210 lines
7.1 KiB
TeX
210 lines
7.1 KiB
TeX
\section{\module{math} ---
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Mathematical functions}
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\declaremodule{builtin}{math}
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\modulesynopsis{Mathematical functions (\function{sin()} etc.).}
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This module is always available. It provides access to the
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mathematical functions defined by the C standard.
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These functions cannot be used with complex numbers; use the functions
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of the same name from the \refmodule{cmath} module if you require
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support for complex numbers. The distinction between functions which
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support complex numbers and those which don't is made since most users
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do not want to learn quite as much mathematics as required to
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understand complex numbers. Receiving an exception instead of a
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complex result allows earlier detection of the unexpected complex
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number used as a parameter, so that the programmer can determine how
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and why it was generated in the first place.
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The following functions are provided by this module. Except
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when explicitly noted otherwise, all return values are floats.
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Number-theoretic and representation functions:
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\begin{funcdesc}{ceil}{x}
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Return the ceiling of \var{x} as a float, the smallest integer value
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greater than or equal to \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{fabs}{x}
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Return the absolute value of \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{floor}{x}
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Return the floor of \var{x} as a float, the largest integer value
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less than or equal to \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{fmod}{x, y}
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Return \code{fmod(\var{x}, \var{y})}, as defined by the platform C library.
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Note that the Python expression \code{\var{x} \%\ \var{y}} may not return
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the same result. The intent of the C standard is that
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\code{fmod(\var{x}, \var{y})} be exactly (mathematically; to infinite
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precision) equal to \code{\var{x} - \var{n}*\var{y}} for some integer
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\var{n} such that the result has the same sign as \var{x} and
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magnitude less than \code{abs(\var{y})}. Python's
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\code{\var{x} \%\ \var{y}} returns a result with the sign of
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\var{y} instead, and may not be exactly computable for float arguments.
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For example, \code{fmod(-1e-100, 1e100)} is \code{-1e-100}, but the
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result of Python's \code{-1e-100 \%\ 1e100} is \code{1e100-1e-100}, which
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cannot be represented exactly as a float, and rounds to the surprising
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\code{1e100}. For this reason, function \function{fmod()} is generally
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preferred when working with floats, while Python's
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\code{\var{x} \%\ \var{y}} is preferred when working with integers.
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\end{funcdesc}
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\begin{funcdesc}{frexp}{x}
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Return the mantissa and exponent of \var{x} as the pair
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\code{(\var{m}, \var{e})}. \var{m} is a float and \var{e} is an
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integer such that \code{\var{x} == \var{m} * 2**\var{e}} exactly.
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If \var{x} is zero, returns \code{(0.0, 0)}, otherwise
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\code{0.5 <= abs(\var{m}) < 1}. This is used to "pick apart" the
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internal representation of a float in a portable way.
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\end{funcdesc}
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\begin{funcdesc}{ldexp}{x, i}
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Return \code{\var{x} * (2**\var{i})}. This is essentially the inverse of
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function \function{frexp()}.
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\end{funcdesc}
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\begin{funcdesc}{modf}{x}
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Return the fractional and integer parts of \var{x}. Both results
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carry the sign of \var{x}, and both are floats.
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\end{funcdesc}
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Note that \function{frexp()} and \function{modf()} have a different
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call/return pattern than their C equivalents: they take a single
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argument and return a pair of values, rather than returning their
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second return value through an `output parameter' (there is no such
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thing in Python).
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For the \function{ceil()}, \function{floor()}, and \function{modf()}
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functions, note that \emph{all} floating-point numbers of sufficiently
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large magnitude are exact integers. Python floats typically carry no more
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than 53 bits of precision (the same as the platform C double type), in
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which case any float \var{x} with \code{abs(\var{x}) >= 2**52}
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necessarily has no fractional bits.
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Power and logarithmic functions:
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\begin{funcdesc}{exp}{x}
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Return \code{e**\var{x}}.
