc/library/cmath.rst:95
"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``."
"Un numero complejo de Python ``z`` se almacena internamente usando "
"coordenadas *rectangulares* o *cartesianas*. Esta determinado completamente "
"por su *parte real* ``z.real`` y su *parte imaginaria* ``z.imag``. Dicho de "
"otra forma::"
#: ../Doc/library/cmath.rst:99
"*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 "
"*Las coordenadas polares* dan una alternativa a la representación de números "
"complejos. En las coordenadas polares, un número complejo *z* se define por "
"los módulos *r* y el ángulo de fase *phi*. El módulo *r* es la distancia "
"desde *z* hasta el origen, mientras que la fase *phi* es el ángulo que va en "
"contra de las agujas del reloj, medido en radianes, desde el eje positivo de "
"las X hasta el segmento de linea que une el origen con *z*."
#: ../Doc/library/cmath.rst:106
"The following functions can be used to convert from the native rectangular "
"coordinates to polar coordinates and back."
"Las siguientes funciones pueden ser usadas para convertir desde coordenadas "
"rectangulares nativas hasta coordenadas polares y viceversa."
#: ../Doc/library/cmath.rst:111
"Return the phase of *z* (also known as the *argument* of *z*), as a float. "
"``phase(z)`` is equivalent to ``math.atan2(z.imag, z.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 ``z.imag``, even when ``z.imag`` is zero::"
"Retorna la fase de *x* (también conocido como el *argumento* de *x*), como "
"un  número de coma  flotante. ``phase(x)`` es equivalente a ``math.atan2(x."
"imag, x.real)``. El resultado se encuentra en el rango [-\\ *π*, *π*], y el "
"tramo para esta operación se sostiene a lo largo del eje de abscisas "
"negativo, continuo desde abajo. En sistemas con soporte para el número 0 con "
"signo (que son las mayoría de ellos en uso vigente), esto significa que el "
"signo del resultado es el mismo como el signo de ``x.imag``, incluso donde "
"``x.imag`` es cero::"
#: ../Doc/library/cmath.rst:117
">>> phase(-1+0j)\n"
"3.141592653589793\n"
">>> phase(-1-0j)\n"
#: ../Doc/library/cmath.rst:125
"The modulus (absolute value) of a complex number *z* can be computed using "
"the built-in :func:`abs` function.  There is no separate :mod:`cmath` module "
"function for this operation."
"El módulo (valor absoluto) de un número complejo *x* puede ser calculado "
"usado la función predeterminada :func:`abs`. No existe otra función aparte "
"del módulo :mod:`cmath` para esta operación."
#: ../Doc/library/cmath.rst:132
"Return the representation of *z* in polar coordinates.  Returns a pair ``(r, "
"phi)`` where *r* is the modulus of *z* and *phi* is the phase of *z*.  "
"``polar(z)`` is equivalent to ``(abs(z), phase(z))``."
"Retorna la representación de *x* en coordenadas polares. Retorna un par "
"``(r, phi)`` donde *r* es el módulo de *x* y phi es la fase de *x*. "
"``polar(x)`` es equivalente a ``(abs(x), phase(x))``."
#: ../Doc/library/cmath.rst:140
"Return the complex number *z* with polar coordinates *r* and *phi*. "
"Equivalent to ``complex(r * math.cos(phi), r * math.sin(phi))``."
"Retorna el número complejo *x* con coordenadas polares *r* y *phi*. Esto es "
"equivalente a ``r*(math.cos(phi) + math.sin(phi)*1j)``."
#: ../Doc/library/cmath.rst:145
msgid "Power and logarithmic functions"
msgstr "Funciones logarítmicas y de potencias"
#: ../Doc/library/cmath.rst:149
"Return *e* raised to the power *z*, where *e* is the base of natural "
"logarithms."
"Retorna *e* elevado a la potencia de *x*, donde *e* es la base de los "
"logaritmos naturales."
#: ../Doc/library/cmath.rst:155
"Return the logarithm of *z* to the given *base*. If the *base* is not "
"specified, returns the natural logarithm of *z*. There is one branch cut, "
"from 0 along the negative real axis to -∞."
"Retorna el logaritmo de *x* dada una *base*. Si la base no se especifica, "
"retorna el logaritmo natural de *x* .No hay tramo, desde 0 en el eje "
"negativo real hasta -∞, continuo desde arriba."
#: ../Doc/library/cmath.rst:162
"Return the base-10 logarithm of *z*. This has the same branch cut as :func:"
"Retorna el logaritmo en base de 10 de *x*. Tiene el mismo tramo que :func:"
#: ../Doc/library/cmath.rst:168
"Return the square root of *z*. This has the same branch cut as :func:`log`."
msgstr "Retorna la raíz cuadrada de *x*. Tiene el mismo tramo que :func:`log`."
#: ../Doc/library/cmath.rst:172
msgid "Trigonometric functions"
msgstr "Funciones trigonométricas"
#: ../Doc/library/cmath.rst:176
"Return the arc cosine of *z*. 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 -∞."
