1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
|
:mod:`cmath` --- Mathematical functions for complex numbers
===========================================================
.. module:: cmath
:synopsis: Mathematical functions for complex numbers.
This module is always available. It provides access to mathematical functions
for complex numbers. The functions in this module accept integers,
floating-point numbers or complex numbers as arguments. They will also accept
any Python object that has either a :meth:`__complex__` or a :meth:`__float__`
method: these methods are used to convert the object to a complex or
floating-point number, respectively, and the function is then applied to the
result of the conversion.
.. note::
On platforms with hardware and system-level support for signed
zeros, functions involving branch cuts are continuous on *both*
sides of the branch cut: the sign of the zero distinguishes one
side of the branch cut from the other. On platforms that do not
support signed zeros the continuity is as specified below.
Complex coordinates
-------------------
Complex numbers can be expressed by two important coordinate systems.
Python's :class:`complex` type uses rectangular coordinates where a number
on the complex plain is defined by two floats, the real part and the imaginary
part.
Definition::
z = x + 1j * y
x := real(z)
y := imag(z)
In engineering the polar coordinate system is popular for complex numbers. In
polar coordinates a complex number is defined by the radius *r* and the phase
angle *φ*. The radius *r* is the absolute value of the complex, which can be
viewed as distance from (0, 0). The radius *r* is always 0 or a positive float.
The phase angle *φ* is the counter clockwise angle from the positive x axis,
e.g. *1* has the angle *0*, *1j* has the angle *π/2* and *-1* the angle *-π*.
.. note::
While :func:`phase` and func:`polar` return *+π* for a negative real they
may return *-π* for a complex with a very small negative imaginary
part, e.g. *-1-1E-300j*.
Definition::
z = r * exp(1j * φ)
z = r * cis(φ)
r := abs(z) := sqrt(real(z)**2 + imag(z)**2)
phi := phase(z) := atan2(imag(z), real(z))
cis(φ) := cos(φ) + 1j * sin(φ)
.. function:: phase(x)
Return phase, also known as the argument, of a complex.
.. function:: polar(x)
Convert a :class:`complex` from rectangular coordinates to polar
coordinates. The function returns a tuple with the two elements
*r* and *phi*. *r* is the distance from 0 and *phi* the phase
angle.
.. function:: rect(r, phi)
Convert from polar coordinates to rectangular coordinates and return
a :class:`complex`.
cmath functions
---------------
.. function:: acos(x)
Return the arc cosine of *x*. There are two branch cuts: One extends right from
1 along the real axis to ∞, continuous from below. The other extends left from
-1 along the real axis to -∞, continuous from above.
.. function:: acosh(x)
Return the hyperbolic arc cosine of *x*. There is one branch cut, extending left
from 1 along the real axis to -∞, continuous from above.
.. function:: asin(x)
Return the arc sine of *x*. This has the same branch cuts as :func:`acos`.
.. function:: asinh(x)
Return the hyperbolic arc sine of *x*. There are two branch cuts:
One extends from ``1j`` along the imaginary axis to ``∞j``,
continuous from the right. The other extends from ``-1j`` along
the imaginary axis to ``-∞j``, continuous from the left.
.. function:: atan(x)
Return the arc tangent of *x*. There are two branch cuts: One extends from
``1j`` along the imaginary axis to ``∞j``, continuous from the right. The
other extends from ``-1j`` along the imaginary axis to ``-∞j``, continuous
from the left.
.. function:: atanh(x)
Return the hyperbolic arc tangent of *x*. There are two branch cuts: One
extends from ``1`` along the real axis to ``∞``, continuous from below. The
other extends from ``-1`` along the real axis to ``-∞``, continuous from
above.
.. function:: cos(x)
Return the cosine of *x*.
.. function:: cosh(x)
Return the hyperbolic cosine of *x*.
.. function:: exp(x)
Return the exponential value ``e**x``.
.. function:: isinf(x)
Return *True* if the real or the imaginary part of x is positive
or negative infinity.
.. function:: isnan(x)
Return *True* if the real or imaginary part of x is not a number (NaN).
.. function:: log(x[, base])
Returns the logarithm of *x* to the given *base*. If the *base* is not
specified, returns the natural logarithm of *x*. There is one branch cut, from 0
along the negative real axis to -∞, continuous from above.
.. function:: log10(x)
Return the base-10 logarithm of *x*. This has the same branch cut as
:func:`log`.
.. function:: sin(x)
Return the sine of *x*.
.. function:: sinh(x)
Return the hyperbolic sine of *x*.
.. function:: sqrt(x)
Return the square root of *x*. This has the same branch cut as :func:`log`.
.. function:: tan(x)
Return the tangent of *x*.
.. function:: tanh(x)
Return the hyperbolic tangent of *x*.
The module also defines two mathematical constants:
.. data:: pi
The mathematical constant *pi*, as a float.
.. data:: e
The mathematical constant *e*, as a float.
.. index:: module: math
Note that the selection of functions is similar, but not identical, to that in
module :mod:`math`. The reason for having two modules is that some users aren't
interested in complex numbers, and perhaps don't even know what they are. They
would rather have ``math.sqrt(-1)`` raise an exception than return a complex
number. Also note that the functions defined in :mod:`cmath` always return a
complex number, even if the answer can be expressed as a real number (in which
case the complex number has an imaginary part of zero).
A note on branch cuts: They are curves along which the given function fails to
be continuous. They are a necessary feature of many complex functions. It is
assumed that if you need to compute with complex functions, you will understand
about branch cuts. Consult almost any (not too elementary) book on complex
variables for enlightenment. For information of the proper choice of branch
cuts for numerical purposes, a good reference should be the following:
.. seealso::
Kahan, W: Branch cuts for complex elementary functions; or, Much ado about
nothing's sign bit. In Iserles, A., and Powell, M. (eds.), The state of the art
in numerical analysis. Clarendon Press (1987) pp165-211.
|
