# Python Cmath Module

The `cmath` 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.

## Importing the Module

```python
import cmath
```

## Constants

The `cmath` module defines constants similar to `math`, but also includes complex infinity and NaN.

*   `cmath.pi`: The mathematical constant π.
*   `cmath.e`: The mathematical constant e.
*   `cmath.tau`: The mathematical constant τ.
*   `cmath.inf`: Floating-point positive infinity.
*   `cmath.infj`: Complex number with zero real part and positive infinity imaginary part.
*   `cmath.nan`: Floating-point "not a number" (NaN) value.
*   `cmath.nanj`: Complex number with zero real part and NaN imaginary part.

## Coordinate Conversions

### `cmath.phase(x)`

Return the phase of `x` (also known as the argument of `x`). This is the angle `phi` such that `x = r * (cos(phi) + 1j * sin(phi))`. The result is in the range [-π, π].

```python
import cmath

z = 1 + 1j
print(cmath.phase(z)) # Output: 0.7853981633974483 (pi/4)
```

### `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`.

```python
import cmath

z = 1 + 1j
r, phi = cmath.polar(z)
print(f"r: {r}, phi: {phi}")
```

### `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)`.

```python
import cmath
import math

z = cmath.rect(math.sqrt(2), math.pi / 4)
print(z) # Output: (1.0000000000000002+1j)
```

## Power and Logarithmic Functions

### `cmath.sqrt(x)`

Return the square root of `x`. This works for negative real numbers as well, returning a complex result.

```python
import cmath

print(cmath.sqrt(4))  # Output: (2+0j)
print(cmath.sqrt(-1)) # Output: 1j
```

### `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`.

```python
import cmath

print(cmath.log(cmath.e)) # Output: (1+0j)
print(cmath.log(-1))      # Output: 3.141592653589793j (pi * 1j)
```

[[programming/python/python]]