Merged revisions 74184,74230 via svnmerge from
svn+ssh://pythondev@svn.python.org/python/trunk ........ r74184 | georg.brandl | 2009-07-23 08:08:58 +0100 (Thu, 23 Jul 2009) | 1 line #6548: dont suggest existence of real and imag functions in cmath. ........ r74230 | mark.dickinson | 2009-07-28 17:12:40 +0100 (Tue, 28 Jul 2009) | 4 lines Issue #6458: Reorganize cmath documentation into sections (similar to the way that the math documentation is organized); clarify section on conversions to and from polar coordinates. ........
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Mark Dickinson
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@ -23,154 +23,78 @@ result of the conversion.
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support signed zeros the continuity is as specified below.
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Complex coordinates
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-------------------
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Conversions to and from polar coordinates
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-----------------------------------------
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Complex numbers can be expressed by two important coordinate systems.
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Python's :class:`complex` type uses rectangular coordinates where a number
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on the complex plain is defined by two floats, the real part and the imaginary
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part.
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A Python complex number ``z`` is stored internally using *rectangular*
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or *Cartesian* coordinates. It is completely determined by its *real
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part* ``z.real`` and its *imaginary part* ``z.imag``. In other
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words::
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Definition::
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z == z.real + z.imag*1j
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z = x + 1j * y
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x := real(z)
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y := imag(z)
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In engineering the polar coordinate system is popular for complex numbers. In
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polar coordinates a complex number is defined by the radius *r* and the phase
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angle *phi*. The radius *r* is the absolute value of the complex, which can be
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viewed as distance from (0, 0). The radius *r* is always 0 or a positive float.
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The phase angle *phi* is the counter clockwise angle from the positive x axis,
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e.g. *1* has the angle *0*, *1j* has the angle *π/2* and *-1* the angle *-π*.
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.. note::
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While :func:`phase` and func:`polar` return *+π* for a negative real they
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may return *-π* for a complex with a very small negative imaginary
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part, e.g. *-1-1E-300j*.
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Definition::
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z = r * exp(1j * phi)
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z = r * cis(phi)
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r := abs(z) := sqrt(real(z)**2 + imag(z)**2)
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phi := phase(z) := atan2(imag(z), real(z))
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cis(phi) := cos(phi) + 1j * sin(phi)
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*Polar coordinates* give an alternative way to represent a complex
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number. In polar coordinates, a complex number *z* is defined by the
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modulus *r* and the phase angle *phi*. The modulus *r* is the distance
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from *z* to the origin, while the phase *phi* is the counterclockwise
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angle from the positive x-axis to the line segment that joins the
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origin to *z*.
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The following functions can be used to convert from the native
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rectangular coordinates to polar coordinates and back.
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.. function:: phase(x)
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Return phase, also known as the argument, of a complex.
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Return the phase of *x* (also known as the *argument* of *x*), as a
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float. ``phase(x)`` is equivalent to ``math.atan2(x.imag,
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x.real)``. The result lies in the range [-π, π], and the branch
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cut for this operation lies along the negative real axis,
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continuous from above. On systems with support for signed zeros
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(which includes most systems in current use), this means that the
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sign of the result is the same as the sign of ``x.imag``, even when
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``x.imag`` is zero::
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>>> phase(complex(-1.0, 0.0))
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3.1415926535897931
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>>> phase(complex(-1.0, -0.0))
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-3.1415926535897931
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.. versionadded:: 2.6
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.. note::
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The modulus (absolute value) of a complex number *x* can be
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computed using the built-in :func:`abs` function. There is no
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separate :mod:`cmath` module function for this operation.
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.. function:: polar(x)
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Convert a :class:`complex` from rectangular coordinates to polar
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coordinates. The function returns a tuple with the two elements
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*r* and *phi*. *r* is the distance from 0 and *phi* the phase
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angle.
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Return the representation of *x* in polar coordinates. Returns a
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pair ``(r, phi)`` where *r* is the modulus of *x* and phi is the
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phase of *x*. ``polar(x)`` is equivalent to ``(abs(x),
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phase(x))``.
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.. versionadded:: 2.6
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.. function:: rect(r, phi)
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Convert from polar coordinates to rectangular coordinates and return
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a :class:`complex`.
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Return the complex number *x* with polar coordinates *r* and *phi*.
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Equivalent to ``r * (math.cos(phi) + math.sin(phi)*1j)``.
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.. versionadded:: 2.6
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cmath functions
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---------------
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.. function:: acos(x)
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Return the arc cosine of *x*. There are two branch cuts: One extends right from
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1 along the real axis to ∞, continuous from below. The other extends left from
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-1 along the real axis to -∞, continuous from above.
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.. function:: acosh(x)
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Return the hyperbolic arc cosine of *x*. There is one branch cut, extending left
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from 1 along the real axis to -∞, continuous from above.
