| 失效链接处理 |
|
A Primer on Scientfic Programming with Python, Third Edition PDF 下载
本站整理下载：
提取码：rjs1
相关截图：
主要内容：
1.6 Complex Numbers
Suppose x2 = 2. Then most of us are able to find out that x = √2 is
a solution to the equation. The more mathematically interested reader
will also remark that x = −√2 is another solution. But faced with the
equation x2 = −2, very few are able to find a proper solution without
any previous knowledge of complex numbers. Such numbers have many
applications in science, and it is therefore important to be able to use
such numbers in our programs.
On the following pages we extend the previous material on computing with real numbers to complex numbers. The text is optional, and
readers without knowledge of complex numbers can safely drop this
section and jump to Chapter 1.7.
A complex number is a pair of real numbers a and b, most often
written as a+bi, or a+ib, where i is called the imaginary unit and acts
as a label for the second term. Mathematically, i = √−1. An important
feature of complex numbers is definitely the ability to compute square
roots of negative numbers. For example, √−2 = √2i (i.e., √2√−1).
The solutions of x2 = −2 are thus x1 = +√2i and x2 = −√2i.
There are rules for addition, subtraction, multiplication, and division between two complex numbers. There are also rules for raising a
complex number to a real power, as well as rules for computing sin z,
cos z, tan z, ez, ln z, sinh z, cosh z, tanh z, etc. for a complex number
z = a + ib. We assume in the following that you are familiar with the
mathematics of complex numbers, at least to the degree encountered
in the program examples.
let u = a + bi and v = c + di
u = v ⇒ a = c, b = d −u = −a − bi
u∗ ≡ a − bi (complex conjugate)
u + v = (a + c)+(b + d)i u − v = (a − c)+(b − d)i
uv = (ac − bd)+(bc + ad)i
u/v = ac + bd
c2 + d2 +
bc − ad
c2 + d2 i
1.6 Complex Numbers 33
|u| = a2 + b2 eiq = cos q + isin q
1.6.1 Complex Arithmetics in Python
Python supports computation with complex numbers. The imaginary
unit is written as j in Python, instead of i as in mathematics. A complex
number 2 − 3i is therefore expressed as (2-3j) in Python. We remark
that the number i is written as 1j, not just j. Below is a sample session
involving definition of complex numbers and some simple arithmetics:
>>> u = 2.5 + 3j # create a complex number
>>> v = 2 # this is an int
>>> w = u + v # complex + int
>>> w
(4.5+3j)
>>> a = -2
>>> b = 0.5
>>> s = a + b*1j # create a complex number from two floats
>>> s = complex(a, b) # alternative creation
>>> s
(-2+0.5j)
>>> s*w # complex*complex
(-10.5-3.75j)
>>> s/w # complex/complex
(-0.25641025641025639+0.28205128205128205j)
A complex object s has functionality for extracting the real and imaginary parts as well as computing the complex conjugate:
>>> s.real
-2.0
>>> s.imag
0.5
>>> s.conjugate()
(-2-0.5j)
1.6.2 Complex Functions in Python
Taking the sine of a complex number does not work:
>>> from math import sin
>>> r = sin(w)
Traceback (most recent call last):
File "<input>", line 1, in ?
TypeError: can’t convert complex to float; use abs(z)
The reason is that the sin function from the math module only works
with real (float) arguments, not complex. A similar module, cmath,
defines functions that take a complex number as argument and return
a complex number as result. As an example of using the cmath module,
we can demonstrate that the relation sin(ai) = isinh a holds:
34 1 Computing with Formulas
>>> from cmath import sin, sinh
>>> r1 = sin(8j)
>>> r1
1490.4788257895502j
>>> r2 = 1j*sinh(8)
>>> r2
1490.4788257895502j
Another relation, eiq = cos q + isin q, is exemplified next:
>>> q = 8 # some arbitrary number
>>> exp(1j*q)
(-0.14550003380861354+0.98935824662338179j)
>>> cos(q) + 1j*sin(q)
(-0.14550003380861354+0.98935824662338179j)
1.6.3 Unified Treatment of Complex and Real Functions
The cmath functions always return complex numbers. It would be nice
to have functions that return a float object if the result is a real
number and a complex object if the result is a complex number. The
Numerical Python package (see more about this package in Chapter 5)
has such versions of the basic mathematical functions known from math
and cmath. By taking a
from numpy.lib.scimath import *
one gets access to these flexible versions of mathematical functions24.
A session will illustrate what we obtain.
Let us first use the sqrt function in the math module:
>>> from math import sqrt
>>> sqrt(4) # float
2.0
>>> sqrt(-1) # illegal
Traceback (most recent call last):
File "<input>", line 1, in ?
ValueError: math domain error
If we now import sqrt from cmath,
>>> from cmath import sqrt
the previous sqrt function is overwritten by the new one. More precisely, the name sqrt was previously bound to a function sqrt from the
math module, but is now bound to another function sqrt from the cmath
module. In this case, any square root results in a complex object:
>>> sqrt(4) # complex
(2+0j)
>>> sqrt(-1) # complex
1j
24 The functions also come into play by a from scipy import * statement or from
scitools.std import *. The latter is used as a standard import later in the book.
1.7 Summary 35
If we now take
>>> from numpy.lib.scimath import *
we import (among other things) a new sqrt function. This function is
slower than the versions from math and cmath, but it has more flexibility
since the returned object is float if that is mathematically possible,
otherwise a complex is returned:
>>> sqrt(4) # float
2.0
>>> sqrt(-1) # complex
1j
As a further illustration of the need for flexible treatment of both complex and real numbers, we may code the formulas for the roots of a
quadratic function f(x) = ax2 + bx + c:
>>> a = 1; b = 2; c = 100 # polynomial coefficients
>>> from numpy.lib.scimath import sqrt
>>> r1 = (-b + sqrt(b**2 - 4*a*c))/(2*a)
>>> r2 = (-b - sqrt(b**2 - 4*a*c))/(2*a)
>>> r1
(-1+9.94987437107j)
>>> r2
(-1-9.94987437107j)
Using the up arrow, we may go back to the definitions of the coefficients
and change them so the roots become real numbers:
>>> a = 1; b = 4; c = 1 # polynomial coefficients
Going back to the computations of r1 and r2 and performing them
again, we get
>>> r1
-0.267949192431
>>> r2
-3.73205080757
That is, the two results are float objects. Had we applied sqrt from
cmath, r1 and r2 would always be complex objects, while sqrt from the
math module would not handle the first (complex) case.
|
| Java1234官方群25： | 
|
| Java1234官方群25： | 838462530 |
苏公网安备 32061202001004号
