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Issue #16199: Backport 2to3 typo fix from 3.2.
author Chris Jerdonek <chris.jerdonek@gmail.com>
date Thu, 11 Oct 2012 16:15:46 -0700
parents b36cb4602e21
children
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1 :mod:`cmath` --- Mathematical functions for complex numbers
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2 ===========================================================
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3
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4 .. module:: cmath
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5 :synopsis: Mathematical functions for complex numbers.
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6
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7
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8 This module is always available. It provides access to mathematical functions
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9 for complex numbers. The functions in this module accept integers,
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10 floating-point numbers or complex numbers as arguments. They will also accept
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11 any Python object that has either a :meth:`__complex__` or a :meth:`__float__`
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12 method: these methods are used to convert the object to a complex or
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13 floating-point number, respectively, and the function is then applied to the
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14 result of the conversion.
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15
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16 .. note::
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17
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18 On platforms with hardware and system-level support for signed
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19 zeros, functions involving branch cuts are continuous on *both*
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20 sides of the branch cut: the sign of the zero distinguishes one
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21 side of the branch cut from the other. On platforms that do not
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22 support signed zeros the continuity is as specified below.
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25 Conversions to and from polar coordinates
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26 -----------------------------------------
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28 A Python complex number ``z`` is stored internally using *rectangular*
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29 or *Cartesian* coordinates. It is completely determined by its *real
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30 part* ``z.real`` and its *imaginary part* ``z.imag``. In other
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31 words::
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33 z == z.real + z.imag*1j
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35 *Polar coordinates* give an alternative way to represent a complex
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36 number. In polar coordinates, a complex number *z* is defined by the
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37 modulus *r* and the phase angle *phi*. The modulus *r* is the distance
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38 from *z* to the origin, while the phase *phi* is the counterclockwise
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39 angle, measured in radians, from the positive x-axis to the line
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40 segment that joins the origin to *z*.
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42 The following functions can be used to convert from the native
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43 rectangular coordinates to polar coordinates and back.
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45 .. function:: phase(x)
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47 Return the phase of *x* (also known as the *argument* of *x*), as a
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48 float. ``phase(x)`` is equivalent to ``math.atan2(x.imag,
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49 x.real)``. The result lies in the range [-π, π], and the branch
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50 cut for this operation lies along the negative real axis,
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51 continuous from above. On systems with support for signed zeros
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52 (which includes most systems in current use), this means that the
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53 sign of the result is the same as the sign of ``x.imag``, even when
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54 ``x.imag`` is zero::
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55
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56 >>> phase(complex(-1.0, 0.0))
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57 3.1415926535897931
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58 >>> phase(complex(-1.0, -0.0))
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59 -3.1415926535897931
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60
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61 .. versionadded:: 2.6
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62
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63
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64 .. note::
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65
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66 The modulus (absolute value) of a complex number *x* can be
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67 computed using the built-in :func:`abs` function. There is no
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68 separate :mod:`cmath` module function for this operation.
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69
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70
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71 .. function:: polar(x)
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73 Return the representation of *x* in polar coordinates. Returns a
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74 pair ``(r, phi)`` where *r* is the modulus of *x* and phi is the
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75 phase of *x*. ``polar(x)`` is equivalent to ``(abs(x),
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76 phase(x))``.
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77
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78 .. versionadded:: 2.6
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81 .. function:: rect(r, phi)
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82
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83 Return the complex number *x* with polar coordinates *r* and *phi*.
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84 Equivalent to ``r * (math.cos(phi) + math.sin(phi)*1j)``.
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86 .. versionadded:: 2.6
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87
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88
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89 Power and logarithmic functions
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90 -------------------------------
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91
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92 .. function:: exp(x)
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93
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94 Return the exponential value ``e**x``.
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95
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96
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97 .. function:: log(x[, base])
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98
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99 Returns the logarithm of *x* to the given *base*. If the *base* is not
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100 specified, returns the natural logarithm of *x*. There is one branch cut, from 0
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101 along the negative real axis to -∞, continuous from above.
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102
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103 .. versionchanged:: 2.4
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104 *base* argument added.
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105
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106
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107 .. function:: log10(x)
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108
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109 Return the base-10 logarithm of *x*. This has the same branch cut as
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110 :func:`log`.
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111
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112
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113 .. function:: sqrt(x)
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114
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115 Return the square root of *x*. This has the same branch cut as :func:`log`.
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116
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117
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118 Trigonometric functions
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119 -----------------------
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120
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121 .. function:: acos(x)
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122
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123 Return the arc cosine of *x*. There are two branch cuts: One extends right from
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124 1 along the real axis to ∞, continuous from below. The other extends left from
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126
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127
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128 .. function:: asin(x)
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129
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130 Return the arc sine of *x*. This has the same branch cuts as :func:`acos`.
