diff -r ffa851df0825 -r 2fb8b9db1c86 symbian-qemu-0.9.1-12/python-2.6.1/Doc/library/cmath.rst --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/symbian-qemu-0.9.1-12/python-2.6.1/Doc/library/cmath.rst Fri Jul 31 15:01:17 2009 +0100 @@ -0,0 +1,250 @@ + +:mod:`cmath` --- Mathematical functions for complex numbers +=========================================================== + +.. module:: cmath + :synopsis: Mathematical functions for complex numbers. + + +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. + +.. note:: + + 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. + + +Complex coordinates +------------------- + +Complex numbers can be expressed by two important coordinate systems. +Python's :class:`complex` type uses rectangular coordinates where a number +on the complex plain is defined by two floats, the real part and the imaginary +part. + +Definition:: + + z = x + 1j * y + + x := real(z) + y := imag(z) + +In engineering the polar coordinate system is popular for complex numbers. In +polar coordinates a complex number is defined by the radius *r* and the phase +angle *phi*. The radius *r* is the absolute value of the complex, which can be +viewed as distance from (0, 0). The radius *r* is always 0 or a positive float. +The phase angle *phi* is the counter clockwise angle from the positive x axis, +e.g. *1* has the angle *0*, *1j* has the angle *π/2* and *-1* the angle *-π*. + +.. note:: + While :func:`phase` and func:`polar` return *+π* for a negative real they + may return *-π* for a complex with a very small negative imaginary + part, e.g. *-1-1E-300j*. + + +Definition:: + + z = r * exp(1j * phi) + z = r * cis(phi) + + r := abs(z) := sqrt(real(z)**2 + imag(z)**2) + phi := phase(z) := atan2(imag(z), real(z)) + cis(phi) := cos(phi) + 1j * sin(phi) + + +.. function:: phase(x) + + Return phase, also known as the argument, of a complex. + + .. versionadded:: 2.6 + + +.. function:: polar(x) + + Convert a :class:`complex` from rectangular coordinates to polar + coordinates. The function returns a tuple with the two elements + *r* and *phi*. *r* is the distance from 0 and *phi* the phase + angle. + + .. versionadded:: 2.6 + + +.. function:: rect(r, phi) + + Convert from polar coordinates to rectangular coordinates and return + a :class:`complex`. + + .. versionadded:: 2.6 + + + +cmath functions +--------------- + +.. function:: acos(x) + + 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. + + +.. function:: acosh(x) + + Return the hyperbolic arc cosine of *x*. There is one branch cut, extending left + from 1 along the real axis to -∞, continuous from above. + + +.. function:: asin(x) + + Return the arc sine of *x*. This has the same branch cuts as :func:`acos`. + + +.. function:: asinh(x) + + Return the hyperbolic arc 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. + + .. versionchanged:: 2.6 + branch cuts moved to match those recommended by the C99 standard + + +.. function:: atan(x) + + 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. + + .. versionchanged:: 2.6 + direction of continuity of upper cut reversed + + +.. function:: atanh(x) + + Return the hyperbolic arc 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. + + .. versionchanged:: 2.6 + direction of continuity of right cut reversed + + +.. function:: cos(x) + + Return the cosine of *x*. + + +.. function:: cosh(x) + + Return the hyperbolic cosine of *x*. + + +.. function:: exp(x) + + Return the exponential value ``e**x``. + + +.. function:: isinf(x) + + Return *True* if the real or the imaginary part of x is positive + or negative infinity. + + .. versionadded:: 2.6 + + +.. function:: isnan(x) + + Return *True* if the real or imaginary part of x is not a number (NaN). + + .. versionadded:: 2.6 + + +.. function:: log(x[, base]) + + 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. + + .. versionchanged:: 2.4 + *base* argument added. + + +.. function:: log10(x) + + Return the base-10 logarithm of *x*. This has the same branch cut as + :func:`log`. + + +.. function:: sin(x) + + Return the sine of *x*. + + +.. function:: sinh(x) + + Return the hyperbolic sine of *x*. + + +.. function:: sqrt(x) + + Return the square root of *x*. This has the same branch cut as :func:`log`. + + +.. function:: tan(x) + + Return the tangent of *x*. + + +.. function:: tanh(x) + + Return the hyperbolic tangent of *x*. + +The module also defines two mathematical constants: + + +.. data:: pi + + The mathematical constant *pi*, as a float. + + +.. data:: e + + The mathematical constant *e*, as a float. + +.. index:: module: math + +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). + +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: + + +.. seealso:: + + 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. + +