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\end{funcdesc}
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\begin{funcdesc}{log}{x\optional{, base}}
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Return the logarithm of \var{x} to the given \var{base}.
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If the \var{base} is not specified, return the natural logarithm of \var{x}
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(that is, the logarithm to base \emph{e}).
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\versionchanged[\var{base} argument added]{2.3}
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\end{funcdesc}
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\begin{funcdesc}{log10}{x}
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Return the base-10 logarithm of \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{pow}{x, y}
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Return \code{\var{x}**\var{y}}.
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\end{funcdesc}
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\begin{funcdesc}{sqrt}{x}
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Return the square root of \var{x}.
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\end{funcdesc}
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Trigonometric functions:
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\begin{funcdesc}{acos}{x}
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Return the arc cosine of \var{x}, in radians.
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\end{funcdesc}
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\begin{funcdesc}{asin}{x}
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Return the arc sine of \var{x}, in radians.
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\end{funcdesc}
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\begin{funcdesc}{atan}{x}
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Return the arc tangent of \var{x}, in radians.
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\end{funcdesc}
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\begin{funcdesc}{atan2}{y, x}
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Return \code{atan(\var{y} / \var{x})}, in radians.
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The result is between \code{-pi} and \code{pi}.
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The vector in the plane from the origin to point \code{(\var{x}, \var{y})}
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makes this angle with the positive X axis.
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The point of \function{atan2()} is that the signs of both inputs are
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known to it, so it can compute the correct quadrant for the angle.
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For example, \code{atan(1}) and \code{atan2(1, 1)} are both \code{pi/4},
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but \code{atan2(-1, -1)} is \code{-3*pi/4}.
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\end{funcdesc}
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\begin{funcdesc}{cos}{x}
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Return the cosine of \var{x} radians.
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\end{funcdesc}
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\begin{funcdesc}{hypot}{x, y}
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Return the Euclidean norm, \code{sqrt(\var{x}*\var{x} + \var{y}*\var{y})}.
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This is the length of the vector from the origin to point
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\code{(\var{x}, \var{y})}.
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\end{funcdesc}
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\begin{funcdesc}{sin}{x}
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Return the sine of \var{x} radians.
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\end{funcdesc}
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\begin{funcdesc}{tan}{x}
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Return the tangent of \var{x} radians.
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\end{funcdesc}
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Angular conversion:
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\begin{funcdesc}{degrees}{x}
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Converts angle \var{x} from radians to degrees.
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\end{funcdesc}
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\begin{funcdesc}{radians}{x}
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Converts angle \var{x} from degrees to radians.
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\end{funcdesc}
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Hyperbolic functions:
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\begin{funcdesc}{cosh}{x}
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Return the hyperbolic cosine of \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{sinh}{x}
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Return the hyperbolic sine of \var{x}.
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\end{funcdesc}
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\begin{funcdesc}{tanh}{x}
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Return the hyperbolic tangent of \var{x}.
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\end{funcdesc}
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The module also defines two mathematical constants:
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\begin{datadesc}{pi}
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The mathematical constant \emph{pi}.
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\end{datadesc}
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\begin{datadesc}{e}
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The mathematical constant \emph{e}.
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\end{datadesc}
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\begin{notice}
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The \module{math} module consists mostly of thin wrappers around
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the platform C math library functions. Behavior in exceptional cases is
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loosely specified by the C standards, and Python inherits much of its
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math-function error-reporting behavior from the platform C
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implementation. As a result,
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the specific exceptions raised in error cases (and even whether some
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arguments are considered to be exceptional at all) are not defined in any
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useful cross-platform or cross-release way. For example, whether
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\code{math.log(0)} returns \code{-Inf} or raises \exception{ValueError} or
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\exception{OverflowError} isn't defined, and in
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cases where \code{math.log(0)} raises \exception{OverflowError},
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\code{math.log(0L)} may raise \exception{ValueError} instead.
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\end{notice}
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\begin{seealso}
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\seemodule{cmath}{Complex number versions of many of these functions.}
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\end{seealso}
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