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2135
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1 # Complex numbers
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2 # ---------------
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3
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4 # This module represents complex numbers as instances of the class Complex.
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5 # A Complex instance z has two data attribues, z.re (the real part) and z.im
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6 # (the imaginary part). In fact, z.re and z.im can have any value -- all
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7 # arithmetic operators work regardless of the type of z.re and z.im (as long
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8 # as they support numerical operations).
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9 #
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10 # The following functions exist (Complex is actually a class):
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11 # Complex([re [,im]) -> creates a complex number from a real and an imaginary part
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12 # IsComplex(z) -> true iff z is a complex number (== has .re and .im attributes)
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13 # Polar([r [,phi [,fullcircle]]]) ->
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14 # the complex number z for which r == z.radius() and phi == z.angle(fullcircle)
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15 # (r and phi default to 0)
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16 #
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17 # Complex numbers have the following methods:
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18 # z.abs() -> absolute value of z
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19 # z.radius() == z.abs()
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20 # z.angle([fullcircle]) -> angle from positive X axis; fullcircle gives units
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21 # z.phi([fullcircle]) == z.angle(fullcircle)
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22 #
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23 # These standard functions and unary operators accept complex arguments:
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24 # abs(z)
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25 # -z
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26 # +z
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27 # not z
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28 # repr(z) == `z`
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29 # str(z)
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30 # hash(z) -> a combination of hash(z.re) and hash(z.im) such that if z.im is zero
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31 # the result equals hash(z.re)
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32 # Note that hex(z) and oct(z) are not defined.
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33 #
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34 # These conversions accept complex arguments only if their imaginary part is zero:
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35 # int(z)
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36 # long(z)
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37 # float(z)
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38 #
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39 # The following operators accept two complex numbers, or one complex number
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40 # and one real number (int, long or float):
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41 # z1 + z2
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42 # z1 - z2
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43 # z1 * z2
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44 # z1 / z2
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45 # pow(z1, z2)
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46 # cmp(z1, z2)
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47 # Note that z1 % z2 and divmod(z1, z2) are not defined,
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48 # nor are shift and mask operations.
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49 #
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50 # The standard module math does not support complex numbers.
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51 # (I suppose it would be easy to implement a cmath module.)
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52 #
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53 # Idea:
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54 # add a class Polar(r, phi) and mixed-mode arithmetic which
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55 # chooses the most appropriate type for the result:
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56 # Complex for +,-,cmp
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57 # Polar for *,/,pow
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58
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59
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60 import types, math
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61
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62 if not hasattr(math, 'hypot'):
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63 def hypot(x, y):
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64 # XXX I know there's a way to compute this without possibly causing
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65 # overflow, but I can't remember what it is right now...
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66 return math.sqrt(x*x + y*y)
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67 math.hypot = hypot
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68
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69 twopi = math.pi*2.0
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70 halfpi = math.pi/2.0
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71
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72 def IsComplex(obj):
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73 return hasattr(obj, 're') and hasattr(obj, 'im')
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74
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75 def Polar(r = 0, phi = 0, fullcircle = twopi):
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76 phi = phi * (twopi / fullcircle)
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77 return Complex(math.cos(phi)*r, math.sin(phi)*r)
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78
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79 class Complex:
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80
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81 def __init__(self, re=0, im=0):
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82 if IsComplex(re):
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83 im = im + re.im
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84 re = re.re
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85 if IsComplex(im):
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86 re = re - im.im
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87 im = im.re
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88 self.re = re
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89 self.im = im
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90
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91 def __setattr__(self, name, value):
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92 if hasattr(self, name):
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93 raise TypeError, "Complex numbers have set-once attributes"
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94 self.__dict__[name] = value
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95
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96 def __repr__(self):
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97 if not self.im:
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98 return 'Complex(%s)' % `self.re`
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99 else:
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100 return 'Complex(%s, %s)' % (`self.re`, `self.im`)
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101
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102 def __str__(self):
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103 if not self.im:
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104 return `self.re`
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105 else:
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106 return 'Complex(%s, %s)' % (`self.re`, `self.im`)
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107
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108 def __coerce__(self, other):
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109 if IsComplex(other):
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110 return self, other
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111 return self, Complex(other) # May fail
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112
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113 def __cmp__(self, other):
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114 return cmp(self.re, other.re) or cmp(self.im, other.im)
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115
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116 def __hash__(self):
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117 if not self.im: return hash(self.re)
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118 mod = sys.maxint + 1L
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119 return int((hash(self.re) + 2L*hash(self.im) + mod) % (2L*mod) - mod)
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120
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121 def __neg__(self):
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122 return Complex(-self.re, -self.im)
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123
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124 def __pos__(self):
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125 return self
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126
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127 def __abs__(self):
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128 return math.hypot(self.re, self.im)
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129 ##return math.sqrt(self.re*self.re + self.im*self.im)
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130
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131
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132 def __int__(self):
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133 if self.im:
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134 raise ValueError, "can't convert Complex with nonzero im to int"
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135 return int(self.re)
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136