"Retorna el arcocoseno de  *x*. Existen dos tramos: Uno que se extiende desde "
"1 sobre todo el eje de abscisas hasta ∞, continuo desde abajo. El otro se "
"extiende desde -1 a lo largo del eje de abscisas hasta -∞, continuo por "
#: ../Doc/library/cmath.rst:183
"Return the arc sine of *z*. This has the same branch cuts as :func:`acos`."
"Retorna el arcoseno de *x*. Este tiene los mismos tramos que :func:`acos`."
#: ../Doc/library/cmath.rst:188
"Return the arc tangent of *z*. 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``."
"Retorna el arcotangente de *x*. Tiene dos tramos. Uno se extiende desde "
"``1j`` a lo largo de el eje de abscisas imaginario ``∞j``,  continuo desde "
"la derecha. El otro extiende desde ``-1j``  a lo largo de el eje de abscisas "
"hasta ``-∞j`` , continuo desde la izquierda."
#: ../Doc/library/cmath.rst:195
msgid "Return the cosine of *z*."
msgstr "Retorna el coseno de *x*."
#: ../Doc/library/cmath.rst:200
msgid "Return the sine of *z*."
msgstr "Retorna el seno de *x*."
#: ../Doc/library/cmath.rst:205
msgid "Return the tangent of *z*."
msgstr "Retorna la tangente de *x*."
#: ../Doc/library/cmath.rst:209
msgid "Hyperbolic functions"
msgstr "Funciones hiperbólicas"
#: ../Doc/library/cmath.rst:213
"Return the inverse hyperbolic cosine of *z*. There is one branch cut, "
"extending left from 1 along the real axis to -∞."
"Retorna el inverso del coseno hiperbólico de *x*. Tiene un tramo, que se "
"extiende desde la izquierda desde 1 a lo largo del eje de abscisas hasta -∞, "
"continuo desde abajo."
#: ../Doc/library/cmath.rst:219
"Return the inverse hyperbolic sine of *z*. 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``."
"Retorna el inverso del seno hiperbólico de *x*. Tiene dos tramos. Uno se "
"extiende desde ``1j`` a lo largo de el eje de abscisas imaginario ``∞j``, "
"continuo desde la  derecha. El otro se extiende desde ``-1j``  a lo largo de "
"el eje de abscisas hasta ``-∞j``, continuo desde la izquierda."
#: ../Doc/library/cmath.rst:226
"Return the inverse hyperbolic tangent of *z*. There are two branch cuts: One "
"extends from ``1`` along the real axis to ``∞``. The other extends from "
"``-1`` along the real axis to ``-∞``."
"Retorna el inverso de la tangente hiperbólica de *x*. Tiene dos tramos: Uno "
"se extiende desde ``1`` a lo largo de las abscisas reales hasta ``∞``,  "
"continuo desde abajo. El otro se extiende desde ``-1`` a lo largo de las "
"abscisas reales hasta ``-∞``, continuo desde arriba."
#: ../Doc/library/cmath.rst:233
msgid "Return the hyperbolic cosine of *z*."
msgstr "Retorna el coseno hiperbólico de *x*."
#: ../Doc/library/cmath.rst:238
msgid "Return the hyperbolic sine of *z*."
msgstr "Retorna el seno hiperbólico de *x*."
#: ../Doc/library/cmath.rst:243
msgid "Return the hyperbolic tangent of *z*."
msgstr "Retorna la tangente hiperbólica de *x*."
#: ../Doc/library/cmath.rst:247
msgid "Classification functions"
msgstr "Funciones de clasificación"
#: ../Doc/library/cmath.rst:251
"Return ``True`` if both the real and imaginary parts of *z* are finite, and "
"``False`` otherwise."
"Retorna ``True`` si tanto la parte imaginaria como real de *x* son finitas, "
"y ``False`` en cualquier otro caso."
#: ../Doc/library/cmath.rst:259
"Return ``True`` if either the real or the imaginary part of *z* is an "
"infinity, and ``False`` otherwise."
"Retorna ``True`` si la parte real o la imaginaria de *x* es un infinito, y "
"``False`` en el caso contrario."
#: ../Doc/library/cmath.rst:265
"Return ``True`` if either the real or the imaginary part of *z* is a NaN, "
"and ``False`` otherwise."
"Retorna ``True`` tanto si la parte real o imaginaria de *x* es NaN, y  "
"``Falso`` en cualquier otro caso."
#: ../Doc/library/cmath.rst:271
"Return ``True`` if the values *a* and *b* are close to each other and "
"``False`` otherwise."
"Retorna ``True`` si los valores *a* y *b* son cercanos el uno al otro y "
"``Falso`` de otro modo."
#: ../Doc/library/cmath.rst:274
"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)``."
"Que dos valores sean o no considerados como cercanos es determinado de "
"acuerdo al valor absoluto y las tolerancias relativas."
