Mercurial > cpython-lru_cache-weakref

annotate Doc/lib/libcmath.tex @ 39042:c02fa187e0f4 legacy-trunk

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Part of SF patch #1484695. This removes dead code. The chksum was already verified in .frombuf() on the lines above. If there was a problem an exception is raised, so there was no way this condition could have been true.
author Neal Norwitz <nnorwitz@gmail.com>
date Mon, 10 Jul 2006 00:23:17 +0000
parents 67f2190e1f6d
children ebe52dbc3553
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1 \section{\module{cmath} ---
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2 Mathematical functions for complex numbers}
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3
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4 \declaremodule{builtin}{cmath}
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5 \modulesynopsis{Mathematical functions for complex numbers.}
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6
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7 This module is always available. It provides access to mathematical
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8 functions for complex numbers. The functions are:
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9
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10 \begin{funcdesc}{acos}{x}
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11 Return the arc cosine of \var{x}.
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12 There are two branch cuts:
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13 One extends right from 1 along the real axis to \infinity, continuous
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14 from below.
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15 The other extends left from -1 along the real axis to -\infinity,
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16 continuous from above.
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17 \end{funcdesc}
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18
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19 \begin{funcdesc}{acosh}{x}
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20 Return the hyperbolic arc cosine of \var{x}.
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21 There is one branch cut, extending left from 1 along the real axis
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22 to -\infinity, continuous from above.
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23 \end{funcdesc}
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24
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25 \begin{funcdesc}{asin}{x}
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26 Return the arc sine of \var{x}.
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27 This has the same branch cuts as \function{acos()}.
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28 \end{funcdesc}
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29
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30 \begin{funcdesc}{asinh}{x}
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31 Return the hyperbolic arc sine of \var{x}.
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32 There are two branch cuts, extending left from \plusminus\code{1j} to
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33 \plusminus-\infinity\code{j}, both continuous from above.
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34 These branch cuts should be considered a bug to be corrected in a
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35 future release.
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36 The correct branch cuts should extend along the imaginary axis,
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37 one from \code{1j} up to \infinity\code{j} and continuous from the
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38 right, and one from -\code{1j} down to -\infinity\code{j} and
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39 continuous from the left.
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40 \end{funcdesc}
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41
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42 \begin{funcdesc}{atan}{x}
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43 Return the arc tangent of \var{x}.
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44 There are two branch cuts:
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45 One extends from \code{1j} along the imaginary axis to
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46 \infinity\code{j}, continuous from the left.
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47 The other extends from -\code{1j} along the imaginary axis to
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48 -\infinity\code{j}, continuous from the left.
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49 (This should probably be changed so the upper cut becomes continuous
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50 from the other side.)
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51 \end{funcdesc}
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52
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53 \begin{funcdesc}{atanh}{x}
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54 Return the hyperbolic arc tangent of \var{x}.
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55 There are two branch cuts:
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56 One extends from 1 along the real axis to \infinity, continuous
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57 from above.
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58 The other extends from -1 along the real axis to -\infinity,
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59 continuous from above.
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60 (This should probably be changed so the right cut becomes continuous from
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61 the other side.)
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62 \end{funcdesc}
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63
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64 \begin{funcdesc}{cos}{x}
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65 Return the cosine of \var{x}.
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66 \end{funcdesc}
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67
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68 \begin{funcdesc}{cosh}{x}
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69 Return the hyperbolic cosine of \var{x}.
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70 \end{funcdesc}
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71
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72 \begin{funcdesc}{exp}{x}
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73 Return the exponential value \code{e**\var{x}}.
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74 \end{funcdesc}
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75
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76 \begin{funcdesc}{log}{x\optional{, base}}
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77 Returns the logarithm of \var{x} to the given \var{base}.
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78 If the \var{base} is not specified, returns the natural logarithm of \var{x}.
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79 There is one branch cut, from 0 along the negative real axis to
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80 -\infinity, continuous from above.
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81 \versionchanged[\var{base} argument added]{2.4}
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82 \end{funcdesc}
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83
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84 \begin{funcdesc}{log10}{x}
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85 Return the base-10 logarithm of \var{x}.
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86 This has the same branch cut as \function{log()}.
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87 \end{funcdesc}
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88
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89 \begin{funcdesc}{sin}{x}
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90 Return the sine of \var{x}.
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91 \end{funcdesc}
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92
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93 \begin{funcdesc}{sinh}{x}
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94 Return the hyperbolic sine of \var{x}.
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95 \end{funcdesc}
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96
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97 \begin{funcdesc}{sqrt}{x}
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98 Return the square root of \var{x}.
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99 This has the same branch cut as \function{log()}.
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100 \end{funcdesc}
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101
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102 \begin{funcdesc}{tan}{x}
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103 Return the tangent of \var{x}.
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104 \end{funcdesc}
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105
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106 \begin{funcdesc}{tanh}{x}
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107 Return the hyperbolic tangent of \var{x}.
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108 \end{funcdesc}
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109
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110 The module also defines two mathematical constants:
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111
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112 \begin{datadesc}{pi}
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113 The mathematical constant \emph{pi}, as a real.
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114 \end{datadesc}
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115
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116 \begin{datadesc}{e}
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117 The mathematical constant \emph{e}, as a real.
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118 \end{datadesc}
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119
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120 Note that the selection of functions is similar, but not identical, to
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121 that in module \refmodule{math}\refbimodindex{math}. The reason for having
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122 two modules is that some users aren't interested in complex numbers,
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123 and perhaps don't even know what they are. They would rather have
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124 \code{math.sqrt(-1)} raise an exception than return a complex number.
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125 Also note that the functions defined in \module{cmath} always return a
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126 complex number, even if the answer can be expressed as a real number
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127 (in which case the complex number has an imaginary part of zero).
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128
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129 A note on branch cuts: They are curves along which the given function
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130 fails to be continuous. They are a necessary feature of many complex
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131 functions. It is assumed that if you need to compute with complex
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132 functions, you will understand about branch cuts. Consult almost any
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133 (not too elementary) book on complex variables for enlightenment. For
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134 information of the proper choice of branch cuts for numerical
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135 purposes, a good reference should be the following:
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136
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137 \begin{seealso}
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138 \seetext{Kahan, W: Branch cuts for complex elementary functions;
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139 or, Much ado about nothing's sign bit. In Iserles, A.,
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140 and Powell, M. (eds.), \citetitle{The state of the art in
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141 numerical analysis}. Clarendon Press (1987) pp165-211.}
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142 \end{seealso}