Copyright | (c) The University of Glasgow 2001 |
---|---|

License | BSD-style (see the file libraries/base/LICENSE) |

Maintainer | libraries@haskell.org |

Stability | provisional |

Portability | portable |

Safe Haskell | Trustworthy |

Language | Haskell2010 |

Complex numbers.

## Synopsis

- data Complex a = !a :+ !a
- realPart :: Complex a -> a
- imagPart :: Complex a -> a
- mkPolar :: Floating a => a -> a -> Complex a
- cis :: Floating a => a -> Complex a
- polar :: RealFloat a => Complex a -> (a, a)
- magnitude :: RealFloat a => Complex a -> a
- phase :: RealFloat a => Complex a -> a
- conjugate :: Num a => Complex a -> Complex a

# Rectangular form

A data type representing complex numbers.

You can read about complex numbers on wikipedia.

In haskell, complex numbers are represented as `a :+ b`

which can be thought of
as representing \(a + bi\). For a complex number `z`

,

is a number with the `abs`

z`magnitude`

of `z`

,
but oriented in the positive real direction, whereas

has the `signum`

z`phase`

of `z`

, but unit `magnitude`

.
Apart from the loss of precision due to IEEE754 floating point numbers,
it holds that `z == `

.`abs`

z * `signum`

z

Note that `Complex`

's instances inherit the deficiencies from the type
parameter's. For example, `Complex Float`

's `Ord`

instance has similar
problems to `Float`

's.

As can be seen in the examples, the `Foldable`

and `Traversable`

instances traverse the real part first.

#### Examples

`>>>`

11.5 :+ 2.5`(5.0 :+ 2.5) + 6.5`

`>>>`

0.0 :+ 0.0`abs (1.0 :+ 1.0) - sqrt 2.0`

`>>>`

1.0 :+ 0.0`abs (signum (4.0 :+ 3.0))`

`>>>`

[1,2]`foldr (:) [] (1 :+ 2)`

`>>>`

1 2`mapM print (1 :+ 2)`

!a :+ !a infix 6 | forms a complex number from its real and imaginary rectangular components. |

#### Instances

realPart :: Complex a -> a Source #

Extracts the real part of a complex number.

#### Examples

`>>>`

5.0`realPart (5.0 :+ 3.0)`

`>>>`

1.0`realPart ((5.0 :+ 3.0) * (2.0 :+ 3.0))`

imagPart :: Complex a -> a Source #

Extracts the imaginary part of a complex number.

#### Examples

`>>>`

3.0`imagPart (5.0 :+ 3.0)`

`>>>`

21.0`imagPart ((5.0 :+ 3.0) * (2.0 :+ 3.0))`

# Polar form

polar :: RealFloat a => Complex a -> (a, a) Source #

The function `polar`

takes a complex number and
returns a (`magnitude`

, `phase`

) pair in canonical form:
the `magnitude`

is non-negative, and the `phase`

in the range `(-`

;
if the `pi`

, `pi`

]`magnitude`

is zero, then so is the `phase`

.

`polar`

z = (`magnitude`

z,`phase`

z)

#### Examples

`>>>`

(1.4142135623730951,0.7853981633974483)`polar (1.0 :+ 1.0)`

`>>>`

(1.0,3.141592653589793)`polar ((-1.0) :+ 0.0)`

`>>>`

(0.0,0.0)`polar (0.0 :+ 0.0)`

magnitude :: RealFloat a => Complex a -> a Source #

The non-negative `magnitude`

of a complex number.

#### Examples

`>>>`

1.4142135623730951`magnitude (1.0 :+ 1.0)`

`>>>`

1.0`magnitude (1.0 + 0.0)`

`>>>`

5.0`magnitude (0.0 :+ (-5.0))`