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131
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132
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133 .. function:: atan(x)
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134
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135 Return the arc tangent of *x*. There are two branch cuts: One extends from
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136 ``1j`` along the imaginary axis to ``∞j``, continuous from the right. The
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137 other extends from ``-1j`` along the imaginary axis to ``-∞j``, continuous
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138 from the left.
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139
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140 .. versionchanged:: 2.6
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141 direction of continuity of upper cut reversed
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142
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143
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144 .. function:: cos(x)
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145
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146 Return the cosine of *x*.
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147
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148
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149 .. function:: sin(x)
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150
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151 Return the sine of *x*.
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152
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153
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154 .. function:: tan(x)
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155
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156 Return the tangent of *x*.
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157
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158
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159 Hyperbolic functions
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160 --------------------
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161
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162 .. function:: acosh(x)
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163
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164 Return the hyperbolic arc cosine of *x*. There is one branch cut, extending left
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165 from 1 along the real axis to -∞, continuous from above.
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166
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167
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168 .. function:: asinh(x)
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169
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170 Return the hyperbolic arc sine of *x*. There are two branch cuts:
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171 One extends from ``1j`` along the imaginary axis to ``∞j``,
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172 continuous from the right. The other extends from ``-1j`` along
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173 the imaginary axis to ``-∞j``, continuous from the left.
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174
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175 .. versionchanged:: 2.6
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176 branch cuts moved to match those recommended by the C99 standard
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177
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178
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179 .. function:: atanh(x)
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180
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181 Return the hyperbolic arc tangent of *x*. There are two branch cuts: One
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182 extends from ``1`` along the real axis to ``∞``, continuous from below. The
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183 other extends from ``-1`` along the real axis to ``-∞``, continuous from
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184 above.
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185
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186 .. versionchanged:: 2.6
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187 direction of continuity of right cut reversed
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188
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189
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190 .. function:: cosh(x)
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191
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192 Return the hyperbolic cosine of *x*.
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193
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194
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195 .. function:: sinh(x)
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196
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197 Return the hyperbolic sine of *x*.
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198
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199
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200 .. function:: tanh(x)
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201
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202 Return the hyperbolic tangent of *x*.
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203
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204
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205 Classification functions
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206 ------------------------
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207
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208 .. function:: isinf(x)
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209
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210 Return *True* if the real or the imaginary part of x is positive
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211 or negative infinity.
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212
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213 .. versionadded:: 2.6
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214
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215
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216 .. function:: isnan(x)
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217
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218 Return *True* if the real or imaginary part of x is not a number (NaN).
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219
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220 .. versionadded:: 2.6
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221
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222
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223 Constants
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224 ---------
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225
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226
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227 .. data:: pi
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228
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229 The mathematical constant *π*, as a float.
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230
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231
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232 .. data:: e
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233
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234 The mathematical constant *e*, as a float.
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235
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236 .. index:: module: math
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237
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238 Note that the selection of functions is similar, but not identical, to that in
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239 module :mod:`math`. The reason for having two modules is that some users aren't
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240 interested in complex numbers, and perhaps don't even know what they are. They
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241 would rather have ``math.sqrt(-1)`` raise an exception than return a complex
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242 number. Also note that the functions defined in :mod:`cmath` always return a
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243 complex number, even if the answer can be expressed as a real number (in which
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244 case the complex number has an imaginary part of zero).
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245
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246 A note on branch cuts: They are curves along which the given function fails to
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247 be continuous. They are a necessary feature of many complex functions. It is
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248 assumed that if you need to compute with complex functions, you will understand
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249 about branch cuts. Consult almost any (not too elementary) book on complex
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250 variables for enlightenment. For information of the proper choice of branch
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251 cuts for numerical purposes, a good reference should be the following:
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252
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253
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254 .. seealso::
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255
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256 Kahan, W: Branch cuts for complex elementary functions; or, Much ado about
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Georg Brandl <georg@python.org>
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257 nothing's sign bit. In Iserles, A., and Powell, M. (eds.), The state of the art
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Georg Brandl <georg@python.org>
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258 in numerical analysis. Clarendon Press (1987) pp165-211.
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Georg Brandl <georg@python.org>
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259
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5bc7b20dc04a I finally got the time to update and merge Mark's and my trunk-math branch. The patch is collaborated work of Mark Dickinson and me. It was mostly done a few months ago. The patch fixes a lot of loose ends and edge cases related to operations with NaN, INF, very small values and complex math.
Christian Heimes <christian@cheimes.de>
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