Python cmath

Question 1

Complex number representation

Answer 1

A Python complex number z is stored internally using rectangular or Cartesian coordinates. It is completely determined by its real part z.real and its imaginary part z.imag. In other words:
z == z.real + z.imag*1j
Polar coordinates give an alternative way to represent a complex number. In polar coordinates, a complex number z is defined by the modulus r and the phase angle phi. The modulus r is the distance from z to the origin, while the phase phi is the counterclockwise angle, measured in radians, from the positive x-axis to the line segment that joins the origin to z.

Question 2

cmath.phase(x)

Answer 2

Return the phase of x (also known as the argument of x), as a float. phase(x) is equivalent to math.atan2(x.imag, x.real). The result lies in the range [-π, π], and the branch cut for this operation lies along the negative real axis, continuous from above. On systems with support for signed zeros (which includes most systems in current use), this means that the sign of the result is the same as the sign of x.imag, even when x.imag is zero:
&raquo_space;> phase(complex(-1.0, 0.0))
3.141592653589793
&raquo_space;> phase(complex(-1.0, -0.0))
-3.141592653589793
Note
The modulus (absolute value) of a complex number x can be computed using the built-in abs() function. There is no separate cmath module function for this operation.

Question 3

cmath.polar(x)

Answer 3

Return the representation of x in polar coordinates. Returns a pair (r, phi) where r is the modulus of x and phi is the phase of x. polar(x) is equivalent to (abs(x), phase(x)).

Question 4

cmath.rect(r, phi)

Answer 4

Return the complex number x with polar coordinates r and phi. Equivalent to r * (math.cos(phi) + math.sin(phi)*1j).

Question 5

cmath.exp(x)

Answer 5

Return e raised to the power x, where e is the base of natural logarithms.

Question 6

cmath.log(x[, base])

Answer 6

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.

Question 7

cmath.log10(x)

Answer 7

Return the base-10 logarithm of x. This has the same branch cut as log().

Question 8

cmath.sqrt(x)

Answer 8

Return the square root of x. This has the same branch cut as log().

Question 9

cmath.acos(x)

Answer 9

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.

Question 10

cmath.asin(x)

Answer 10

Return the arc sine of x. This has the same branch cuts as acos().

Question 11

cmath.atan(x)

Answer 11

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.

Question 12

cmath.cos(x)

Answer 12

Return the cosine of x.

Question 13

cmath.sin(x)

Answer 13

Return the sine of x.

Question 14

cmath.tan(x)

Answer 14

Return the tangent of x.

Question 15

cmath.acosh(x)

Answer 15

Return the inverse hyperbolic cosine of x. There is one branch cut, extending left from 1 along the real axis to -∞, continuous from above.

Question 16

cmath.asinh(x)

Answer 16

Return the inverse hyperbolic 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.

Question 17

cmath.atanh(x)

Answer 17

Return the inverse hyperbolic 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.

Question 18

cmath.cosh(x)

Answer 18

Return the hyperbolic cosine of x.

Question 19

cmath.sinh(x)

Answer 19

Return the hyperbolic sine of x.

Question 20

cmath.tanh(x)

Answer 20

Return the hyperbolic tangent of x.

Question 21

cmath.isfinite(x)

Answer 21

Return True if both the real and imaginary parts of x are finite, and False otherwise.

Question 22

cmath.isinf(x)

Answer 22

Return True if either the real or the imaginary part of x is an infinity, and False otherwise.

Question 23

cmath.isnan(x)

Answer 23

Return True if either the real or the imaginary part of x is a NaN, and False otherwise.

Question 24

cmath.isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)

Answer 24

Return True if the values a and b are close to each other and False otherwise.
Whether or not two values are considered close is determined according to given absolute and relative tolerances.
rel_tol is the relative tolerance – it is the maximum allowed difference between a and b, relative to the larger absolute value of a or b. For example, to set a tolerance of 5%, pass rel_tol=0.05. The default tolerance is 1e-09, which assures that the two values are the same within about 9 decimal digits. rel_tol must be greater than zero.
abs_tol is the minimum absolute tolerance – useful for comparisons near zero. abs_tol must be at least zero.
If no errors occur, the result will be: abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol).
The IEEE 754 special values of NaN, inf, and -inf will be handled according to IEEE rules. Specifically, NaN is not considered close to any other value, including NaN. inf and -inf are only considered close to themselves.

Question 25

cmath.pi

Answer 25

The mathematical constant π, as a float.

Question 26

cmath.e

Answer 26

The mathematical constant e, as a float.

Question 27

cmath.tau

Answer 27

The mathematical constant τ, as a float.

Question 28

cmath.inf

Answer 28

Floating-point positive infinity. Equivalent to float(‘inf’).

Question 29

cmath.infj

Answer 29

Complex number with zero real part and positive infinity imaginary part. Equivalent to complex(0.0, float(‘inf’)).

Question 30

cmath.nan

Answer 30

A floating-point “not a number” (NaN) value. Equivalent to float(‘nan’).

Question 31

cmath.nanj

Answer 31

Complex number with zero real part and NaN imaginary part. Equivalent to complex(0.0, float(‘nan’)).

Question 32

cmath notes

Answer 32

Note that the selection of functions is similar, but not identical, to that in module 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 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).