#: ../Doc/library/cmath.rst:278
"*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 "
"*rel_tol* es la tolerancia relativa -- es el máximo valor permitido de la "
"resta entre *a* y *b*, relativo al valor absoluto más grande de *a* o *b*. "
"Por ejemplo, para fijar una tolerancia del 5%, usar ``rel_tol=0.05``. El "
"valor de tolerancia por defecto es ``1e-09``, lo que asegura que los dos "
"valores son los mismos en aproximadamente 9 dígitos decimales. *rel_tol* "
"debe ser mayor que cero."
#: ../Doc/library/cmath.rst:285
"*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."
#: ../Doc/library/cmath.rst:291
"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."
"Los valores especiales IEEE 754 de ``NaN``, ``inf`` y ``-inf`` serán "
"manejados de acuerdo al estándar de IEEE. Especialmente, ``NaN`` no se "
"considera cercano a ningún otro valor, incluido ``NaN``. ``inf`` y ``-inf`` "
"solo son considerados cercanos a sí mismos."
#: ../Doc/library/cmath.rst:300
msgid ":pep:`485` -- A function for testing approximate equality"
msgstr ":pep:`485` -- Una función para probar igualdad aproximada."
#: ../Doc/library/cmath.rst:304
msgid "Constants"
msgstr "Constantes"
#: ../Doc/library/cmath.rst:308
msgid "The mathematical constant *π*, as a float."
msgstr "La constante matemática *π*, como número de coma flotante."
#: ../Doc/library/cmath.rst:313
msgid "The mathematical constant *e*, as a float."
msgstr "La constante matemática *e*, como número de coma flotante."
#: ../Doc/library/cmath.rst:318
msgid "The mathematical constant *τ*, as a float."
msgstr "La constante matemática *τ*, como número de coma flotante."
#: ../Doc/library/cmath.rst:325
msgid "Floating-point positive infinity. Equivalent to ``float('inf')``."
msgstr "Números de coma flotante de +infinito. Equivalente a ``float('inf')``."
#: ../Doc/library/cmath.rst:332
"Complex number with zero real part and positive infinity imaginary part. "
"Equivalent to ``complex(0.0, float('inf'))``."
"Números complejos con la parte real cero y números positivos infinitos como "
"la parte imaginaria. Equivalente a ``complex(0.0, float('inf'))``."
#: ../Doc/library/cmath.rst:340
"A floating-point \"not a number\" (NaN) value.  Equivalent to "
"``float('nan')``."
"El valor del número de coma flotante \"not a number\" (NaN) . Equivalente a "
"``float('nan')``."
#: ../Doc/library/cmath.rst:348
"Complex number with zero real part and NaN imaginary part. Equivalent to "
"``complex(0.0, float('nan'))``."
"Números complejos con parte real cero y como parte imaginaria NaN. "
"Equivalente a ``complex(0.0, float('nan'))``."
#: ../Doc/library/cmath.rst:356
"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 "
"Nótese que la selección de funciones es similar, pero no idéntica, a la del "
"módulo  :mod:`math`. El motivo de tener dos módulos se halla en que algunos "
"usuarios no se encuentran interesados en números complejos, y quizás ni "
"siquiera sepan que son. Preferirían que ``math.sqrt(-1)`` lance una "
"excepción a que retorne un número complejo. Además fíjese que las funciones "
"definidas en :mod:`cmath` siempre retornan un número complejo, incluso si la "
"respuesta puede ser expresada como un número real  (en cuyo caso el número "
"complejo tiene una parte imaginaria de cero)."
#: ../Doc/library/cmath.rst:364
"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:"
"Un apunte en los tramos: Se tratan de curvas en las cuales las funciones "
"fallan a ser continua. Son un complemento necesario de muchas funciones "
"complejas. Se asume que si se necesitan cálculos con funciones complejas, "
"usted entenderá sobre tramos. Consulte casi cualquier(no muy elemental) "
"libro sobre variables complejas para saber más. Para más información en la "
"correcta elección de los tramos para propósitos numéricos, se recomienda la "
"siguiente bibliografía:"
#: ../Doc/library/cmath.rst:374
"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."
"*Kahan, W:  Branch cuts for complex elementary functions; o, Much ado about "
"nothing's sign bit.  En Iserles, A., and Powell, M. (eds.), The state of the "
"art in numerical analysis. Clarendon Press (1987) pp165--211*."
#: ../Doc/library/cmath.rst:354
msgid "module"
#: ../Doc/library/cmath.rst:354
msgid "math"
#~ msgid ""
#~ "*abs_tol* is the minimum absolute tolerance -- useful for comparisons "
#~ "near zero. *abs_tol* must be at least zero."
#~ msgstr ""
#~ "*abs_tol* es la tolerancia mínima absoluta -- útil a la hora de hacer "
#~ "comparaciones cercanas al cero. *abs_tol* debe ser al menos cero."
#~ msgid ""
#~ "If no errors occur, the result will be: ``abs(a-b) <= max(rel_tol * "
#~ "max(abs(a), abs(b)), abs_tol)``."
#~ msgstr ""
#~ "Si no ocurren errores, el resultado será: ``abs(a-b) <= max(rel_tol * "
#~ "max(abs(a), abs(b)), abs_tol)``."
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