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.. function:: asin(x)
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Return the arc sine of *x*. This has the same branch cuts as :func:`acos`.
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.. function:: asinh(x)
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Return the hyperbolic arc sine of *x*. There are two branch cuts:
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One extends from ``1j`` along the imaginary axis to ``∞j``,
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continuous from the right. The other extends from ``-1j`` along
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the imaginary axis to ``-∞j``, continuous from the left.
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.. versionchanged:: 2.6
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branch cuts moved to match those recommended by the C99 standard
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.. function:: atan(x)
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Return the arc tangent of *x*. There are two branch cuts: One extends from
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``1j`` along the imaginary axis to ``∞j``, continuous from the right. The
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other extends from ``-1j`` along the imaginary axis to ``-∞j``, continuous
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from the left.
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.. versionchanged:: 2.6
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direction of continuity of upper cut reversed
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.. function:: atanh(x)
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Return the hyperbolic arc tangent of *x*. There are two branch cuts: One
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extends from ``1`` along the real axis to ``∞``, continuous from below. The
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other extends from ``-1`` along the real axis to ``-∞``, continuous from
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above.
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.. versionchanged:: 2.6
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direction of continuity of right cut reversed
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.. function:: cos(x)
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Return the cosine of *x*.
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.. function:: cosh(x)
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Return the hyperbolic cosine of *x*.
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Power and logarithmic functions
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-------------------------------
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.. function:: exp(x)
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Return the exponential value ``e**x``.
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.. function:: isinf(x)
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Return *True* if the real or the imaginary part of x is positive
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or negative infinity.
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.. versionadded:: 2.6
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.. function:: isnan(x)
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Return *True* if the real or imaginary part of x is not a number (NaN).
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.. versionadded:: 2.6
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.. function:: log(x[, base])
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Returns the logarithm of *x* to the given *base*. If the *base* is not
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@ -187,36 +111,123 @@ cmath functions
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:func:`log`.
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.. function:: sin(x)
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Return the sine of *x*.
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.. function:: sinh(x)
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Return the hyperbolic sine of *x*.
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.. function:: sqrt(x)
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Return the square root of *x*. This has the same branch cut as :func:`log`.
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Trigonometric functions
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-----------------------
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.. function:: acos(x)
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Return the arc cosine of *x*. There are two branch cuts: One extends right from
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1 along the real axis to ∞, continuous from below. The other extends left from
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-1 along the real axis to -∞, continuous from above.
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.. function:: asin(x)
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Return the arc sine of *x*. This has the same branch cuts as :func:`acos`.
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.. function:: atan(x)
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Return the arc tangent of *x*. There are two branch cuts: One extends from
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``1j`` along the imaginary axis to ``∞j``, continuous from the right. The
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other extends from ``-1j`` along the imaginary axis to ``-∞j``, continuous
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from the left.
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.. versionchanged:: 2.6
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direction of continuity of upper cut reversed
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.. function:: cos(x)
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Return the cosine of *x*.
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.. function:: sin(x)
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Return the sine of *x*.
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.. function:: tan(x)
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Return the tangent of *x*.
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Hyperbolic functions
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--------------------
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.. function:: acosh(x)
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Return the hyperbolic arc cosine of *x*. There is one branch cut, extending left
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from 1 along the real axis to -∞, continuous from above.
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.. function:: asinh(x)
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Return the hyperbolic arc sine of *x*. There are two branch cuts:
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One extends from ``1j`` along the imaginary axis to ``∞j``,
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continuous from the right. The other extends from ``-1j`` along
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the imaginary axis to ``-∞j``, continuous from the left.
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.. versionchanged:: 2.6
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branch cuts moved to match those recommended by the C99 standard
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.. function:: atanh(x)
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Return the hyperbolic arc tangent of *x*. There are two branch cuts: One
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extends from ``1`` along the real axis to ``∞``, continuous from below. The
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other extends from ``-1`` along the real axis to ``-∞``, continuous from
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above.
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.. versionchanged:: 2.6
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direction of continuity of right cut reversed
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.. function:: cosh(x)
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Return the hyperbolic cosine of *x*.
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.. function:: sinh(x)
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Return the hyperbolic sine of *x*.
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.. function:: tanh(x)
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Return the hyperbolic tangent of *x*.
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The module also defines two mathematical constants:
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Classification functions
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------------------------
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.. function:: isinf(x)
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Return *True* if the real or the imaginary part of x is positive
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or negative infinity.
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.. versionadded:: 2.6
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.. function:: isnan(x)
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Return *True* if the real or imaginary part of x is not a number (NaN).
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.. versionadded:: 2.6
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Constants
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---------
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.. data:: pi
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The mathematical constant *pi*, as a float.
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The mathematical constant *π*, as a float.
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.. data:: e
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