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137 def __long__(self):
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138 if self.im:
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139 raise ValueError, "can't convert Complex with nonzero im to long"
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140 return long(self.re)
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141
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142 def __float__(self):
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143 if self.im:
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144 raise ValueError, "can't convert Complex with nonzero im to float"
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145 return float(self.re)
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146
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147 def __nonzero__(self):
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148 return not (self.re == self.im == 0)
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149
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150 abs = radius = __abs__
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151
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152 def angle(self, fullcircle = twopi):
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153 return (fullcircle/twopi) * ((halfpi - math.atan2(self.re, self.im)) % twopi)
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154
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155 phi = angle
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156
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157 def __add__(self, other):
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158 return Complex(self.re + other.re, self.im + other.im)
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159
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160 __radd__ = __add__
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161
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162 def __sub__(self, other):
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163 return Complex(self.re - other.re, self.im - other.im)
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164
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165 def __rsub__(self, other):
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166 return Complex(other.re - self.re, other.im - self.im)
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167
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168 def __mul__(self, other):
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169 return Complex(self.re*other.re - self.im*other.im,
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170 self.re*other.im + self.im*other.re)
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171
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172 __rmul__ = __mul__
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173
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174 def __div__(self, other):
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175 # Deviating from the general principle of not forcing re or im
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176 # to be floats, we cast to float here, otherwise division
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177 # of Complex numbers with integer re and im parts would use
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178 # the (truncating) integer division
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179 d = float(other.re*other.re + other.im*other.im)
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180 if not d: raise ZeroDivisionError, 'Complex division'
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181 return Complex((self.re*other.re + self.im*other.im) / d,
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182 (self.im*other.re - self.re*other.im) / d)
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183
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184 def __rdiv__(self, other):
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185 return other / self
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186
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187 def __pow__(self, n, z=None):
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188 if z is not None:
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189 raise TypeError, 'Complex does not support ternary pow()'
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190 if IsComplex(n):
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191 if n.im: raise TypeError, 'Complex to the Complex power'
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192 n = n.re
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193 r = pow(self.abs(), n)
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194 phi = n*self.angle()
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195 return Complex(math.cos(phi)*r, math.sin(phi)*r)
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196
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197 def __rpow__(self, base):
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198 return pow(base, self)
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199
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200
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201 # Everything below this point is part of the test suite
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202
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203 def checkop(expr, a, b, value, fuzz = 1e-6):
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204 import sys
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205 print ' ', a, 'and', b,
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206 try:
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207 result = eval(expr)
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208 except:
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209 result = sys.exc_type
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210 print '->', result
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211 if (type(result) == type('') or type(value) == type('')):
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212 ok = result == value
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213 else:
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214 ok = abs(result - value) <= fuzz
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215 if not ok:
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216 print '!!\t!!\t!! should be', value, 'diff', abs(result - value)
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217
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218
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219 def test():
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220 testsuite = {
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221 'a+b': [
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222 (1, 10, 11),
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223 (1, Complex(0,10), Complex(1,10)),
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224 (Complex(0,10), 1, Complex(1,10)),
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225 (Complex(0,10), Complex(1), Complex(1,10)),
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226 (Complex(1), Complex(0,10), Complex(1,10)),
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227 ],
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228 'a-b': [
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229 (1, 10, -9),
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230 (1, Complex(0,10), Complex(1,-10)),
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231 (Complex(0,10), 1, Complex(-1,10)),
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232 (Complex(0,10), Complex(1), Complex(-1,10)),
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233 (Complex(1), Complex(0,10), Complex(1,-10)),
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234 ],
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235 'a*b': [
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236 (1, 10, 10),
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237 (1, Complex(0,10), Complex(0, 10)),
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238 (Complex(0,10), 1, Complex(0,10)),
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239 (Complex(0,10), Complex(1), Complex(0,10)),
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240 (Complex(1), Complex(0,10), Complex(0,10)),
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241 ],
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242 'a/b': [
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243 (1., 10, 0.1),
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244 (1, Complex(0,10), Complex(0, -0.1)),
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245 (Complex(0, 10), 1, Complex(0, 10)),
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246 (Complex(0, 10), Complex(1), Complex(0, 10)),
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247 (Complex(1), Complex(0,10), Complex(0, -0.1)),
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248 ],
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249 'pow(a,b)': [
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250 (1, 10, 1),
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251 (1, Complex(0,10), 'TypeError'),
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252 (Complex(0,10), 1, Complex(0,10)),
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253 (Complex(0,10), Complex(1), Complex(0,10)),
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254 (Complex(1), Complex(0,10), 'TypeError'),
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255 (2, Complex(4,0), 16),
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256 ],
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257 'cmp(a,b)': [
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258 (1, 10, -1),
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259 (1, Complex(0,10), 1),
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260 (Complex(0,10), 1, -1),
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261 (Complex(0,10), Complex(1), -1),
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262 (Complex(1), Complex(0,10), 1),
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263 ],
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264 }
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265 exprs = testsuite.keys()
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266 exprs.sort()
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267 for expr in exprs:
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268 print expr + ':'
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269 t = (expr,)
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270 for item in testsuite[expr]:
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271 apply(checkop, t+item)
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272
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273
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274 if __name__ == '__main__':
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275 test()
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