# `cmath` --- Mathematical functions for complex numbers¶

This module 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 `__complex__()` or a `__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.

For functions involving branch cuts, we have the problem of deciding how to define those functions on the cut itself. Following Kahan's "Branch cuts for complex elementary functions" paper, as well as Annex G of C99 and later C standards, we use the sign of zero to distinguish one side of the branch cut from the other: for a branch cut along (a portion of) the real axis we look at the sign of the imaginary part, while for a branch cut along the imaginary axis we look at the sign of the real part.

For example, the `cmath.sqrt()` function has a branch cut along the negative real axis. An argument of `complex(-2.0, -0.0)` is treated as though it lies below the branch cut, and so gives a result on the negative imaginary axis:

```>>> cmath.sqrt(complex(-2.0, -0.0))
-1.4142135623730951j
```

But an argument of `complex(-2.0, 0.0)` is treated as though it lies above the branch cut:

```>>> cmath.sqrt(complex(-2.0, 0.0))
1.4142135623730951j
```

## Conversions to and from polar coordinates¶

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.

The following functions can be used to convert from the native rectangular coordinates to polar coordinates and back.

cmath.phase(x)

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. The sign of the result is the same as the sign of `x.imag`, even when `x.imag` is zero:

```>>> phase(complex(-1.0, 0.0))
3.141592653589793
>>> phase(complex(-1.0, -0.0))
-3.141592653589793
```

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.

cmath.polar(x)

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))```.

cmath.rect(r, phi)

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

## Power and logarithmic functions¶

cmath.exp(x)

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

cmath.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 -∞.

cmath.log10(x)

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

cmath.sqrt(x)

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

## Trigonometric functions¶

cmath.acos(x)

Return the arc cosine of x. There are two branch cuts: One extends right from 1 along the real axis to ∞. The other extends left from -1 along the real axis to -∞.

cmath.asin(x)

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

cmath.atan(x)

Return the arc tangent of x. There are two branch cuts: One extends from `1j` along the imaginary axis to `∞j`. The other extends from `-1j` along the imaginary axis to `-∞j`.

cmath.cos(x)

Return the cosine of x.

cmath.sin(x)

Return the sine of x.

cmath.tan(x)

Return the tangent of x.

## Hyperbolic functions¶

cmath.acosh(x)

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

cmath.asinh(x)

Return the inverse hyperbolic sine of x. There are two branch cuts: One extends from `1j` along the imaginary axis to `∞j`. The other extends from `-1j` along the imaginary axis to `-∞j`.

cmath.atanh(x)

Return the inverse hyperbolic tangent of x. There are two branch cuts: One extends from `1` along the real axis to `∞`. The other extends from `-1` along the real axis to `-∞`.

cmath.cosh(x)

Return the hyperbolic cosine of x.

cmath.sinh(x)

Return the hyperbolic sine of x.

cmath.tanh(x)

Return the hyperbolic tangent of x.

## Classification functions¶

cmath.isfinite(x)

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

cmath.isinf(x)

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

cmath.isnan(x)

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

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

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.

PEP 485 -- A function for testing approximate equality

## 常數¶

cmath.pi

The mathematical constant π, as a float.

cmath.e

The mathematical constant e, as a float.

cmath.tau

The mathematical constant τ, as a float.

cmath.inf

Floating-point positive infinity. Equivalent to `float('inf')`.

cmath.infj

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

cmath.nan

A floating-point "not a number" (NaN) value. Equivalent to `float('nan')`.

cmath.nanj

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

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).

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:

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.