Mercurial > cpython-lru_cache-weakref

annotate Doc/lib/libcmath.tex @ 25838:8540887bcec6 2.2

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Backport of 2.183: Fix memory leak in add_subclass() found by NealN with valgrind.
author Guido van Rossum <guido@python.org>
date Fri, 18 Oct 2002 13:42:21 +0000
parents d923b3c8b152
children 11c31a98dfd6
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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}
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77 Return the natural logarithm of \var{x}.
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78 There is one branch cut, from 0 along the negative real axis to
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79 -\infinity, continuous from above.
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80 \end{funcdesc}
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81
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82 \begin{funcdesc}{log10}{x}
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83 Return the base-10 logarithm of \var{x}.
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84 This has the same branch cut as \function{log()}.
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85 \end{funcdesc}
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86
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87 \begin{funcdesc}{sin}{x}
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88 Return the sine of \var{x}.
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89 \end{funcdesc}
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90
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91 \begin{funcdesc}{sinh}{x}
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92 Return the hyperbolic sine of \var{x}.
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93 \end{funcdesc}
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94
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95 \begin{funcdesc}{sqrt}{x}
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96 Return the square root of \var{x}.
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97 This has the same branch cut as \function{log()}.
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98 \end{funcdesc}
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99
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100 \begin{funcdesc}{tan}{x}
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101 Return the tangent of \var{x}.
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102 \end{funcdesc}
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103
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104 \begin{funcdesc}{tanh}{x}
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105 Return the hyperbolic tangent of \var{x}.
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106 \end{funcdesc}
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107
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108 The module also defines two mathematical constants:
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109
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110 \begin{datadesc}{pi}
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111 The mathematical constant \emph{pi}, as a real.
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112 \end{datadesc}
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113
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114 \begin{datadesc}{e}
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115 The mathematical constant \emph{e}, as a real.
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116 \end{datadesc}
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117
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118 Note that the selection of functions is similar, but not identical, to
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119 that in module \refmodule{math}\refbimodindex{math}. The reason for having
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120 two modules is that some users aren't interested in complex numbers,
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121 and perhaps don't even know what they are. They would rather have
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122 \code{math.sqrt(-1)} raise an exception than return a complex number.
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123 Also note that the functions defined in \module{cmath} always return a
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124 complex number, even if the answer can be expressed as a real number
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125 (in which case the complex number has an imaginary part of zero).
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126
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127 A note on branch cuts: They are curves along which the given function
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128 fails to be continuous. They are a necessary feature of many complex
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129 functions. It is assumed that if you need to compute with complex
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130 functions, you will understand about branch cuts. Consult almost any
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131 (not too elementary) book on complex variables for enlightenment. For
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132 information of the proper choice of branch cuts for numerical
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133 purposes, a good reference should be the following:
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134
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135 \begin{seealso}
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136 \seetext{Kahan, W: Branch cuts for complex elementary functions;
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137 or, Much ado about nothings's sign bit. In Iserles, A.,
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138 and Powell, M. (eds.), \citetitle{The state of the art in
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139 numerical analysis}. Clarendon Press (1987) pp165-211.}
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140 \end{seealso}